text:
@testset "1.1.2.4 (e x)^m (a+b x^2)^p (c+d x^2)^q" begin
    (A, B, a, b, c, d, e, m, p, q, x, ) = @variables A B a b c d e m p q x

    #= ::Package:: =#

    #= ::Title:: =#
    #=Integrands*of*the*form*(e*x)^m*(a+b*x^2)^p*(c+d*x^2)^q=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(e*x)^m*(a+b*x^2)^p*(c+d*x^2)^q=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^2)^p*(c+d*x^2)^1=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^2*(a + b*x^2)*(A + B*x^2), x, 2, (a*A*x^3)/3 + ((A*b + a*B)*x^5)/5 + (b*B*x^7)/7]
    @test_int [x^1*(a + b*x^2)*(A + B*x^2), x, 3, (a*A*x^2)/2 + ((A*b + a*B)*x^4)/4 + (b*B*x^6)/6]
    @test_int [x^0*(a + b*x^2)*(A + B*x^2), x, 2, a*A*x + (1/3)*(A*b + a*B)*x^3 + (1/5)*b*B*x^5]

    @test_int [((a + b*x^2)*(A + B*x^2))/x^1, x, 3, ((A*b + a*B)*x^2)/2 + (b*B*x^4)/4 + a*A*log(x)]
    @test_int [((a + b*x^2)*(A + B*x^2))/x^2, x, 2, -((a*A)/x) + (A*b + a*B)*x + (b*B*x^3)/3]
    @test_int [((a + b*x^2)*(A + B*x^2))/x^3, x, 3, -(a*A)/(2*x^2) + (b*B*x^2)/2 + (A*b + a*B)*log(x)]
    @test_int [((a + b*x^2)*(A + B*x^2))/x^4, x, 2, -(a*A)/(3*x^3) - (A*b + a*B)/x + b*B*x]
    @test_int [((a + b*x^2)*(A + B*x^2))/x^5, x, 3, -((a*A)/(4*x^4)) - (A*b + a*B)/(2*x^2) + b*B*log(x)]

    @test_int [((a + b*x^2)*(A + B*x^2))/x^6, x, 2, -(a*A)/(5*x^5) - (A*b + a*B)/(3*x^3) - (b*B)/x]
    @test_int [((a + b*x^2)*(A + B*x^2))/x^7, x, 3, -(a*A)/(6*x^6) - (A*b + a*B)/(4*x^4) - (b*B)/(2*x^2)]

    @test_int [x^2*(a + b*x^2)^2*(A + B*x^2), x, 2, (a^2*A*x^3)/3 + (a*(2*A*b + a*B)*x^5)/5 + (b*(A*b + 2*a*B)*x^7)/7 + (b^2*B*x^9)/9]
    @test_int [x^1*(a + b*x^2)^2*(A + B*x^2), x, 3, ((A*b - a*B)*(a + b*x^2)^3)/(6*b^2) + (B*(a + b*x^2)^4)/(8*b^2)]
    @test_int [x^0*(a + b*x^2)^2*(A + B*x^2), x, 2, a^2*A*x + (1/3)*a*(2*A*b + a*B)*x^3 + (1/5)*b*(A*b + 2*a*B)*x^5 + (1/7)*b^2*B*x^7]

    @test_int [((a + b*x^2)^2*(A + B*x^2))/x^1, x, 4, a*A*b*x^2 + (1/4)*A*b^2*x^4 + (B*(a + b*x^2)^3)/(6*b) + a^2*A*log(x)]
    @test_int [((a + b*x^2)^2*(A + B*x^2))/x^2, x, 2, -(a^2*A)/x + a*(2*A*b + a*B)*x + (b*(A*b + 2*a*B)*x^3)/3 + (b^2*B*x^5)/5]
    @test_int [((a + b*x^2)^2*(A + B*x^2))/x^3, x, 3, -(a^2*A)/(2*x^2) + (1/2)*b*(A*b + 2*a*B)*x^2 + (1/4)*b^2*B*x^4 + a*(2*A*b + a*B)*log(x)]
    @test_int [((a + b*x^2)^2*(A + B*x^2))/x^4, x, 2, -(a^2*A)/(3*x^3) - (a*(2*A*b + a*B))/x + b*(A*b + 2*a*B)*x + (1/3)*b^2*B*x^3]
    @test_int [((a + b*x^2)^2*(A + B*x^2))/x^5, x, 3, -(a^2*A)/(4*x^4) - (a*(2*A*b + a*B))/(2*x^2) + (1/2)*b^2*B*x^2 + b*(A*b + 2*a*B)*log(x)]
    @test_int [((a + b*x^2)^2*(A + B*x^2))/x^6, x, 2, -(a^2*A)/(5*x^5) - (a*(2*A*b + a*B))/(3*x^3) - (b*(A*b + 2*a*B))/x + b^2*B*x]
    @test_int [((a + b*x^2)^2*(A + B*x^2))/x^7, x, 3, -((a^2*A)/(6*x^6)) - (a*(2*A*b + a*B))/(4*x^4) - (b*(A*b + 2*a*B))/(2*x^2) + b^2*B*log(x)]

    @test_int [((a + b*x^2)^2*(A + B*x^2))/x^8, x, 2, -((a^2*A)/(7*x^7)) - (a*(2*A*b + a*B))/(5*x^5) - (b*(A*b + 2*a*B))/(3*x^3) - (b^2*B)/x]
    @test_int [((a + b*x^2)^2*(A + B*x^2))/x^9, x, 3, -((A*(a + b*x^2)^3)/(8*a*x^8)) + ((A*b - 4*a*B)*(a + b*x^2)^3)/(24*a^2*x^6)]

    @test_int [x^9*(a + b*x^2)^5*(A + B*x^2), x, 3, (a^5*A*x^10)/10 + (a^4*(5*A*b + a*B)*x^12)/12 + (5*a^3*b*(2*A*b + a*B)*x^14)/14 + (5*a^2*b^2*(A*b + a*B)*x^16)/8 + (5*a*b^3*(A*b + 2*a*B)*x^18)/18 + (b^4*(A*b + 5*a*B)*x^20)/20 + (b^5*B*x^22)/22]
    @test_int [x^8*(a + b*x^2)^5*(A + B*x^2), x, 2, (a^5*A*x^9)/9 + (a^4*(5*A*b + a*B)*x^11)/11 + (5*a^3*b*(2*A*b + a*B)*x^13)/13 + (2*a^2*b^2*(A*b + a*B)*x^15)/3 + (5*a*b^3*(A*b + 2*a*B)*x^17)/17 + (b^4*(A*b + 5*a*B)*x^19)/19 + (b^5*B*x^21)/21]
    @test_int [x^7*(a + b*x^2)^5*(A + B*x^2), x, 3, -((a^3*(A*b - a*B)*(a + b*x^2)^6)/(12*b^5)) + (a^2*(3*A*b - 4*a*B)*(a + b*x^2)^7)/(14*b^5) - (3*a*(A*b - 2*a*B)*(a + b*x^2)^8)/(16*b^5) + ((A*b - 4*a*B)*(a + b*x^2)^9)/(18*b^5) + (B*(a + b*x^2)^10)/(20*b^5)]

    @test_int [x^6*(a + b*x^2)^5*(A + B*x^2), x, 2, (a^5*A*x^7)/7 + (a^4*(5*A*b + a*B)*x^9)/9 + (5*a^3*b*(2*A*b + a*B)*x^11)/11 + (10*a^2*b^2*(A*b + a*B)*x^13)/13 + (a*b^3*(A*b + 2*a*B)*x^15)/3 + (b^4*(A*b + 5*a*B)*x^17)/17 + (b^5*B*x^19)/19]
    @test_int [x^5*(a + b*x^2)^5*(A + B*x^2), x, 3, (a^2*(A*b - a*B)*(a + b*x^2)^6)/(12*b^4) - (a*(2*A*b - 3*a*B)*(a + b*x^2)^7)/(14*b^4) + ((A*b - 3*a*B)*(a + b*x^2)^8)/(16*b^4) + (B*(a + b*x^2)^9)/(18*b^4)]
    @test_int [x^4*(a + b*x^2)^5*(A + B*x^2), x, 2, (a^5*A*x^5)/5 + (a^4*(5*A*b + a*B)*x^7)/7 + (5*a^3*b*(2*A*b + a*B)*x^9)/9 + (10*a^2*b^2*(A*b + a*B)*x^11)/11 + (5*a*b^3*(A*b + 2*a*B)*x^13)/13 + (b^4*(A*b + 5*a*B)*x^15)/15 + (b^5*B*x^17)/17]
    @test_int [x^3*(a + b*x^2)^5*(A + B*x^2), x, 3, -(a*(A*b - a*B)*(a + b*x^2)^6)/(12*b^3) + ((A*b - 2*a*B)*(a + b*x^2)^7)/(14*b^3) + (B*(a + b*x^2)^8)/(16*b^3)]
    @test_int [x^2*(a + b*x^2)^5*(A + B*x^2), x, 2, (a^5*A*x^3)/3 + (a^4*(5*A*b + a*B)*x^5)/5 + (5*a^3*b*(2*A*b + a*B)*x^7)/7 + (10*a^2*b^2*(A*b + a*B)*x^9)/9 + (5*a*b^3*(A*b + 2*a*B)*x^11)/11 + (b^4*(A*b + 5*a*B)*x^13)/13 + (b^5*B*x^15)/15]
    @test_int [x^1*(a + b*x^2)^5*(A + B*x^2), x, 3, ((A*b - a*B)*(a + b*x^2)^6)/(12*b^2) + (B*(a + b*x^2)^7)/(14*b^2)]
    @test_int [x^0*(a + b*x^2)^5*(A + B*x^2), x, 2, a^5*A*x + (1/3)*a^4*(5*A*b + a*B)*x^3 + a^3*b*(2*A*b + a*B)*x^5 + (10/7)*a^2*b^2*(A*b + a*B)*x^7 + (5/9)*a*b^3*(A*b + 2*a*B)*x^9 + (1/11)*b^4*(A*b + 5*a*B)*x^11 + (1/13)*b^5*B*x^13]

    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^1, x, 4, (5*a^4*A*b*x^2)/2 + (5*a^3*A*b^2*x^4)/2 + (5*a^2*A*b^3*x^6)/3 + (5*a*A*b^4*x^8)/8 + (A*b^5*x^10)/10 + (B*(a + b*x^2)^6)/(12*b) + a^5*A*log(x)]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^2, x, 2, -((a^5*A)/x) + a^4*(5*A*b + a*B)*x + (5*a^3*b*(2*A*b + a*B)*x^3)/3 + 2*a^2*b^2*(A*b + a*B)*x^5 + (5*a*b^3*(A*b + 2*a*B)*x^7)/7 + (b^4*(A*b + 5*a*B)*x^9)/9 + (b^5*B*x^11)/11]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^3, x, 3, -(a^5*A)/(2*x^2) + (5*a^3*b*(2*A*b + a*B)*x^2)/2 + (5*a^2*b^2*(A*b + a*B)*x^4)/2 + (5*a*b^3*(A*b + 2*a*B)*x^6)/6 + (b^4*(A*b + 5*a*B)*x^8)/8 + (b^5*B*x^10)/10 + a^4*(5*A*b + a*B)*log(x)]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^4, x, 2, -(a^5*A)/(3*x^3) - (a^4*(5*A*b + a*B))/x + 5*a^3*b*(2*A*b + a*B)*x + (10*a^2*b^2*(A*b + a*B)*x^3)/3 + a*b^3*(A*b + 2*a*B)*x^5 + (b^4*(A*b + 5*a*B)*x^7)/7 + (b^5*B*x^9)/9]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^5, x, 3, -(a^5*A)/(4*x^4) - (a^4*(5*A*b + a*B))/(2*x^2) + 5*a^2*b^2*(A*b + a*B)*x^2 + (5*a*b^3*(A*b + 2*a*B)*x^4)/4 + (b^4*(A*b + 5*a*B)*x^6)/6 + (b^5*B*x^8)/8 + 5*a^3*b*(2*A*b + a*B)*log(x)]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^6, x, 2, -(a^5*A)/(5*x^5) - (a^4*(5*A*b + a*B))/(3*x^3) - (5*a^3*b*(2*A*b + a*B))/x + 10*a^2*b^2*(A*b + a*B)*x + (5*a*b^3*(A*b + 2*a*B)*x^3)/3 + (b^4*(A*b + 5*a*B)*x^5)/5 + (b^5*B*x^7)/7]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^7, x, 3, -(a^5*A)/(6*x^6) - (a^4*(5*A*b + a*B))/(4*x^4) - (5*a^3*b*(2*A*b + a*B))/(2*x^2) + (5*a*b^3*(A*b + 2*a*B)*x^2)/2 + (b^4*(A*b + 5*a*B)*x^4)/4 + (b^5*B*x^6)/6 + 10*a^2*b^2*(A*b + a*B)*log(x)]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^8, x, 2, -(a^5*A)/(7*x^7) - (a^4*(5*A*b + a*B))/(5*x^5) - (5*a^3*b*(2*A*b + a*B))/(3*x^3) - (10*a^2*b^2*(A*b + a*B))/x + 5*a*b^3*(A*b + 2*a*B)*x + (b^4*(A*b + 5*a*B)*x^3)/3 + (b^5*B*x^5)/5]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^9, x, 3, -(a^5*A)/(8*x^8) - (a^4*(5*A*b + a*B))/(6*x^6) - (5*a^3*b*(2*A*b + a*B))/(4*x^4) - (5*a^2*b^2*(A*b + a*B))/x^2 + (b^4*(A*b + 5*a*B)*x^2)/2 + (b^5*B*x^4)/4 + 5*a*b^3*(A*b + 2*a*B)*log(x)]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^10, x, 2, -(a^5*A)/(9*x^9) - (a^4*(5*A*b + a*B))/(7*x^7) - (a^3*b*(2*A*b + a*B))/x^5 - (10*a^2*b^2*(A*b + a*B))/(3*x^3) - (5*a*b^3*(A*b + 2*a*B))/x + b^4*(A*b + 5*a*B)*x + (b^5*B*x^3)/3]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^11, x, 3, -(a^5*A)/(10*x^10) - (a^4*(5*A*b + a*B))/(8*x^8) - (5*a^3*b*(2*A*b + a*B))/(6*x^6) - (5*a^2*b^2*(A*b + a*B))/(2*x^4) - (5*a*b^3*(A*b + 2*a*B))/(2*x^2) + (b^5*B*x^2)/2 + b^4*(A*b + 5*a*B)*log(x)]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^12, x, 2, -(a^5*A)/(11*x^11) - (a^4*(5*A*b + a*B))/(9*x^9) - (5*a^3*b*(2*A*b + a*B))/(7*x^7) - (2*a^2*b^2*(A*b + a*B))/x^5 - (5*a*b^3*(A*b + 2*a*B))/(3*x^3) - (b^4*(A*b + 5*a*B))/x + b^5*B*x]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^13, x, 4, -((a^5*B)/(10*x^10)) - (5*a^4*b*B)/(8*x^8) - (5*a^3*b^2*B)/(3*x^6) - (5*a^2*b^3*B)/(2*x^4) - (5*a*b^4*B)/(2*x^2) - (A*(a + b*x^2)^6)/(12*a*x^12) + b^5*B*log(x)]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^14, x, 2, -(a^5*A)/(13*x^13) - (a^4*(5*A*b + a*B))/(11*x^11) - (5*a^3*b*(2*A*b + a*B))/(9*x^9) - (10*a^2*b^2*(A*b + a*B))/(7*x^7) - (a*b^3*(A*b + 2*a*B))/x^5 - (b^4*(A*b + 5*a*B))/(3*x^3) - (b^5*B)/x]

    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^15, x, 3, -(A*(a + b*x^2)^6)/(14*a*x^14) + ((A*b - 7*a*B)*(a + b*x^2)^6)/(84*a^2*x^12)]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^16, x, 2, -(a^5*A)/(15*x^15) - (a^4*(5*A*b + a*B))/(13*x^13) - (5*a^3*b*(2*A*b + a*B))/(11*x^11) - (10*a^2*b^2*(A*b + a*B))/(9*x^9) - (5*a*b^3*(A*b + 2*a*B))/(7*x^7) - (b^4*(A*b + 5*a*B))/(5*x^5) - (b^5*B)/(3*x^3)]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^17, x, 4, -(A*(a + b*x^2)^6)/(16*a*x^16) + ((A*b - 4*a*B)*(a + b*x^2)^6)/(56*a^2*x^14) - (b*(A*b - 4*a*B)*(a + b*x^2)^6)/(336*a^3*x^12)]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^18, x, 2, -(a^5*A)/(17*x^17) - (a^4*(5*A*b + a*B))/(15*x^15) - (5*a^3*b*(2*A*b + a*B))/(13*x^13) - (10*a^2*b^2*(A*b + a*B))/(11*x^11) - (5*a*b^3*(A*b + 2*a*B))/(9*x^9) - (b^4*(A*b + 5*a*B))/(7*x^7) - (b^5*B)/(5*x^5)]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^19, x, 3, -(a^5*A)/(18*x^18) - (a^4*(5*A*b + a*B))/(16*x^16) - (5*a^3*b*(2*A*b + a*B))/(14*x^14) - (5*a^2*b^2*(A*b + a*B))/(6*x^12) - (a*b^3*(A*b + 2*a*B))/(2*x^10) - (b^4*(A*b + 5*a*B))/(8*x^8) - (b^5*B)/(6*x^6)]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^20, x, 2, -(a^5*A)/(19*x^19) - (a^4*(5*A*b + a*B))/(17*x^17) - (a^3*b*(2*A*b + a*B))/(3*x^15) - (10*a^2*b^2*(A*b + a*B))/(13*x^13) - (5*a*b^3*(A*b + 2*a*B))/(11*x^11) - (b^4*(A*b + 5*a*B))/(9*x^9) - (b^5*B)/(7*x^7)]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^21, x, 3, -(a^5*A)/(20*x^20) - (a^4*(5*A*b + a*B))/(18*x^18) - (5*a^3*b*(2*A*b + a*B))/(16*x^16) - (5*a^2*b^2*(A*b + a*B))/(7*x^14) - (5*a*b^3*(A*b + 2*a*B))/(12*x^12) - (b^4*(A*b + 5*a*B))/(10*x^10) - (b^5*B)/(8*x^8)]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^22, x, 2, -(a^5*A)/(21*x^21) - (a^4*(5*A*b + a*B))/(19*x^19) - (5*a^3*b*(2*A*b + a*B))/(17*x^17) - (2*a^2*b^2*(A*b + a*B))/(3*x^15) - (5*a*b^3*(A*b + 2*a*B))/(13*x^13) - (b^4*(A*b + 5*a*B))/(11*x^11) - (b^5*B)/(9*x^9)]
    @test_int [((a + b*x^2)^5*(A + B*x^2))/x^23, x, 3, -(a^5*A)/(22*x^22) - (a^4*(5*A*b + a*B))/(20*x^20) - (5*a^3*b*(2*A*b + a*B))/(18*x^18) - (5*a^2*b^2*(A*b + a*B))/(8*x^16) - (5*a*b^3*(A*b + 2*a*B))/(14*x^14) - (b^4*(A*b + 5*a*B))/(12*x^12) - (b^5*B)/(10*x^10)]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(x^6*(A + B*x^2))/(a + b*x^2), x, 4, (a^2*(A*b - a*B)*x)/b^4 - (a*(A*b - a*B)*x^3)/(3*b^3) + ((A*b - a*B)*x^5)/(5*b^2) + (B*x^7)/(7*b) - (a^(5/2)*(A*b - a*B)*atan((sqrt(b)*x)/sqrt(a)))/b^(9/2)]
    @test_int [(x^5*(A + B*x^2))/(a + b*x^2), x, 3, -((a*(A*b - a*B)*x^2)/(2*b^3)) + ((A*b - a*B)*x^4)/(4*b^2) + (B*x^6)/(6*b) + (a^2*(A*b - a*B)*log(a + b*x^2))/(2*b^4)]
    @test_int [(x^4*(A + B*x^2))/(a + b*x^2), x, 4, -((a*(A*b - a*B)*x)/b^3) + ((A*b - a*B)*x^3)/(3*b^2) + (B*x^5)/(5*b) + (a^(3/2)*(A*b - a*B)*atan((sqrt(b)*x)/sqrt(a)))/b^(7/2)]
    @test_int [(x^3*(A + B*x^2))/(a + b*x^2), x, 3, ((A*b - a*B)*x^2)/(2*b^2) + (B*x^4)/(4*b) - (a*(A*b - a*B)*log(a + b*x^2))/(2*b^3)]
    @test_int [(x^2*(A + B*x^2))/(a + b*x^2), x, 3, ((A*b - a*B)*x)/b^2 + (B*x^3)/(3*b) - (sqrt(a)*(A*b - a*B)*atan((sqrt(b)*x)/sqrt(a)))/b^(5/2)]
    @test_int [(x^1*(A + B*x^2))/(a + b*x^2), x, 3, (B*x^2)/(2*b) + ((A*b - a*B)*log(a + b*x^2))/(2*b^2)]
    @test_int [(A + B*x^2)/(a + b*x^2), x, 2, (B*x)/b + ((A*b - a*B)*atan((sqrt(b)*x)/sqrt(a)))/(sqrt(a)*b^(3/2))]
    @test_int [(A + B*x^2)/(x^1*(a + b*x^2)), x, 3, (A*log(x))/a - ((A*b - a*B)*log(a + b*x^2))/(2*a*b)]
    @test_int [(A + B*x^2)/(x^2*(a + b*x^2)), x, 2, -(A/(a*x)) - ((A*b - a*B)*atan((sqrt(b)*x)/sqrt(a)))/(a^(3/2)*sqrt(b))]
    @test_int [(A + B*x^2)/(x^3*(a + b*x^2)), x, 3, -A/(2*a*x^2) - ((A*b - a*B)*log(x))/a^2 + ((A*b - a*B)*log(a + b*x^2))/(2*a^2)]
    @test_int [(A + B*x^2)/(x^4*(a + b*x^2)), x, 3, -A/(3*a*x^3) + (A*b - a*B)/(a^2*x) + (sqrt(b)*(A*b - a*B)*atan((sqrt(b)*x)/sqrt(a)))/a^(5/2)]
    @test_int [(A + B*x^2)/(x^5*(a + b*x^2)), x, 3, -A/(4*a*x^4) + (A*b - a*B)/(2*a^2*x^2) + (b*(A*b - a*B)*log(x))/a^3 - (b*(A*b - a*B)*log(a + b*x^2))/(2*a^3)]
    @test_int [(A + B*x^2)/(x^6*(a + b*x^2)), x, 4, -A/(5*a*x^5) + (A*b - a*B)/(3*a^2*x^3) - (b*(A*b - a*B))/(a^3*x) - (b^(3/2)*(A*b - a*B)*atan((sqrt(b)*x)/sqrt(a)))/a^(7/2)]
    @test_int [(A + B*x^2)/(x^7*(a + b*x^2)), x, 3, -A/(6*a*x^6) + (A*b - a*B)/(4*a^2*x^4) - (b*(A*b - a*B))/(2*a^3*x^2) - (b^2*(A*b - a*B)*log(x))/a^4 + (b^2*(A*b - a*B)*log(a + b*x^2))/(2*a^4)]
    @test_int [(A + B*x^2)/(x^8*(a + b*x^2)), x, 5, -A/(7*a*x^7) + (A*b - a*B)/(5*a^2*x^5) - (b*(A*b - a*B))/(3*a^3*x^3) + (b^2*(A*b - a*B))/(a^4*x) + (b^(5/2)*(A*b - a*B)*atan((sqrt(b)*x)/sqrt(a)))/a^(9/2)]

    @test_int [(x^9*(A + B*x^2))/(a + b*x^2)^2, x, 3, (a^2*(3*A*b - 4*a*B)*x^2)/(2*b^5) - (a*(2*A*b - 3*a*B)*x^4)/(4*b^4) + ((A*b - 2*a*B)*x^6)/(6*b^3) + (B*x^8)/(8*b^2) - (a^4*(A*b - a*B))/(2*b^6*(a + b*x^2)) - (a^3*(4*A*b - 5*a*B)*log(a + b*x^2))/(2*b^6)]
    @test_int [(x^8*(A + B*x^2))/(a + b*x^2)^2, x, 4, (a^2*(3*A*b - 4*a*B)*x)/b^5 - (a*(2*A*b - 3*a*B)*x^3)/(3*b^4) + ((A*b - 2*a*B)*x^5)/(5*b^3) + (B*x^7)/(7*b^2) + (a^3*(A*b - a*B)*x)/(2*b^5*(a + b*x^2)) - (a^(5/2)*(7*A*b - 9*a*B)*atan((sqrt(b)*x)/sqrt(a)))/(2*b^(11/2))]
    @test_int [(x^7*(A + B*x^2))/(a + b*x^2)^2, x, 3, -((a*(2*A*b - 3*a*B)*x^2)/(2*b^4)) + ((A*b - 2*a*B)*x^4)/(4*b^3) + (B*x^6)/(6*b^2) + (a^3*(A*b - a*B))/(2*b^5*(a + b*x^2)) + (a^2*(3*A*b - 4*a*B)*log(a + b*x^2))/(2*b^5)]
    @test_int [(x^6*(A + B*x^2))/(a + b*x^2)^2, x, 4, -((a*(2*A*b - 3*a*B)*x)/b^4) + ((A*b - 2*a*B)*x^3)/(3*b^3) + (B*x^5)/(5*b^2) - (a^2*(A*b - a*B)*x)/(2*b^4*(a + b*x^2)) + (a^(3/2)*(5*A*b - 7*a*B)*atan((sqrt(b)*x)/sqrt(a)))/(2*b^(9/2))]
    @test_int [(x^5*(A + B*x^2))/(a + b*x^2)^2, x, 3, ((A*b - 2*a*B)*x^2)/(2*b^3) + (B*x^4)/(4*b^2) - (a^2*(A*b - a*B))/(2*b^4*(a + b*x^2)) - (a*(2*A*b - 3*a*B)*log(a + b*x^2))/(2*b^4)]
    @test_int [(x^4*(A + B*x^2))/(a + b*x^2)^2, x, 4, ((A*b - 2*a*B)*x)/b^3 + (B*x^3)/(3*b^2) + (a*(A*b - a*B)*x)/(2*b^3*(a + b*x^2)) - (sqrt(a)*(3*A*b - 5*a*B)*atan((sqrt(b)*x)/sqrt(a)))/(2*b^(7/2))]
    @test_int [(x^3*(A + B*x^2))/(a + b*x^2)^2, x, 3, (B*x^2)/(2*b^2) + (a*(A*b - a*B))/(2*b^3*(a + b*x^2)) + ((A*b - 2*a*B)*log(a + b*x^2))/(2*b^3)]
    @test_int [(x^2*(A + B*x^2))/(a + b*x^2)^2, x, 3, (B*x)/b^2 - ((A*b - a*B)*x)/(2*b^2*(a + b*x^2)) + ((A*b - 3*a*B)*atan((sqrt(b)*x)/sqrt(a)))/(2*sqrt(a)*b^(5/2))]
    @test_int [(x^1*(A + B*x^2))/(a + b*x^2)^2, x, 3, -((A*b - a*B)/(2*b^2*(a + b*x^2))) + (B*log(a + b*x^2))/(2*b^2)]
    @test_int [(A + B*x^2)/(a + b*x^2)^2, x, 2, ((A*b - a*B)*x)/(2*a*b*(a + b*x^2)) + ((A*b + a*B)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(3/2)*b^(3/2))]
    @test_int [(A + B*x^2)/(x^1*(a + b*x^2)^2), x, 3, (A*b - a*B)/(2*a*b*(a + b*x^2)) + (A*log(x))/a^2 - (A*log(a + b*x^2))/(2*a^2)]
    @test_int [(A + B*x^2)/(x^2*(a + b*x^2)^2), x, 3, -(A/(a^2*x)) - ((A*b - a*B)*x)/(2*a^2*(a + b*x^2)) - ((3*A*b - a*B)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(5/2)*sqrt(b))]
    @test_int [(A + B*x^2)/(x^3*(a + b*x^2)^2), x, 3, -(A/(2*a^2*x^2)) - (A*b - a*B)/(2*a^2*(a + b*x^2)) - ((2*A*b - a*B)*log(x))/a^3 + ((2*A*b - a*B)*log(a + b*x^2))/(2*a^3)]
    @test_int [(A + B*x^2)/(x^4*(a + b*x^2)^2), x, 4, -(A/(3*a^2*x^3)) + (2*A*b - a*B)/(a^3*x) + (b*(A*b - a*B)*x)/(2*a^3*(a + b*x^2)) + (sqrt(b)*(5*A*b - 3*a*B)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(7/2))]
    @test_int [(A + B*x^2)/(x^5*(a + b*x^2)^2), x, 3, -(A/(4*a^2*x^4)) + (2*A*b - a*B)/(2*a^3*x^2) + (b*(A*b - a*B))/(2*a^3*(a + b*x^2)) + (b*(3*A*b - 2*a*B)*log(x))/a^4 - (b*(3*A*b - 2*a*B)*log(a + b*x^2))/(2*a^4)]
    @test_int [(A + B*x^2)/(x^6*(a + b*x^2)^2), x, 4, -(A/(5*a^2*x^5)) + (2*A*b - a*B)/(3*a^3*x^3) - (b*(3*A*b - 2*a*B))/(a^4*x) - (b^2*(A*b - a*B)*x)/(2*a^4*(a + b*x^2)) - (b^(3/2)*(7*A*b - 5*a*B)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(9/2))]
    @test_int [(A + B*x^2)/(x^7*(a + b*x^2)^2), x, 3, -(A/(6*a^2*x^6)) + (2*A*b - a*B)/(4*a^3*x^4) - (b*(3*A*b - 2*a*B))/(2*a^4*x^2) - (b^2*(A*b - a*B))/(2*a^4*(a + b*x^2)) - (b^2*(4*A*b - 3*a*B)*log(x))/a^5 + (b^2*(4*A*b - 3*a*B)*log(a + b*x^2))/(2*a^5)]

    @test_int [(x^11*(A + B*x^2))/(a + b*x^2)^3, x, 3, (a^2*(3*A*b - 5*a*B)*x^2)/b^6 - (3*a*(A*b - 2*a*B)*x^4)/(4*b^5) + ((A*b - 3*a*B)*x^6)/(6*b^4) + (B*x^8)/(8*b^3) + (a^5*(A*b - a*B))/(4*b^7*(a + b*x^2)^2) - (a^4*(5*A*b - 6*a*B))/(2*b^7*(a + b*x^2)) - (5*a^3*(2*A*b - 3*a*B)*log(a + b*x^2))/(2*b^7)]
    @test_int [(x^9*(A + B*x^2))/(a + b*x^2)^3, x, 3, (-3*a*(A*b - 2*a*B)*x^2)/(2*b^5) + ((A*b - 3*a*B)*x^4)/(4*b^4) + (B*x^6)/(6*b^3) - (a^4*(A*b - a*B))/(4*b^6*(a + b*x^2)^2) + (a^3*(4*A*b - 5*a*B))/(2*b^6*(a + b*x^2)) + (a^2*(3*A*b - 5*a*B)*log(a + b*x^2))/b^6]
    @test_int [(x^7*(A + B*x^2))/(a + b*x^2)^3, x, 3, ((A*b - 3*a*B)*x^2)/(2*b^4) + (B*x^4)/(4*b^3) + (a^3*(A*b - a*B))/(4*b^5*(a + b*x^2)^2) - (a^2*(3*A*b - 4*a*B))/(2*b^5*(a + b*x^2)) - (3*a*(A*b - 2*a*B)*log(a + b*x^2))/(2*b^5)]
    @test_int [(x^5*(A + B*x^2))/(a + b*x^2)^3, x, 3, (B*x^2)/(2*b^3) - (a^2*(A*b - a*B))/(4*b^4*(a + b*x^2)^2) + (a*(2*A*b - 3*a*B))/(2*b^4*(a + b*x^2)) + ((A*b - 3*a*B)*log(a + b*x^2))/(2*b^4)]
    @test_int [(x^3*(A + B*x^2))/(a + b*x^2)^3, x, 3, (a*(A*b - a*B))/(4*b^3*(a + b*x^2)^2) - (A*b - 2*a*B)/(2*b^3*(a + b*x^2)) + (B*log(a + b*x^2))/(2*b^3)]
    @test_int [(x^1*(A + B*x^2))/(a + b*x^2)^3, x, 2, -(A + B*x^2)^2/(4*(A*b - a*B)*(a + b*x^2)^2)]
    @test_int [(A + B*x^2)/(x^1*(a + b*x^2)^3), x, 3, (A*b - a*B)/(4*a*b*(a + b*x^2)^2) + A/(2*a^2*(a + b*x^2)) + (A*log(x))/a^3 - (A*log(a + b*x^2))/(2*a^3)]
    @test_int [(A + B*x^2)/(x^3*(a + b*x^2)^3), x, 3, -(A/(2*a^3*x^2)) - (A*b - a*B)/(4*a^2*(a + b*x^2)^2) - (2*A*b - a*B)/(2*a^3*(a + b*x^2)) - ((3*A*b - a*B)*log(x))/a^4 + ((3*A*b - a*B)*log(a + b*x^2))/(2*a^4)]
    @test_int [(A + B*x^2)/(x^5*(a + b*x^2)^3), x, 3, -(A/(4*a^3*x^4)) + (3*A*b - a*B)/(2*a^4*x^2) + (b*(A*b - a*B))/(4*a^3*(a + b*x^2)^2) + (b*(3*A*b - 2*a*B))/(2*a^4*(a + b*x^2)) + (3*b*(2*A*b - a*B)*log(x))/a^5 - (3*b*(2*A*b - a*B)*log(a + b*x^2))/(2*a^5)]
    @test_int [(A + B*x^2)/(x^7*(a + b*x^2)^3), x, 3, -(A/(6*a^3*x^6)) + (3*A*b - a*B)/(4*a^4*x^4) - (3*b*(2*A*b - a*B))/(2*a^5*x^2) - (b^2*(A*b - a*B))/(4*a^4*(a + b*x^2)^2) - (b^2*(4*A*b - 3*a*B))/(2*a^5*(a + b*x^2)) - (2*b^2*(5*A*b - 3*a*B)*log(x))/a^6 + (b^2*(5*A*b - 3*a*B)*log(a + b*x^2))/a^6]

    @test_int [(x^10*(A + B*x^2))/(a + b*x^2)^3, x, 5, (2*a^2*(3*A*b - 5*a*B)*x)/b^6 - (a*(A*b - 2*a*B)*x^3)/b^5 + ((A*b - 3*a*B)*x^5)/(5*b^4) + (B*x^7)/(7*b^3) - (a^4*(A*b - a*B)*x)/(4*b^6*(a + b*x^2)^2) + (a^3*(17*A*b - 21*a*B)*x)/(8*b^6*(a + b*x^2)) - (9*a^(5/2)*(7*A*b - 11*a*B)*atan((sqrt(b)*x)/sqrt(a)))/(8*b^(13/2))]
    @test_int [(x^8*(A + B*x^2))/(a + b*x^2)^3, x, 5, -((3*a*(A*b - 2*a*B)*x)/b^5) + ((A*b - 3*a*B)*x^3)/(3*b^4) + (B*x^5)/(5*b^3) + (a^3*(A*b - a*B)*x)/(4*b^5*(a + b*x^2)^2) - (a^2*(13*A*b - 17*a*B)*x)/(8*b^5*(a + b*x^2)) + (7*a^(3/2)*(5*A*b - 9*a*B)*atan((sqrt(b)*x)/sqrt(a)))/(8*b^(11/2))]
    @test_int [(x^6*(A + B*x^2))/(a + b*x^2)^3, x, 5, ((A*b - 3*a*B)*x)/b^4 + (B*x^3)/(3*b^3) - (a^2*(A*b - a*B)*x)/(4*b^4*(a + b*x^2)^2) + (a*(9*A*b - 13*a*B)*x)/(8*b^4*(a + b*x^2)) - (5*sqrt(a)*(3*A*b - 7*a*B)*atan((sqrt(b)*x)/sqrt(a)))/(8*b^(9/2))]
    @test_int [(x^4*(A + B*x^2))/(a + b*x^2)^3, x, 4, (B*x)/b^3 + (a*(A*b - a*B)*x)/(4*b^3*(a + b*x^2)^2) - ((5*A*b - 9*a*B)*x)/(8*b^3*(a + b*x^2)) + (3*(A*b - 5*a*B)*atan((sqrt(b)*x)/sqrt(a)))/(8*sqrt(a)*b^(7/2))]
    @test_int [(x^2*(A + B*x^2))/(a + b*x^2)^3, x, 3, -(((A*b - a*B)*x)/(4*b^2*(a + b*x^2)^2)) + ((A*b - 5*a*B)*x)/(8*a*b^2*(a + b*x^2)) + ((A*b + 3*a*B)*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(3/2)*b^(5/2))]
    @test_int [(x^0*(A + B*x^2))/(a + b*x^2)^3, x, 3, ((A*b - a*B)*x)/(4*a*b*(a + b*x^2)^2) + ((3*A*b + a*B)*x)/(8*a^2*b*(a + b*x^2)) + ((3*A*b + a*B)*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(5/2)*b^(3/2))]
    @test_int [(A + B*x^2)/(x^2*(a + b*x^2)^3), x, 4, -(A/(a^3*x)) - ((A*b - a*B)*x)/(4*a^2*(a + b*x^2)^2) - ((7*A*b - 3*a*B)*x)/(8*a^3*(a + b*x^2)) - (3*(5*A*b - a*B)*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(7/2)*sqrt(b))]
    @test_int [(A + B*x^2)/(x^4*(a + b*x^2)^3), x, 5, -(A/(3*a^3*x^3)) + (3*A*b - a*B)/(a^4*x) + (b*(A*b - a*B)*x)/(4*a^3*(a + b*x^2)^2) + (b*(11*A*b - 7*a*B)*x)/(8*a^4*(a + b*x^2)) + (5*sqrt(b)*(7*A*b - 3*a*B)*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(9/2))]
    @test_int [(A + B*x^2)/(x^6*(a + b*x^2)^3), x, 5, -(A/(5*a^3*x^5)) + (3*A*b - a*B)/(3*a^4*x^3) - (3*b*(2*A*b - a*B))/(a^5*x) - (b^2*(A*b - a*B)*x)/(4*a^4*(a + b*x^2)^2) - (b^2*(15*A*b - 11*a*B)*x)/(8*a^5*(a + b*x^2)) - (7*b^(3/2)*(9*A*b - 5*a*B)*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(11/2))]

    @test_int [(a + b*x^2)/(1 + x^2), x, 2, b*x + (a - b)*atan(x)]
    @test_int [(a + b*x^2)/(1 - x^2), x, 2, (-b)*x + (a + b)*atanh(x)]

    @test_int [(1 + x^2)/(-1 + x^2)^2, x, 1, x/(1 - x^2)]
    @test_int [(1 - x^2)/(1 + x^2)^2, x, 1, x/(1 + x^2)]
    @test_int [(3 + 2*x^2)/(1 + x^2)^2, x, 2, x/(2*(1 + x^2)) + (5*atan(x))/2]
    @test_int [(-2 + x^2)/(1 + x^2)^2, x, 2, (-3*x)/(2*(1 + x^2)) - atan(x)/2]
    @test_int [(3 + x^2)/(1 + x^2)^2, x, 2, x/(1 + x^2) + 2*atan(x)]
    @test_int [(a + b*x^2)/(-a + b*x^2)^2, x, 1, x/(a - b*x^2)]
    @test_int [(a + b*x^2)/(a - b*x^2)^2, x, 1, x/(a - b*x^2)]

    @test_int [(A + B*x^2)/(a - b*x^2), x, 2, -((B*x)/b) + ((A*b + a*B)*atanh((sqrt(b)*x)/sqrt(a)))/(sqrt(a)*b^(3/2))]

    @test_int [(1 + x^2)/(16 + x^2)^3, x, 3, -((15*x)/(64*(16 + x^2)^2)) + (19*x)/(2048*(16 + x^2)) + (19*atan(x/4))/8192]

    @test_int [(1 + 2*x^2)/(x^5*(1 + x^2)^3), x, 2, -1/(4*x^4*(1 + x^2)^2)]

    @test_int [(1 - x^2)^2/(-1 + x^2)^2, x, 2, x]

    @test_int [(x^3*(a*c + b*c*x^2))/(a + b*x^2), x, 2, (c*x^4)/4]
    @test_int [(x^2*(a*c + b*c*x^2))/(a + b*x^2), x, 2, (c*x^3)/3]
    @test_int [(x*(a*c + b*c*x^2))/(a + b*x^2), x, 2, (c*x^2)/2]
    @test_int [(a*c + b*c*x^2)/(a + b*x^2), x, 2, c*x]
    @test_int [(a*c + b*c*x^2)/(x*(a + b*x^2)), x, 2, c*log(x)]
    @test_int [(a*c + b*c*x^2)/(x^2*(a + b*x^2)), x, 2, -(c/x)]
    @test_int [(a*c + b*c*x^2)/(x^3*(a + b*x^2)), x, 2, -c/(2*x^2)]

    @test_int [(x^3*(a*c + b*c*x^2))/(a + b*x^2)^2, x, 4, (c*x^2)/(2*b) - (a*c*log(a + b*x^2))/(2*b^2)]
    @test_int [(x^2*(a*c + b*c*x^2))/(a + b*x^2)^2, x, 3, (c*x)/b - (sqrt(a)*c*atan((sqrt(b)*x)/sqrt(a)))/b^(3/2)]
    @test_int [(x*(a*c + b*c*x^2))/(a + b*x^2)^2, x, 2, (c*log(a + b*x^2))/(2*b)]
    @test_int [(a*c + b*c*x^2)/(a + b*x^2)^2, x, 2, (c*atan((sqrt(b)*x)/sqrt(a)))/(sqrt(a)*sqrt(b))]
    @test_int [(a*c + b*c*x^2)/(x*(a + b*x^2)^2), x, 5, (c*log(x))/a - (c*log(a + b*x^2))/(2*a)]
    @test_int [(a*c + b*c*x^2)/(x^2*(a + b*x^2)^2), x, 3, -(c/(a*x)) - (sqrt(b)*c*atan((sqrt(b)*x)/sqrt(a)))/a^(3/2)]
    @test_int [(a*c + b*c*x^2)/(x^3*(a + b*x^2)^2), x, 4, -c/(2*a*x^2) - (b*c*log(x))/a^2 + (b*c*log(a + b*x^2))/(2*a^2)]

    @test_int [(x^3*(a*c + b*c*x^2))/(a + b*x^2)^3, x, 4, (a*c)/(2*b^2*(a + b*x^2)) + (c*log(a + b*x^2))/(2*b^2)]
    @test_int [(x^2*(a*c + b*c*x^2))/(a + b*x^2)^3, x, 3, -(c*x)/(2*b*(a + b*x^2)) + (c*atan((sqrt(b)*x)/sqrt(a)))/(2*sqrt(a)*b^(3/2))]
    @test_int [(x*(a*c + b*c*x^2))/(a + b*x^2)^3, x, 2, -c/(2*b*(a + b*x^2))]
    @test_int [(a*c + b*c*x^2)/(a + b*x^2)^3, x, 3, (c*x)/(2*a*(a + b*x^2)) + (c*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(3/2)*sqrt(b))]
    @test_int [(a*c + b*c*x^2)/(x*(a + b*x^2)^3), x, 4, c/(2*a*(a + b*x^2)) + (c*log(x))/a^2 - (c*log(a + b*x^2))/(2*a^2)]
    @test_int [(a*c + b*c*x^2)/(x^2*(a + b*x^2)^3), x, 4, (-3*c)/(2*a^2*x) + c/(2*a*x*(a + b*x^2)) - (3*sqrt(b)*c*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(5/2))]
    @test_int [(a*c + b*c*x^2)/(x^3*(a + b*x^2)^3), x, 4, -(c/(2*a^2*x^2)) - (b*c)/(2*a^2*(a + b*x^2)) - (2*b*c*log(x))/a^3 + (b*c*log(a + b*x^2))/a^3]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^2)^p*(c+d*x^2)^2=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^4*(a + b*x^2)^2*(c + d*x^2), x, 2, (a^2*c*x^5)/5 + (a*(2*b*c + a*d)*x^7)/7 + (b*(b*c + 2*a*d)*x^9)/9 + (b^2*d*x^11)/11]
    @test_int [x^3*(a + b*x^2)^2*(c + d*x^2), x, 3, (a^2*c*x^4)/4 + (a*(2*b*c + a*d)*x^6)/6 + (b*(b*c + 2*a*d)*x^8)/8 + (b^2*d*x^10)/10]
    @test_int [x^2*(a + b*x^2)^2*(c + d*x^2), x, 2, (a^2*c*x^3)/3 + (a*(2*b*c + a*d)*x^5)/5 + (b*(b*c + 2*a*d)*x^7)/7 + (b^2*d*x^9)/9]
    @test_int [x^1*(a + b*x^2)^2*(c + d*x^2), x, 3, ((b*c - a*d)*(a + b*x^2)^3)/(6*b^2) + (d*(a + b*x^2)^4)/(8*b^2)]
    @test_int [(a + b*x^2)^2*(c + d*x^2), x, 2, a^2*c*x + (a*(2*b*c + a*d)*x^3)/3 + (b*(b*c + 2*a*d)*x^5)/5 + (b^2*d*x^7)/7]
    @test_int [((a + b*x^2)^2*(c + d*x^2))/x^1, x, 4, a*b*c*x^2 + (b^2*c*x^4)/4 + (d*(a + b*x^2)^3)/(6*b) + a^2*c*log(x)]
    @test_int [((a + b*x^2)^2*(c + d*x^2))/x^2, x, 2, -((a^2*c)/x) + a*(2*b*c + a*d)*x + (b*(b*c + 2*a*d)*x^3)/3 + (b^2*d*x^5)/5]
    @test_int [((a + b*x^2)^2*(c + d*x^2))/x^3, x, 3, -(a^2*c)/(2*x^2) + (b*(b*c + 2*a*d)*x^2)/2 + (b^2*d*x^4)/4 + a*(2*b*c + a*d)*log(x)]
    @test_int [((a + b*x^2)^2*(c + d*x^2))/x^4, x, 2, -(a^2*c)/(3*x^3) - (a*(2*b*c + a*d))/x + b*(b*c + 2*a*d)*x + (b^2*d*x^3)/3]

    @test_int [x^4*(a + b*x^2)^2*(c + d*x^2)^2, x, 2, (1/5)*a^2*c^2*x^5 + (2/7)*a*c*(b*c + a*d)*x^7 + (1/9)*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^9 + (2/11)*b*d*(b*c + a*d)*x^11 + (1/13)*b^2*d^2*x^13]
    @test_int [x^3*(a + b*x^2)^2*(c + d*x^2)^2, x, 3, (1/4)*a^2*c^2*x^4 + (1/3)*a*c*(b*c + a*d)*x^6 + (1/8)*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^8 + (1/5)*b*d*(b*c + a*d)*x^10 + (1/12)*b^2*d^2*x^12]
    @test_int [x^2*(a + b*x^2)^2*(c + d*x^2)^2, x, 2, (1/3)*a^2*c^2*x^3 + (2/5)*a*c*(b*c + a*d)*x^5 + (1/7)*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^7 + (2/9)*b*d*(b*c + a*d)*x^9 + (1/11)*b^2*d^2*x^11]
    @test_int [x^1*(a + b*x^2)^2*(c + d*x^2)^2, x, 3, ((b*c - a*d)^2*(a + b*x^2)^3)/(6*b^3) + (d*(b*c - a*d)*(a + b*x^2)^4)/(4*b^3) + (d^2*(a + b*x^2)^5)/(10*b^3)]
    @test_int [(a + b*x^2)^2*(c + d*x^2)^2, x, 2, a^2*c^2*x + (2*a*c*(b*c + a*d)*x^3)/3 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^5)/5 + (2*b*d*(b*c + a*d)*x^7)/7 + (b^2*d^2*x^9)/9]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^2)/x^1, x, 3, a*c*(b*c + a*d)*x^2 + (1/4)*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 + (1/3)*b*d*(b*c + a*d)*x^6 + (1/8)*b^2*d^2*x^8 + a^2*c^2*log(x)]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^2)/x^2, x, 2, -((a^2*c^2)/x) + 2*a*c*(b*c + a*d)*x + (1/3)*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^3 + (2/5)*b*d*(b*c + a*d)*x^5 + (1/7)*b^2*d^2*x^7]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^2)/x^3, x, 3, -((a^2*c^2)/(2*x^2)) + (1/2)*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^2 + (1/2)*b*d*(b*c + a*d)*x^4 + (1/6)*b^2*d^2*x^6 + 2*a*c*(b*c + a*d)*log(x)]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^2)/x^4, x, 2, -((a^2*c^2)/(3*x^3)) - (2*a*c*(b*c + a*d))/x + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x + (2/3)*b*d*(b*c + a*d)*x^3 + (1/5)*b^2*d^2*x^5]

    @test_int [x^4*(a + b*x^2)^2*(c + d*x^2)^3, x, 2, (1/5)*a^2*c^3*x^5 + (1/7)*a*c^2*(2*b*c + 3*a*d)*x^7 + (1/9)*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^9 + (1/11)*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^11 + (1/13)*b*d^2*(3*b*c + 2*a*d)*x^13 + (1/15)*b^2*d^3*x^15]
    @test_int [x^3*(a + b*x^2)^2*(c + d*x^2)^3, x, 3, -((c*(b*c - a*d)^2*(c + d*x^2)^4)/(8*d^4)) + ((b*c - a*d)*(3*b*c - a*d)*(c + d*x^2)^5)/(10*d^4) - (b*(3*b*c - 2*a*d)*(c + d*x^2)^6)/(12*d^4) + (b^2*(c + d*x^2)^7)/(14*d^4)]
    @test_int [x^2*(a + b*x^2)^2*(c + d*x^2)^3, x, 2, (1/3)*a^2*c^3*x^3 + (1/5)*a*c^2*(2*b*c + 3*a*d)*x^5 + (1/7)*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^7 + (1/9)*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^9 + (1/11)*b*d^2*(3*b*c + 2*a*d)*x^11 + (1/13)*b^2*d^3*x^13]
    @test_int [x^1*(a + b*x^2)^2*(c + d*x^2)^3, x, 3, ((b*c - a*d)^2*(c + d*x^2)^4)/(8*d^3) - (b*(b*c - a*d)*(c + d*x^2)^5)/(5*d^3) + (b^2*(c + d*x^2)^6)/(12*d^3)]
    @test_int [(a + b*x^2)^2*(c + d*x^2)^3, x, 2, a^2*c^3*x + (a*c^2*(2*b*c + 3*a*d)*x^3)/3 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^5)/5 + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^7)/7 + (b*d^2*(3*b*c + 2*a*d)*x^9)/9 + (b^2*d^3*x^11)/11]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^3)/x^1, x, 3, (1/2)*a*c^2*(2*b*c + 3*a*d)*x^2 + (1/4)*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^4 + (1/6)*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^6 + (1/8)*b*d^2*(3*b*c + 2*a*d)*x^8 + (1/10)*b^2*d^3*x^10 + a^2*c^3*log(x)]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^3)/x^2, x, 2, -((a^2*c^3)/x) + a*c^2*(2*b*c + 3*a*d)*x + (1/3)*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^3 + (1/5)*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^5 + (1/7)*b*d^2*(3*b*c + 2*a*d)*x^7 + (1/9)*b^2*d^3*x^9]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^3)/x^3, x, 3, -((a^2*c^3)/(2*x^2)) + (1/2)*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^2 + (1/4)*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^4 + (1/6)*b*d^2*(3*b*c + 2*a*d)*x^6 + (1/8)*b^2*d^3*x^8 + a*c^2*(2*b*c + 3*a*d)*log(x)]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^3)/x^4, x, 2, -((a^2*c^3)/(3*x^3)) - (a*c^2*(2*b*c + 3*a*d))/x + c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x + (1/3)*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^3 + (1/5)*b*d^2*(3*b*c + 2*a*d)*x^5 + (1/7)*b^2*d^3*x^7]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(x^4*(a + b*x^2)^2)/(c + d*x^2), x, 3, -((c*(b*c - a*d)^2*x)/d^4) + ((b*c - a*d)^2*x^3)/(3*d^3) - (b*(b*c - 2*a*d)*x^5)/(5*d^2) + (b^2*x^7)/(7*d) + (c^(3/2)*(b*c - a*d)^2*atan((sqrt(d)*x)/sqrt(c)))/d^(9/2)]
    @test_int [(x^3*(a + b*x^2)^2)/(c + d*x^2), x, 3, ((b*c - a*d)^2*x^2)/(2*d^3) - (b*(b*c - 2*a*d)*x^4)/(4*d^2) + (b^2*x^6)/(6*d) - (c*(b*c - a*d)^2*log(c + d*x^2))/(2*d^4)]
    @test_int [(x^2*(a + b*x^2)^2)/(c + d*x^2), x, 3, ((b*c - a*d)^2*x)/d^3 - (b*(b*c - 2*a*d)*x^3)/(3*d^2) + (b^2*x^5)/(5*d) - (sqrt(c)*(b*c - a*d)^2*atan((sqrt(d)*x)/sqrt(c)))/d^(7/2)]
    @test_int [(x^1*(a + b*x^2)^2)/(c + d*x^2), x, 3, -((b*(b*c - a*d)*x^2)/(2*d^2)) + (a + b*x^2)^2/(4*d) + ((b*c - a*d)^2*log(c + d*x^2))/(2*d^3)]
    @test_int [(a + b*x^2)^2/(c + d*x^2), x, 3, -((b*(b*c - 2*a*d)*x)/d^2) + (b^2*x^3)/(3*d) + ((b*c - a*d)^2*atan((sqrt(d)*x)/sqrt(c)))/(sqrt(c)*d^(5/2))]
    @test_int [(a + b*x^2)^2/(x^1*(c + d*x^2)), x, 3, (b^2*x^2)/(2*d) + (a^2*log(x))/c - ((b*c - a*d)^2*log(c + d*x^2))/(2*c*d^2)]
    @test_int [(a + b*x^2)^2/(x^2*(c + d*x^2)), x, 3, -(a^2/(c*x)) + (b^2*x)/d - ((b*c - a*d)^2*atan((sqrt(d)*x)/sqrt(c)))/(c^(3/2)*d^(3/2))]
    @test_int [(a + b*x^2)^2/(x^3*(c + d*x^2)), x, 3, -(a^2/(2*c*x^2)) + (a*(2*b*c - a*d)*log(x))/c^2 + ((b*c - a*d)^2*log(c + d*x^2))/(2*c^2*d)]
    @test_int [(a + b*x^2)^2/(x^4*(c + d*x^2)), x, 3, -(a^2/(3*c*x^3)) - (a*(2*b*c - a*d))/(c^2*x) + ((b*c - a*d)^2*atan((sqrt(d)*x)/sqrt(c)))/(c^(5/2)*sqrt(d))]
    @test_int [(a + b*x^2)^2/(x^5*(c + d*x^2)), x, 3, -(a^2/(4*c*x^4)) - (a*(2*b*c - a*d))/(2*c^2*x^2) + ((b*c - a*d)^2*log(x))/c^3 - ((b*c - a*d)^2*log(c + d*x^2))/(2*c^3)]
    @test_int [(a + b*x^2)^2/(x^6*(c + d*x^2)), x, 3, -(a^2/(5*c*x^5)) - (a*(2*b*c - a*d))/(3*c^2*x^3) - (b*c - a*d)^2/(c^3*x) - (sqrt(d)*(b*c - a*d)^2*atan((sqrt(d)*x)/sqrt(c)))/c^(7/2)]
    @test_int [(a + b*x^2)^2/(x^7*(c + d*x^2)), x, 3, -(a^2/(6*c*x^6)) - (a*(2*b*c - a*d))/(4*c^2*x^4) - (b*c - a*d)^2/(2*c^3*x^2) - (d*(b*c - a*d)^2*log(x))/c^4 + (d*(b*c - a*d)^2*log(c + d*x^2))/(2*c^4)]

    @test_int [(x^4*(a + b*x^2)^2)/(c + d*x^2)^2, x, 5, ((7*b*c - 3*a*d)*(b*c - a*d)*x)/(2*d^4) - ((7*b*c - 3*a*d)*(b*c - a*d)*x^3)/(6*c*d^3) + (b^2*x^5)/(5*d^2) + ((b*c - a*d)^2*x^5)/(2*c*d^2*(c + d*x^2)) - (sqrt(c)*(7*b*c - 3*a*d)*(b*c - a*d)*atan((sqrt(d)*x)/sqrt(c)))/(2*d^(9/2))]
    @test_int [(x^3*(a + b*x^2)^2)/(c + d*x^2)^2, x, 3, -((b*(b*c - a*d)*x^2)/d^3) + (b^2*x^4)/(4*d^2) + (c*(b*c - a*d)^2)/(2*d^4*(c + d*x^2)) + ((b*c - a*d)*(3*b*c - a*d)*log(c + d*x^2))/(2*d^4)]
    @test_int [(x^2*(a + b*x^2)^2)/(c + d*x^2)^2, x, 4, -(((b*c - a*d)*(5*b*c - a*d)*x)/(2*c*d^3)) + (b^2*x^3)/(3*d^2) + ((b*c - a*d)^2*x^3)/(2*c*d^2*(c + d*x^2)) + ((b*c - a*d)*(5*b*c - a*d)*atan((sqrt(d)*x)/sqrt(c)))/(2*sqrt(c)*d^(7/2))]
    @test_int [(x^1*(a + b*x^2)^2)/(c + d*x^2)^2, x, 3, (b^2*x^2)/(2*d^2) - (b*c - a*d)^2/(2*d^3*(c + d*x^2)) - (b*(b*c - a*d)*log(c + d*x^2))/d^3]
    @test_int [(a + b*x^2)^2/(c + d*x^2)^2, x, 4, (b^2*x)/d^2 + ((b*c - a*d)^2*x)/(2*c*d^2*(c + d*x^2)) - ((b*c - a*d)*(3*b*c + a*d)*atan((sqrt(d)*x)/sqrt(c)))/(2*c^(3/2)*d^(5/2))]
    @test_int [(a + b*x^2)^2/(x^1*(c + d*x^2)^2), x, 3, (b*c - a*d)^2/(2*c*d^2*(c + d*x^2)) + (a^2*log(x))/c^2 - (1/2)*(a^2/c^2 - b^2/d^2)*log(c + d*x^2)]
    @test_int [(a + b*x^2)^2/(x^2*(c + d*x^2)^2), x, 3, -(a^2/(c*x*(c + d*x^2))) - ((b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2)*x)/(2*c^2*d*(c + d*x^2)) + ((b*c - a*d)*(b*c + 3*a*d)*atan((sqrt(d)*x)/sqrt(c)))/(2*c^(5/2)*d^(3/2))]
    @test_int [(a + b*x^2)^2/(x^3*(c + d*x^2)^2), x, 3, -(a^2/(2*c^2*x^2)) - (b*c - a*d)^2/(2*c^2*d*(c + d*x^2)) + (2*a*(b*c - a*d)*log(x))/c^3 - (a*(b*c - a*d)*log(c + d*x^2))/c^3]
    @test_int [(a + b*x^2)^2/(x^4*(c + d*x^2)^2), x, 4, -((a*(6*b*c - 5*a*d))/(3*c^3*x)) - a^2/(3*c*x^3*(c + d*x^2)) + ((3*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*x)/(6*c^3*(c + d*x^2)) + ((b*c - 5*a*d)*(b*c - a*d)*atan((sqrt(d)*x)/sqrt(c)))/(2*c^(7/2)*sqrt(d))]

    @test_int [(x^4*(a + b*x^2)^2)/(c + d*x^2)^3, x, 5, -(((13*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*x)/(4*c*d^4)) + (b^2*x^3)/(3*d^3) + ((b*c - a*d)^2*x^5)/(4*c*d^2*(c + d*x^2)^2) - ((b*c - a*d)*(9*b*c - a*d)*x)/(8*d^4*(c + d*x^2)) + ((35*b^2*c^2 - 30*a*b*c*d + 3*a^2*d^2)*atan((sqrt(d)*x)/sqrt(c)))/(8*sqrt(c)*d^(9/2))]
    @test_int [(x^3*(a + b*x^2)^2)/(c + d*x^2)^3, x, 3, (b^2*x^2)/(2*d^3) + (c*(b*c - a*d)^2)/(4*d^4*(c + d*x^2)^2) - ((b*c - a*d)*(3*b*c - a*d))/(2*d^4*(c + d*x^2)) - (b*(3*b*c - 2*a*d)*log(c + d*x^2))/(2*d^4)]
    @test_int [(x^2*(a + b*x^2)^2)/(c + d*x^2)^3, x, 4, (b^2*x)/d^3 + ((b*c - a*d)^2*x^3)/(4*c*d^2*(c + d*x^2)^2) + ((b*c - a*d)*(7*b*c + a*d)*x)/(8*c*d^3*(c + d*x^2)) - ((15*b^2*c^2 - 6*a*b*c*d - a^2*d^2)*atan((sqrt(d)*x)/sqrt(c)))/(8*c^(3/2)*d^(7/2))]
    @test_int [(x^1*(a + b*x^2)^2)/(c + d*x^2)^3, x, 3, -((b*c - a*d)^2/(4*d^3*(c + d*x^2)^2)) + (b*(b*c - a*d))/(d^3*(c + d*x^2)) + (b^2*log(c + d*x^2))/(2*d^3)]
    @test_int [(a + b*x^2)^2/(c + d*x^2)^3, x, 3, -(((b*c - a*d)*x*(a + b*x^2))/(4*c*d*(c + d*x^2)^2)) + (3*(a^2/c^2 - b^2/d^2)*x)/(8*(c + d*x^2)) + ((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*atan((sqrt(d)*x)/sqrt(c)))/(8*c^(5/2)*d^(5/2))]
    @test_int [(a + b*x^2)^2/(x^1*(c + d*x^2)^3), x, 3, (b*c - a*d)^2/(4*c*d^2*(c + d*x^2)^2) + (a^2/c^2 - b^2/d^2)/(2*(c + d*x^2)) + (a^2*log(x))/c^3 - (a^2*log(c + d*x^2))/(2*c^3)]
    @test_int [(a + b*x^2)^2/(x^2*(c + d*x^2)^3), x, 4, -(a^2/(c*x*(c + d*x^2)^2)) - ((b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*x)/(4*c^2*d*(c + d*x^2)^2) + ((b^2*c^2 + 3*a*d*(2*b*c - 5*a*d))*x)/(8*c^3*d*(c + d*x^2)) + ((b^2*c^2 + 3*a*d*(2*b*c - 5*a*d))*atan((sqrt(d)*x)/sqrt(c)))/(8*c^(7/2)*d^(3/2))]
    @test_int [(a + b*x^2)^2/(x^3*(c + d*x^2)^3), x, 3, -(a^2/(2*c^3*x^2)) - (b*c - a*d)^2/(4*c^2*d*(c + d*x^2)^2) + (a*(b*c - a*d))/(c^3*(c + d*x^2)) + (a*(2*b*c - 3*a*d)*log(x))/c^4 - (a*(2*b*c - 3*a*d)*log(c + d*x^2))/(2*c^4)]
    @test_int [(a + b*x^2)^2/(x^4*(c + d*x^2)^3), x, 5, -((a*(6*b*c - 7*a*d))/(3*c^4*x)) - a^2/(3*c*x^3*(c + d*x^2)^2) + ((3*b^2*c^2 - 6*a*b*c*d + 7*a^2*d^2)*x)/(12*c^3*(c + d*x^2)^2) + ((3*b*c - 7*a*d)^2*x)/(24*c^4*(c + d*x^2)) + ((3*b^2*c^2 - 30*a*b*c*d + 35*a^2*d^2)*atan((sqrt(d)*x)/sqrt(c)))/(8*c^(9/2)*sqrt(d))]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^2)^p*(c+d*x^2)^3=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^2)^p / (c+d*x^2)^1=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(x^5*(c + d*x^2))/(a + b*x^2), x, 3, -((a*(b*c - a*d)*x^2)/(2*b^3)) + ((b*c - a*d)*x^4)/(4*b^2) + (d*x^6)/(6*b) + (a^2*(b*c - a*d)*log(a + b*x^2))/(2*b^4)]
    @test_int [(x^4*(c + d*x^2))/(a + b*x^2), x, 4, -((a*(b*c - a*d)*x)/b^3) + ((b*c - a*d)*x^3)/(3*b^2) + (d*x^5)/(5*b) + (a^(3/2)*(b*c - a*d)*atan((sqrt(b)*x)/sqrt(a)))/b^(7/2)]
    @test_int [(x^3*(c + d*x^2))/(a + b*x^2), x, 3, ((b*c - a*d)*x^2)/(2*b^2) + (d*x^4)/(4*b) - (a*(b*c - a*d)*log(a + b*x^2))/(2*b^3)]
    @test_int [(x^2*(c + d*x^2))/(a + b*x^2), x, 3, ((b*c - a*d)*x)/b^2 + (d*x^3)/(3*b) - (sqrt(a)*(b*c - a*d)*atan((sqrt(b)*x)/sqrt(a)))/b^(5/2)]
    @test_int [(x^1*(c + d*x^2))/(a + b*x^2), x, 3, (d*x^2)/(2*b) + ((b*c - a*d)*log(a + b*x^2))/(2*b^2)]
    @test_int [(c + d*x^2)/(a + b*x^2), x, 2, (d*x)/b + ((b*c - a*d)*atan((sqrt(b)*x)/sqrt(a)))/(sqrt(a)*b^(3/2))]
    @test_int [(c + d*x^2)/(x^1*(a + b*x^2)), x, 3, (c*log(x))/a - ((b*c - a*d)*log(a + b*x^2))/(2*a*b)]
    @test_int [(c + d*x^2)/(x^2*(a + b*x^2)), x, 2, -(c/(a*x)) - ((b*c - a*d)*atan((sqrt(b)*x)/sqrt(a)))/(a^(3/2)*sqrt(b))]
    @test_int [(c + d*x^2)/(x^3*(a + b*x^2)), x, 3, -c/(2*a*x^2) - ((b*c - a*d)*log(x))/a^2 + ((b*c - a*d)*log(a + b*x^2))/(2*a^2)]
    @test_int [(c + d*x^2)/(x^4*(a + b*x^2)), x, 3, -c/(3*a*x^3) + (b*c - a*d)/(a^2*x) + (sqrt(b)*(b*c - a*d)*atan((sqrt(b)*x)/sqrt(a)))/a^(5/2)]

    @test_int [(x^5*(c + d*x^2)^2)/(a + b*x^2), x, 3, -((a*(b*c - a*d)^2*x^2)/(2*b^4)) + ((b*c - a*d)^2*x^4)/(4*b^3) + (d*(2*b*c - a*d)*x^6)/(6*b^2) + (d^2*x^8)/(8*b) + (a^2*(b*c - a*d)^2*log(a + b*x^2))/(2*b^5)]
    @test_int [(x^4*(c + d*x^2)^2)/(a + b*x^2), x, 3, -((a*(b*c - a*d)^2*x)/b^4) + ((b*c - a*d)^2*x^3)/(3*b^3) + (d*(2*b*c - a*d)*x^5)/(5*b^2) + (d^2*x^7)/(7*b) + (a^(3/2)*(b*c - a*d)^2*atan((sqrt(b)*x)/sqrt(a)))/b^(9/2)]
    @test_int [(x^3*(c + d*x^2)^2)/(a + b*x^2), x, 3, ((b*c - a*d)^2*x^2)/(2*b^3) + (d*(2*b*c - a*d)*x^4)/(4*b^2) + (d^2*x^6)/(6*b) - (a*(b*c - a*d)^2*log(a + b*x^2))/(2*b^4)]
    @test_int [(x^2*(c + d*x^2)^2)/(a + b*x^2), x, 3, ((b*c - a*d)^2*x)/b^3 + (d*(2*b*c - a*d)*x^3)/(3*b^2) + (d^2*x^5)/(5*b) - (sqrt(a)*(b*c - a*d)^2*atan((sqrt(b)*x)/sqrt(a)))/b^(7/2)]
    @test_int [(x^1*(c + d*x^2)^2)/(a + b*x^2), x, 3, (d*(b*c - a*d)*x^2)/(2*b^2) + (c + d*x^2)^2/(4*b) + ((b*c - a*d)^2*log(a + b*x^2))/(2*b^3)]
    @test_int [(c + d*x^2)^2/(a + b*x^2), x, 3, (d*(2*b*c - a*d)*x)/b^2 + (d^2*x^3)/(3*b) + ((b*c - a*d)^2*atan((sqrt(b)*x)/sqrt(a)))/(sqrt(a)*b^(5/2))]
    @test_int [(c + d*x^2)^2/(x^1*(a + b*x^2)), x, 3, (d^2*x^2)/(2*b) + (c^2*log(x))/a - ((b*c - a*d)^2*log(a + b*x^2))/(2*a*b^2)]
    @test_int [(c + d*x^2)^2/(x^2*(a + b*x^2)), x, 3, -(c^2/(a*x)) + (d^2*x)/b - ((b*c - a*d)^2*atan((sqrt(b)*x)/sqrt(a)))/(a^(3/2)*b^(3/2))]
    @test_int [(c + d*x^2)^2/(x^3*(a + b*x^2)), x, 3, -c^2/(2*a*x^2) - (c*(b*c - 2*a*d)*log(x))/a^2 + ((b*c - a*d)^2*log(a + b*x^2))/(2*a^2*b)]
    @test_int [(c + d*x^2)^2/(x^4*(a + b*x^2)), x, 3, -c^2/(3*a*x^3) + (c*(b*c - 2*a*d))/(a^2*x) + ((b*c - a*d)^2*atan((sqrt(b)*x)/sqrt(a)))/(a^(5/2)*sqrt(b))]

    @test_int [(x^5*(c + d*x^2)^3)/(a + b*x^2), x, 3, -((a*(b*c - a*d)^3*x^2)/(2*b^5)) + ((b*c - a*d)^3*x^4)/(4*b^4) + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^6)/(6*b^3) + (d^2*(3*b*c - a*d)*x^8)/(8*b^2) + (d^3*x^10)/(10*b) + (a^2*(b*c - a*d)^3*log(a + b*x^2))/(2*b^6)]
    @test_int [(x^4*(c + d*x^2)^3)/(a + b*x^2), x, 3, -((a*(b*c - a*d)^3*x)/b^5) + ((b*c - a*d)^3*x^3)/(3*b^4) + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^5)/(5*b^3) + (d^2*(3*b*c - a*d)*x^7)/(7*b^2) + (d^3*x^9)/(9*b) + (a^(3/2)*(b*c - a*d)^3*atan((sqrt(b)*x)/sqrt(a)))/b^(11/2)]
    @test_int [(x^3*(c + d*x^2)^3)/(a + b*x^2), x, 3, ((b*c - a*d)^3*x^2)/(2*b^4) + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^4)/(4*b^3) + (d^2*(3*b*c - a*d)*x^6)/(6*b^2) + (d^3*x^8)/(8*b) - (a*(b*c - a*d)^3*log(a + b*x^2))/(2*b^5)]
    @test_int [(x^2*(c + d*x^2)^3)/(a + b*x^2), x, 3, ((b*c - a*d)^3*x)/b^4 + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^3)/(3*b^3) + (d^2*(3*b*c - a*d)*x^5)/(5*b^2) + (d^3*x^7)/(7*b) - (sqrt(a)*(b*c - a*d)^3*atan((sqrt(b)*x)/sqrt(a)))/b^(9/2)]
    @test_int [(x^1*(c + d*x^2)^3)/(a + b*x^2), x, 3, (d*(b*c - a*d)^2*x^2)/(2*b^3) + ((b*c - a*d)*(c + d*x^2)^2)/(4*b^2) + (c + d*x^2)^3/(6*b) + ((b*c - a*d)^3*log(a + b*x^2))/(2*b^4)]
    @test_int [(c + d*x^2)^3/(a + b*x^2), x, 3, (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x)/b^3 + (d^2*(3*b*c - a*d)*x^3)/(3*b^2) + (d^3*x^5)/(5*b) + ((b*c - a*d)^3*atan((sqrt(b)*x)/sqrt(a)))/(sqrt(a)*b^(7/2))]
    @test_int [(c + d*x^2)^3/(x^1*(a + b*x^2)), x, 3, (d^2*(3*b*c - a*d)*x^2)/(2*b^2) + (d^3*x^4)/(4*b) + (c^3*log(x))/a - ((b*c - a*d)^3*log(a + b*x^2))/(2*a*b^3)]
    @test_int [(c + d*x^2)^3/(x^2*(a + b*x^2)), x, 3, -(c^3/(a*x)) + (d^2*(3*b*c - a*d)*x)/b^2 + (d^3*x^3)/(3*b) - ((b*c - a*d)^3*atan((sqrt(b)*x)/sqrt(a)))/(a^(3/2)*b^(5/2))]
    @test_int [(c + d*x^2)^3/(x^3*(a + b*x^2)), x, 3, -c^3/(2*a*x^2) + (d^3*x^2)/(2*b) - (c^2*(b*c - 3*a*d)*log(x))/a^2 + ((b*c - a*d)^3*log(a + b*x^2))/(2*a^2*b^2)]
    @test_int [(c + d*x^2)^3/(x^4*(a + b*x^2)), x, 3, -c^3/(3*a*x^3) + (c^2*(b*c - 3*a*d))/(a^2*x) + (d^3*x)/b + ((b*c - a*d)^3*atan((sqrt(b)*x)/sqrt(a)))/(a^(5/2)*b^(3/2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^5/((a + b*x^2)*(c + d*x^2)), x, 3, x^2/(2*b*d) + (a^2*log(a + b*x^2))/(2*b^2*(b*c - a*d)) - (c^2*log(c + d*x^2))/(2*d^2*(b*c - a*d))]
    @test_int [x^4/((a + b*x^2)*(c + d*x^2)), x, 4, x/(b*d) + (a^(3/2)*atan((sqrt(b)*x)/sqrt(a)))/(b^(3/2)*(b*c - a*d)) - (c^(3/2)*atan((sqrt(d)*x)/sqrt(c)))/(d^(3/2)*(b*c - a*d))]
    @test_int [x^3/((a + b*x^2)*(c + d*x^2)), x, 3, -(a*log(a + b*x^2))/(2*b*(b*c - a*d)) + (c*log(c + d*x^2))/(2*d*(b*c - a*d))]
    @test_int [x^2/((a + b*x^2)*(c + d*x^2)), x, 3, -((sqrt(a)*atan((sqrt(b)*x)/sqrt(a)))/(sqrt(b)*(b*c - a*d))) + (sqrt(c)*atan((sqrt(d)*x)/sqrt(c)))/(sqrt(d)*(b*c - a*d))]
    @test_int [x^1/((a + b*x^2)*(c + d*x^2)), x, 4, log(a + b*x^2)/(2*(b*c - a*d)) - log(c + d*x^2)/(2*(b*c - a*d))]
    @test_int [1/((a + b*x^2)*(c + d*x^2)), x, 3, (sqrt(b)*atan((sqrt(b)*x)/sqrt(a)))/(sqrt(a)*(b*c - a*d)) - (sqrt(d)*atan((sqrt(d)*x)/sqrt(c)))/(sqrt(c)*(b*c - a*d))]
    @test_int [1/(x^1*(a + b*x^2)*(c + d*x^2)), x, 3, log(x)/(a*c) - (b*log(a + b*x^2))/(2*a*(b*c - a*d)) + (d*log(c + d*x^2))/(2*c*(b*c - a*d))]
    @test_int [1/(x^2*(a + b*x^2)*(c + d*x^2)), x, 4, -(1/(a*c*x)) - (b^(3/2)*atan((sqrt(b)*x)/sqrt(a)))/(a^(3/2)*(b*c - a*d)) + (d^(3/2)*atan((sqrt(d)*x)/sqrt(c)))/(c^(3/2)*(b*c - a*d))]
    @test_int [1/(x^3*(a + b*x^2)*(c + d*x^2)), x, 3, -1/(2*a*c*x^2) - ((b*c + a*d)*log(x))/(a^2*c^2) + (b^2*log(a + b*x^2))/(2*a^2*(b*c - a*d)) - (d^2*log(c + d*x^2))/(2*c^2*(b*c - a*d))]
    @test_int [1/(x^4*(a + b*x^2)*(c + d*x^2)), x, 5, -(1/(3*a*c*x^3)) + (b*c + a*d)/(a^2*c^2*x) + (b^(5/2)*atan((sqrt(b)*x)/sqrt(a)))/(a^(5/2)*(b*c - a*d)) - (d^(5/2)*atan((sqrt(d)*x)/sqrt(c)))/(c^(5/2)*(b*c - a*d))]
    @test_int [1/(x^5*(a + b*x^2)*(c + d*x^2)), x, 3, -1/(4*a*c*x^4) + (b*c + a*d)/(2*a^2*c^2*x^2) + ((b^2*c^2 + a*b*c*d + a^2*d^2)*log(x))/(a^3*c^3) - (b^3*log(a + b*x^2))/(2*a^3*(b*c - a*d)) + (d^3*log(c + d*x^2))/(2*c^3*(b*c - a*d))]
    @test_int [1/(x^6*(a + b*x^2)*(c + d*x^2)), x, 6, -(1/(5*a*c*x^5)) + (b*c + a*d)/(3*a^2*c^2*x^3) - (b^2*c^2 + a*b*c*d + a^2*d^2)/(a^3*c^3*x) - (b^(7/2)*atan((sqrt(b)*x)/sqrt(a)))/(a^(7/2)*(b*c - a*d)) + (d^(7/2)*atan((sqrt(d)*x)/sqrt(c)))/(c^(7/2)*(b*c - a*d))]
    @test_int [1/(x^7*(a + b*x^2)*(c + d*x^2)), x, 3, -(1/(6*a*c*x^6)) + (b*c + a*d)/(4*a^2*c^2*x^4) - (b^2*c^2 + a*b*c*d + a^2*d^2)/(2*a^3*c^3*x^2) - ((b*c + a*d)*(b^2*c^2 + a^2*d^2)*log(x))/(a^4*c^4) + (b^4*log(a + b*x^2))/(2*a^4*(b*c - a*d)) - (d^4*log(c + d*x^2))/(2*c^4*(b*c - a*d))]

    @test_int [x^5/((a + b*x^2)^2*(c + d*x^2)), x, 3, -a^2/(2*b^2*(b*c - a*d)*(a + b*x^2)) - (a*(2*b*c - a*d)*log(a + b*x^2))/(2*b^2*(b*c - a*d)^2) + (c^2*log(c + d*x^2))/(2*d*(b*c - a*d)^2)]
    @test_int [x^4/((a + b*x^2)*(c + d*x^2)^2), x, 4, -((c*x)/(2*d*(b*c - a*d)*(c + d*x^2))) + (a^(3/2)*atan((sqrt(b)*x)/sqrt(a)))/(sqrt(b)*(b*c - a*d)^2) + (sqrt(c)*(b*c - 3*a*d)*atan((sqrt(d)*x)/sqrt(c)))/(2*d^(3/2)*(b*c - a*d)^2)]
    @test_int [x^3/((a + b*x^2)*(c + d*x^2)^2), x, 3, -c/(2*d*(b*c - a*d)*(c + d*x^2)) - (a*log(a + b*x^2))/(2*(b*c - a*d)^2) + (a*log(c + d*x^2))/(2*(b*c - a*d)^2)]
    @test_int [x^2/((a + b*x^2)*(c + d*x^2)^2), x, 4, x/(2*(b*c - a*d)*(c + d*x^2)) - (sqrt(a)*sqrt(b)*atan((sqrt(b)*x)/sqrt(a)))/(b*c - a*d)^2 + ((b*c + a*d)*atan((sqrt(d)*x)/sqrt(c)))/(2*sqrt(c)*sqrt(d)*(b*c - a*d)^2)]
    @test_int [x^1/((a + b*x^2)*(c + d*x^2)^2), x, 3, 1/(2*(b*c - a*d)*(c + d*x^2)) + (b*log(a + b*x^2))/(2*(b*c - a*d)^2) - (b*log(c + d*x^2))/(2*(b*c - a*d)^2)]
    @test_int [1/((a + b*x^2)*(c + d*x^2)^2), x, 4, -((d*x)/(2*c*(b*c - a*d)*(c + d*x^2))) + (b^(3/2)*atan((sqrt(b)*x)/sqrt(a)))/(sqrt(a)*(b*c - a*d)^2) - (sqrt(d)*(3*b*c - a*d)*atan((sqrt(d)*x)/sqrt(c)))/(2*c^(3/2)*(b*c - a*d)^2)]
    @test_int [1/(x^1*(a + b*x^2)*(c + d*x^2)^2), x, 3, -d/(2*c*(b*c - a*d)*(c + d*x^2)) + log(x)/(a*c^2) - (b^2*log(a + b*x^2))/(2*a*(b*c - a*d)^2) + (d*(2*b*c - a*d)*log(c + d*x^2))/(2*c^2*(b*c - a*d)^2)]
    @test_int [1/(x^2*(a + b*x^2)*(c + d*x^2)^2), x, 5, -((2*b*c - 3*a*d)/(2*a*c^2*(b*c - a*d)*x)) - d/(2*c*(b*c - a*d)*x*(c + d*x^2)) - (b^(5/2)*atan((sqrt(b)*x)/sqrt(a)))/(a^(3/2)*(b*c - a*d)^2) + (d^(3/2)*(5*b*c - 3*a*d)*atan((sqrt(d)*x)/sqrt(c)))/(2*c^(5/2)*(b*c - a*d)^2)]
    @test_int [1/(x^3*(a + b*x^2)*(c + d*x^2)^2), x, 3, -1/(2*a*c^2*x^2) + d^2/(2*c^2*(b*c - a*d)*(c + d*x^2)) - ((b*c + 2*a*d)*log(x))/(a^2*c^3) + (b^3*log(a + b*x^2))/(2*a^2*(b*c - a*d)^2) - (d^2*(3*b*c - 2*a*d)*log(c + d*x^2))/(2*c^3*(b*c - a*d)^2)]
    @test_int [1/(x^4*(a + b*x^2)*(c + d*x^2)^2), x, 6, -((2*b*c - 5*a*d)/(6*a*c^2*(b*c - a*d)*x^3)) + (2*b^2*c^2 + 2*a*b*c*d - 5*a^2*d^2)/(2*a^2*c^3*(b*c - a*d)*x) - d/(2*c*(b*c - a*d)*x^3*(c + d*x^2)) + (b^(7/2)*atan((sqrt(b)*x)/sqrt(a)))/(a^(5/2)*(b*c - a*d)^2) - (d^(5/2)*(7*b*c - 5*a*d)*atan((sqrt(d)*x)/sqrt(c)))/(2*c^(7/2)*(b*c - a*d)^2)]

    @test_int [x^5/((a + b*x^2)^3*(c + d*x^2)), x, 3, -a^2/(4*b^2*(b*c - a*d)*(a + b*x^2)^2) + (a*(2*b*c - a*d))/(2*b^2*(b*c - a*d)^2*(a + b*x^2)) + (c^2*log(a + b*x^2))/(2*(b*c - a*d)^3) - (c^2*log(c + d*x^2))/(2*(b*c - a*d)^3)]
    @test_int [x^4/((a + b*x^2)*(c + d*x^2)^3), x, 5, -((c*x)/(4*d*(b*c - a*d)*(c + d*x^2)^2)) + ((b*c - 5*a*d)*x)/(8*d*(b*c - a*d)^2*(c + d*x^2)) + (a^(3/2)*sqrt(b)*atan((sqrt(b)*x)/sqrt(a)))/(b*c - a*d)^3 + ((b^2*c^2 - 6*a*b*c*d - 3*a^2*d^2)*atan((sqrt(d)*x)/sqrt(c)))/(8*sqrt(c)*d^(3/2)*(b*c - a*d)^3)]
    @test_int [x^3/((a + b*x^2)*(c + d*x^2)^3), x, 3, -c/(4*d*(b*c - a*d)*(c + d*x^2)^2) - a/(2*(b*c - a*d)^2*(c + d*x^2)) - (a*b*log(a + b*x^2))/(2*(b*c - a*d)^3) + (a*b*log(c + d*x^2))/(2*(b*c - a*d)^3)]
    @test_int [x^2/((a + b*x^2)*(c + d*x^2)^3), x, 5, x/(4*(b*c - a*d)*(c + d*x^2)^2) + ((3*b*c + a*d)*x)/(8*c*(b*c - a*d)^2*(c + d*x^2)) - (sqrt(a)*b^(3/2)*atan((sqrt(b)*x)/sqrt(a)))/(b*c - a*d)^3 + ((3*b^2*c^2 + 6*a*b*c*d - a^2*d^2)*atan((sqrt(d)*x)/sqrt(c)))/(8*c^(3/2)*sqrt(d)*(b*c - a*d)^3)]
    @test_int [x^1/((a + b*x^2)*(c + d*x^2)^3), x, 3, 1/(4*(b*c - a*d)*(c + d*x^2)^2) + b/(2*(b*c - a*d)^2*(c + d*x^2)) + (b^2*log(a + b*x^2))/(2*(b*c - a*d)^3) - (b^2*log(c + d*x^2))/(2*(b*c - a*d)^3)]
    @test_int [1/((a + b*x^2)*(c + d*x^2)^3), x, 5, -((d*x)/(4*c*(b*c - a*d)*(c + d*x^2)^2)) - (d*(7*b*c - 3*a*d)*x)/(8*c^2*(b*c - a*d)^2*(c + d*x^2)) + (b^(5/2)*atan((sqrt(b)*x)/sqrt(a)))/(sqrt(a)*(b*c - a*d)^3) - (sqrt(d)*(15*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*atan((sqrt(d)*x)/sqrt(c)))/(8*c^(5/2)*(b*c - a*d)^3)]
    @test_int [1/(x^1*(a + b*x^2)*(c + d*x^2)^3), x, 3, -d/(4*c*(b*c - a*d)*(c + d*x^2)^2) - (d*(2*b*c - a*d))/(2*c^2*(b*c - a*d)^2*(c + d*x^2)) + log(x)/(a*c^3) - (b^3*log(a + b*x^2))/(2*a*(b*c - a*d)^3) + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*log(c + d*x^2))/(2*c^3*(b*c - a*d)^3)]
    @test_int [1/(x^2*(a + b*x^2)*(c + d*x^2)^3), x, 6, -((8*b^2*c^2 - 27*a*b*c*d + 15*a^2*d^2)/(8*a*c^3*(b*c - a*d)^2*x)) - d/(4*c*(b*c - a*d)*x*(c + d*x^2)^2) - (d*(9*b*c - 5*a*d))/(8*c^2*(b*c - a*d)^2*x*(c + d*x^2)) - (b^(7/2)*atan((sqrt(b)*x)/sqrt(a)))/(a^(3/2)*(b*c - a*d)^3) + (d^(3/2)*(35*b^2*c^2 - 42*a*b*c*d + 15*a^2*d^2)*atan((sqrt(d)*x)/sqrt(c)))/(8*c^(7/2)*(b*c - a*d)^3)]
    @test_int [1/(x^3*(a + b*x^2)*(c + d*x^2)^3), x, 3, -1/(2*a*c^3*x^2) + d^2/(4*c^2*(b*c - a*d)*(c + d*x^2)^2) + (d^2*(3*b*c - 2*a*d))/(2*c^3*(b*c - a*d)^2*(c + d*x^2)) - ((b*c + 3*a*d)*log(x))/(a^2*c^4) + (b^4*log(a + b*x^2))/(2*a^2*(b*c - a*d)^3) - (d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*log(c + d*x^2))/(2*c^4*(b*c - a*d)^3)]
    @test_int [1/(x^4*(a + b*x^2)*(c + d*x^2)^3), x, 7, -((8*b^2*c^2 - 55*a*b*c*d + 35*a^2*d^2)/(24*a*c^3*(b*c - a*d)^2*x^3)) + (8*b^3*c^3 + 8*a*b^2*c^2*d - 55*a^2*b*c*d^2 + 35*a^3*d^3)/(8*a^2*c^4*(b*c - a*d)^2*x) - d/(4*c*(b*c - a*d)*x^3*(c + d*x^2)^2) - (d*(11*b*c - 7*a*d))/(8*c^2*(b*c - a*d)^2*x^3*(c + d*x^2)) + (b^(9/2)*atan((sqrt(b)*x)/sqrt(a)))/(a^(5/2)*(b*c - a*d)^3) - (d^(5/2)*(63*b^2*c^2 - 90*a*b*c*d + 35*a^2*d^2)*atan((sqrt(d)*x)/sqrt(c)))/(8*c^(9/2)*(b*c - a*d)^3)]

    @test_int [x/((1 + x^2)*(4 + x^2)), x, 4, (1/6)*log(1 + x^2) - (1/6)*log(4 + x^2)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^2)^p / (c+d*x^2)^2=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(x^4*(c + d*x^2))/(a + b*x^2)^2, x, 4, ((b*c - 2*a*d)*x)/b^3 + (d*x^3)/(3*b^2) + (a*(b*c - a*d)*x)/(2*b^3*(a + b*x^2)) - (sqrt(a)*(3*b*c - 5*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*b^(7/2))]
    @test_int [(x^3*(c + d*x^2))/(a + b*x^2)^2, x, 3, (d*x^2)/(2*b^2) + (a*(b*c - a*d))/(2*b^3*(a + b*x^2)) + ((b*c - 2*a*d)*log(a + b*x^2))/(2*b^3)]
    @test_int [(x^2*(c + d*x^2))/(a + b*x^2)^2, x, 3, (d*x)/b^2 - ((b*c - a*d)*x)/(2*b^2*(a + b*x^2)) + ((b*c - 3*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*sqrt(a)*b^(5/2))]
    @test_int [(x^1*(c + d*x^2))/(a + b*x^2)^2, x, 3, -(b*c - a*d)/(2*b^2*(a + b*x^2)) + (d*log(a + b*x^2))/(2*b^2)]
    @test_int [(c + d*x^2)/(a + b*x^2)^2, x, 2, ((b*c - a*d)*x)/(2*a*b*(a + b*x^2)) + ((b*c + a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(3/2)*b^(3/2))]
    @test_int [(c + d*x^2)/(x^1*(a + b*x^2)^2), x, 3, (b*c - a*d)/(2*a*b*(a + b*x^2)) + (c*log(x))/a^2 - (c*log(a + b*x^2))/(2*a^2)]
    @test_int [(c + d*x^2)/(x^2*(a + b*x^2)^2), x, 3, -(c/(a^2*x)) - ((b*c - a*d)*x)/(2*a^2*(a + b*x^2)) - ((3*b*c - a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(5/2)*sqrt(b))]
    @test_int [(c + d*x^2)/(x^3*(a + b*x^2)^2), x, 3, -(c/(2*a^2*x^2)) - (b*c - a*d)/(2*a^2*(a + b*x^2)) - ((2*b*c - a*d)*log(x))/a^3 + ((2*b*c - a*d)*log(a + b*x^2))/(2*a^3)]
    @test_int [(c + d*x^2)/(x^4*(a + b*x^2)^2), x, 4, -(c/(3*a^2*x^3)) + (2*b*c - a*d)/(a^3*x) + (b*(b*c - a*d)*x)/(2*a^3*(a + b*x^2)) + (sqrt(b)*(5*b*c - 3*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(7/2))]

    @test_int [(x^4*(c + d*x^2)^2)/(a + b*x^2)^2, x, 5, ((3*b*c - 7*a*d)*(b*c - a*d)*x)/(2*b^4) - ((3*b*c - 7*a*d)*(b*c - a*d)*x^3)/(6*a*b^3) + (d^2*x^5)/(5*b^2) + ((b*c - a*d)^2*x^5)/(2*a*b^2*(a + b*x^2)) - (sqrt(a)*(3*b*c - 7*a*d)*(b*c - a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*b^(9/2))]
    @test_int [(x^3*(c + d*x^2)^2)/(a + b*x^2)^2, x, 3, (d*(b*c - a*d)*x^2)/b^3 + (d^2*x^4)/(4*b^2) + (a*(b*c - a*d)^2)/(2*b^4*(a + b*x^2)) + ((b*c - 3*a*d)*(b*c - a*d)*log(a + b*x^2))/(2*b^4)]
    @test_int [(x^2*(c + d*x^2)^2)/(a + b*x^2)^2, x, 4, -(((b*c - 5*a*d)*(b*c - a*d)*x)/(2*a*b^3)) + (d^2*x^3)/(3*b^2) + ((b*c - a*d)^2*x^3)/(2*a*b^2*(a + b*x^2)) + ((b*c - 5*a*d)*(b*c - a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*sqrt(a)*b^(7/2))]
    @test_int [(x^1*(c + d*x^2)^2)/(a + b*x^2)^2, x, 3, (d^2*x^2)/(2*b^2) - (b*c - a*d)^2/(2*b^3*(a + b*x^2)) + (d*(b*c - a*d)*log(a + b*x^2))/b^3]
    @test_int [(c + d*x^2)^2/(a + b*x^2)^2, x, 4, (d^2*x)/b^2 + ((b*c - a*d)^2*x)/(2*a*b^2*(a + b*x^2)) + ((b*c - a*d)*(b*c + 3*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(3/2)*b^(5/2))]
    @test_int [(c + d*x^2)^2/(x^1*(a + b*x^2)^2), x, 3, (b*c - a*d)^2/(2*a*b^2*(a + b*x^2)) + (c^2*log(x))/a^2 - (1/2)*(c^2/a^2 - d^2/b^2)*log(a + b*x^2)]
    @test_int [(c + d*x^2)^2/(x^2*(a + b*x^2)^2), x, 3, -(c^2/(a*x*(a + b*x^2))) - (((3*b*c^2)/a - 2*c*d + (a*d^2)/b)*x)/(2*a*(a + b*x^2)) - ((b*c - a*d)*(3*b*c + a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(5/2)*b^(3/2))]
    @test_int [(c + d*x^2)^2/(x^3*(a + b*x^2)^2), x, 3, -(c^2/(2*a^2*x^2)) - (b*c - a*d)^2/(2*a^2*b*(a + b*x^2)) - (2*c*(b*c - a*d)*log(x))/a^3 + (c*(b*c - a*d)*log(a + b*x^2))/a^3]
    @test_int [(c + d*x^2)^2/(x^4*(a + b*x^2)^2), x, 4, (c*(5*b*c - 6*a*d))/(3*a^3*x) - c^2/(3*a*x^3*(a + b*x^2)) + ((5*b^2*c^2 - 6*a*b*c*d + 3*a^2*d^2)*x)/(6*a^3*(a + b*x^2)) + ((b*c - a*d)*(5*b*c - a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(7/2)*sqrt(b))]

    @test_int [(x^4*(c + d*x^2)^3)/(a + b*x^2)^2, x, 4, (3*(b*c - 3*a*d)*(b*c - a*d)^2*x)/(2*b^5) + (d*(5*b^2*c^2 - 7*a*b*c*d + 3*a^2*d^2)*x^3)/(2*b^4) + (3*d^2*(7*b*c - 3*a*d)*x^5)/(10*b^3) + (9*d^3*x^7)/(14*b^2) - (x^3*(c + d*x^2)^3)/(2*b*(a + b*x^2)) - (3*sqrt(a)*(b*c - 3*a*d)*(b*c - a*d)^2*atan((sqrt(b)*x)/sqrt(a)))/(2*b^(11/2))]
    @test_int [(x^3*(c + d*x^2)^3)/(a + b*x^2)^2, x, 3, (3*d*(b*c - a*d)^2*x^2)/(2*b^4) + (d^2*(3*b*c - 2*a*d)*x^4)/(4*b^3) + (d^3*x^6)/(6*b^2) + (a*(b*c - a*d)^3)/(2*b^5*(a + b*x^2)) + ((b*c - 4*a*d)*(b*c - a*d)^2*log(a + b*x^2))/(2*b^5)]
    @test_int [(x^2*(c + d*x^2)^3)/(a + b*x^2)^2, x, 5, (d*(81*b^2*c^2 - 190*a*b*c*d + 105*a^2*d^2)*x)/(30*b^4) + (d*(33*b*c - 35*a*d)*x*(c + d*x^2))/(30*b^3) + (7*d*x*(c + d*x^2)^2)/(10*b^2) - (x*(c + d*x^2)^3)/(2*b*(a + b*x^2)) + ((b*c - 7*a*d)*(b*c - a*d)^2*atan((sqrt(b)*x)/sqrt(a)))/(2*sqrt(a)*b^(9/2))]
    @test_int [(x^1*(c + d*x^2)^3)/(a + b*x^2)^2, x, 3, (d^2*(3*b*c - 2*a*d)*x^2)/(2*b^3) + (d^3*x^4)/(4*b^2) - (b*c - a*d)^3/(2*b^4*(a + b*x^2)) + (3*d*(b*c - a*d)^2*log(a + b*x^2))/(2*b^4)]
    @test_int [(c + d*x^2)^3/(a + b*x^2)^2, x, 4, (d^2*(3*b*c - 2*a*d)*x)/b^3 + (d^3*x^3)/(3*b^2) + ((b*c - a*d)^3*x)/(2*a*b^3*(a + b*x^2)) + ((b*c - a*d)^2*(b*c + 5*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(3/2)*b^(7/2))]
    @test_int [(c + d*x^2)^3/(x^1*(a + b*x^2)^2), x, 3, (d^3*x^2)/(2*b^2) + (b*c - a*d)^3/(2*a*b^3*(a + b*x^2)) + (c^3*log(x))/a^2 - ((b*c - a*d)^2*(b*c + 2*a*d)*log(a + b*x^2))/(2*a^2*b^3)]
    @test_int [(c + d*x^2)^3/(x^2*(a + b*x^2)^2), x, 4, -((c^2*(3*b*c - a*d))/(2*a^2*b*x)) - (d^2*(b*c - 3*a*d)*x)/(2*a*b^2) + ((b*c - a*d)*(c + d*x^2)^2)/(2*a*b*x*(a + b*x^2)) - (3*(b*c - a*d)^2*(b*c + a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(5/2)*b^(5/2))]
    @test_int [(c + d*x^2)^3/(x^3*(a + b*x^2)^2), x, 3, -c^3/(2*a^2*x^2) - (b*c - a*d)^3/(2*a^2*b^2*(a + b*x^2)) - (c^2*(2*b*c - 3*a*d)*log(x))/a^3 + ((b*c - a*d)^2*(2*b*c + a*d)*log(a + b*x^2))/(2*a^3*b^2)]
    @test_int [(c + d*x^2)^3/(x^4*(a + b*x^2)^2), x, 4, -((c^2*(5*b*c - 3*a*d))/(6*a^2*b*x^3)) + (c*(5*b^2*c^2 - 9*a*b*c*d + 2*a^2*d^2))/(2*a^3*b*x) + ((b*c - a*d)*(c + d*x^2)^2)/(2*a*b*x^3*(a + b*x^2)) + ((b*c - a*d)^2*(5*b*c + a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(7/2)*b^(3/2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^4/((a + b*x^2)^2*(c + d*x^2)), x, 4, (a*x)/(2*b*(b*c - a*d)*(a + b*x^2)) - (sqrt(a)*(3*b*c - a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*b^(3/2)*(b*c - a*d)^2) + (c^(3/2)*atan((sqrt(d)*x)/sqrt(c)))/(sqrt(d)*(b*c - a*d)^2)]
    @test_int [x^3/((a + b*x^2)^2*(c + d*x^2)), x, 3, a/(2*b*(b*c - a*d)*(a + b*x^2)) + (c*log(a + b*x^2))/(2*(b*c - a*d)^2) - (c*log(c + d*x^2))/(2*(b*c - a*d)^2)]
    @test_int [x^2/((a + b*x^2)^2*(c + d*x^2)), x, 4, -(x/(2*(b*c - a*d)*(a + b*x^2))) + ((b*c + a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*sqrt(a)*sqrt(b)*(b*c - a*d)^2) - (sqrt(c)*sqrt(d)*atan((sqrt(d)*x)/sqrt(c)))/(b*c - a*d)^2]
    @test_int [x^1/((a + b*x^2)^2*(c + d*x^2)), x, 3, -1/(2*(b*c - a*d)*(a + b*x^2)) - (d*log(a + b*x^2))/(2*(b*c - a*d)^2) + (d*log(c + d*x^2))/(2*(b*c - a*d)^2)]
    @test_int [1/((a + b*x^2)^2*(c + d*x^2)), x, 4, (b*x)/(2*a*(b*c - a*d)*(a + b*x^2)) + (sqrt(b)*(b*c - 3*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(3/2)*(b*c - a*d)^2) + (d^(3/2)*atan((sqrt(d)*x)/sqrt(c)))/(sqrt(c)*(b*c - a*d)^2)]
    @test_int [1/(x^1*(a + b*x^2)^2*(c + d*x^2)), x, 3, b/(2*a*(b*c - a*d)*(a + b*x^2)) + log(x)/(a^2*c) - (b*(b*c - 2*a*d)*log(a + b*x^2))/(2*a^2*(b*c - a*d)^2) - (d^2*log(c + d*x^2))/(2*c*(b*c - a*d)^2)]
    @test_int [1/(x^2*(a + b*x^2)^2*(c + d*x^2)), x, 5, -((3*b*c - 2*a*d)/(2*a^2*c*(b*c - a*d)*x)) + b/(2*a*(b*c - a*d)*x*(a + b*x^2)) - (b^(3/2)*(3*b*c - 5*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(5/2)*(b*c - a*d)^2) - (d^(5/2)*atan((sqrt(d)*x)/sqrt(c)))/(c^(3/2)*(b*c - a*d)^2)]
    @test_int [1/(x^3*(a + b*x^2)^2*(c + d*x^2)), x, 3, -1/(2*a^2*c*x^2) - b^2/(2*a^2*(b*c - a*d)*(a + b*x^2)) - ((2*b*c + a*d)*log(x))/(a^3*c^2) + (b^2*(2*b*c - 3*a*d)*log(a + b*x^2))/(2*a^3*(b*c - a*d)^2) + (d^3*log(c + d*x^2))/(2*c^2*(b*c - a*d)^2)]
    @test_int [1/(x^4*(a + b*x^2)^2*(c + d*x^2)), x, 6, -((5*b*c - 2*a*d)/(6*a^2*c*(b*c - a*d)*x^3)) + (5*b^2*c^2 - 2*a*b*c*d - 2*a^2*d^2)/(2*a^3*c^2*(b*c - a*d)*x) + b/(2*a*(b*c - a*d)*x^3*(a + b*x^2)) + (b^(5/2)*(5*b*c - 7*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(7/2)*(b*c - a*d)^2) + (d^(7/2)*atan((sqrt(d)*x)/sqrt(c)))/(c^(5/2)*(b*c - a*d)^2)]
    @test_int [1/(x^5*(a + b*x^2)^2*(c + d*x^2)), x, 3, -1/(4*a^2*c*x^4) + (2*b*c + a*d)/(2*a^3*c^2*x^2) + b^3/(2*a^3*(b*c - a*d)*(a + b*x^2)) + ((3*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*log(x))/(a^4*c^3) - (b^3*(3*b*c - 4*a*d)*log(a + b*x^2))/(2*a^4*(b*c - a*d)^2) - (d^4*log(c + d*x^2))/(2*c^3*(b*c - a*d)^2)]
    @test_int [1/(x^6*(a + b*x^2)^2*(c + d*x^2)), x, 7, -((7*b*c - 2*a*d)/(10*a^2*c*(b*c - a*d)*x^5)) + (7*b^2*c^2 - 2*a*b*c*d - 2*a^2*d^2)/(6*a^3*c^2*(b*c - a*d)*x^3) - (7*b^3*c^3 - 2*a*b^2*c^2*d - 2*a^2*b*c*d^2 - 2*a^3*d^3)/(2*a^4*c^3*(b*c - a*d)*x) + b/(2*a*(b*c - a*d)*x^5*(a + b*x^2)) - (b^(7/2)*(7*b*c - 9*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(9/2)*(b*c - a*d)^2) - (d^(9/2)*atan((sqrt(d)*x)/sqrt(c)))/(c^(7/2)*(b*c - a*d)^2)]
    @test_int [1/(x^7*(a + b*x^2)^2*(c + d*x^2)), x, 3, -1/(6*a^2*c*x^6) + (2*b*c + a*d)/(4*a^3*c^2*x^4) - (3*b^2*c^2 + 2*a*b*c*d + a^2*d^2)/(2*a^4*c^3*x^2) - b^4/(2*a^4*(b*c - a*d)*(a + b*x^2)) - ((4*b^3*c^3 + 3*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3)*log(x))/(a^5*c^4) + (b^4*(4*b*c - 5*a*d)*log(a + b*x^2))/(2*a^5*(b*c - a*d)^2) + (d^5*log(c + d*x^2))/(2*c^4*(b*c - a*d)^2)]

    @test_int [x^4/((a + b*x^2)^2*(c + d*x^2)^2), x, 5, ((b*c + a*d)*x)/(2*b*(b*c - a*d)^2*(c + d*x^2)) + (a*x)/(2*b*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)) - (sqrt(a)*(3*b*c + a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*sqrt(b)*(b*c - a*d)^3) + (sqrt(c)*(b*c + 3*a*d)*atan((sqrt(d)*x)/sqrt(c)))/(2*sqrt(d)*(b*c - a*d)^3)]
    @test_int [x^3/((a + b*x^2)^2*(c + d*x^2)^2), x, 3, a/(2*(b*c - a*d)^2*(a + b*x^2)) + c/(2*(b*c - a*d)^2*(c + d*x^2)) + ((b*c + a*d)*log(a + b*x^2))/(2*(b*c - a*d)^3) - ((b*c + a*d)*log(c + d*x^2))/(2*(b*c - a*d)^3)]
    @test_int [x^2/((a + b*x^2)^2*(c + d*x^2)^2), x, 5, -((d*x)/((b*c - a*d)^2*(c + d*x^2))) - x/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)) + (sqrt(b)*(b*c + 3*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*sqrt(a)*(b*c - a*d)^3) - (sqrt(d)*(3*b*c + a*d)*atan((sqrt(d)*x)/sqrt(c)))/(2*sqrt(c)*(b*c - a*d)^3)]
    @test_int [x^1/((a + b*x^2)^2*(c + d*x^2)^2), x, 3, -b/(2*(b*c - a*d)^2*(a + b*x^2)) - d/(2*(b*c - a*d)^2*(c + d*x^2)) - (b*d*log(a + b*x^2))/(b*c - a*d)^3 + (b*d*log(c + d*x^2))/(b*c - a*d)^3]
    @test_int [1/((a + b*x^2)^2*(c + d*x^2)^2), x, 5, (d*(b*c + a*d)*x)/(2*a*c*(b*c - a*d)^2*(c + d*x^2)) + (b*x)/(2*a*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)) + (b^(3/2)*(b*c - 5*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(3/2)*(b*c - a*d)^3) + (d^(3/2)*(5*b*c - a*d)*atan((sqrt(d)*x)/sqrt(c)))/(2*c^(3/2)*(b*c - a*d)^3)]
    @test_int [1/(x^1*(a + b*x^2)^2*(c + d*x^2)^2), x, 3, b^2/(2*a*(b*c - a*d)^2*(a + b*x^2)) + d^2/(2*c*(b*c - a*d)^2*(c + d*x^2)) + log(x)/(a^2*c^2) - (b^2*(b*c - 3*a*d)*log(a + b*x^2))/(2*a^2*(b*c - a*d)^3) - (d^2*(3*b*c - a*d)*log(c + d*x^2))/(2*c^2*(b*c - a*d)^3)]
    @test_int [1/(x^2*(a + b*x^2)^2*(c + d*x^2)^2), x, 6, -((3*b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)/(2*a^2*c^2*(b*c - a*d)^2*x)) + (d*(b*c + a*d))/(2*a*c*(b*c - a*d)^2*x*(c + d*x^2)) + b/(2*a*(b*c - a*d)*x*(a + b*x^2)*(c + d*x^2)) - (b^(5/2)*(3*b*c - 7*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(5/2)*(b*c - a*d)^3) - (d^(5/2)*(7*b*c - 3*a*d)*atan((sqrt(d)*x)/sqrt(c)))/(2*c^(5/2)*(b*c - a*d)^3)]
    @test_int [1/(x^3*(a + b*x^2)^2*(c + d*x^2)^2), x, 3, -1/(2*a^2*c^2*x^2) - b^3/(2*a^2*(b*c - a*d)^2*(a + b*x^2)) - d^3/(2*c^2*(b*c - a*d)^2*(c + d*x^2)) - (2*(b*c + a*d)*log(x))/(a^3*c^3) + (b^3*(b*c - 2*a*d)*log(a + b*x^2))/(a^3*(b*c - a*d)^3) + (d^3*(2*b*c - a*d)*log(c + d*x^2))/(c^3*(b*c - a*d)^3)]
    @test_int [1/(x^4*(a + b*x^2)^2*(c + d*x^2)^2), x, 7, -((5*b^2*c^2 - 4*a*b*c*d + 5*a^2*d^2)/(6*a^2*c^2*(b*c - a*d)^2*x^3)) + ((b*c + a*d)*(5*b^2*c^2 - 9*a*b*c*d + 5*a^2*d^2))/(2*a^3*c^3*(b*c - a*d)^2*x) + (d*(b*c + a*d))/(2*a*c*(b*c - a*d)^2*x^3*(c + d*x^2)) + b/(2*a*(b*c - a*d)*x^3*(a + b*x^2)*(c + d*x^2)) + (b^(7/2)*(5*b*c - 9*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(7/2)*(b*c - a*d)^3) + (d^(7/2)*(9*b*c - 5*a*d)*atan((sqrt(d)*x)/sqrt(c)))/(2*c^(7/2)*(b*c - a*d)^3)]

    @test_int [x^4/((a + b*x^2)^2*(c + d*x^2)^3), x, 6, ((b*c + 2*a*d)*x)/(4*b*(b*c - a*d)^2*(c + d*x^2)^2) + (a*x)/(2*b*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) + (3*(b*c + 3*a*d)*x)/(8*(b*c - a*d)^3*(c + d*x^2)) - (3*sqrt(a)*sqrt(b)*(b*c + a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*(b*c - a*d)^4) + (3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*atan((sqrt(d)*x)/sqrt(c)))/(8*sqrt(c)*sqrt(d)*(b*c - a*d)^4)]
    @test_int [x^3/((a + b*x^2)^2*(c + d*x^2)^3), x, 3, (a*b)/(2*(b*c - a*d)^3*(a + b*x^2)) + c/(4*(b*c - a*d)^2*(c + d*x^2)^2) + (b*c + a*d)/(2*(b*c - a*d)^3*(c + d*x^2)) + (b*(b*c + 2*a*d)*log(a + b*x^2))/(2*(b*c - a*d)^4) - (b*(b*c + 2*a*d)*log(c + d*x^2))/(2*(b*c - a*d)^4)]
    @test_int [x^2/((a + b*x^2)^2*(c + d*x^2)^3), x, 6, -((3*d*x)/(4*(b*c - a*d)^2*(c + d*x^2)^2)) - x/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) - (d*(11*b*c + a*d)*x)/(8*c*(b*c - a*d)^3*(c + d*x^2)) + (b^(3/2)*(b*c + 5*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*sqrt(a)*(b*c - a*d)^4) - (sqrt(d)*(15*b^2*c^2 + 10*a*b*c*d - a^2*d^2)*atan((sqrt(d)*x)/sqrt(c)))/(8*c^(3/2)*(b*c - a*d)^4)]
    @test_int [x^1/((a + b*x^2)^2*(c + d*x^2)^3), x, 3, -b^2/(2*(b*c - a*d)^3*(a + b*x^2)) - d/(4*(b*c - a*d)^2*(c + d*x^2)^2) - (b*d)/((b*c - a*d)^3*(c + d*x^2)) - (3*b^2*d*log(a + b*x^2))/(2*(b*c - a*d)^4) + (3*b^2*d*log(c + d*x^2))/(2*(b*c - a*d)^4)]
    @test_int [1/((a + b*x^2)^2*(c + d*x^2)^3), x, 6, (d*(2*b*c + a*d)*x)/(4*a*c*(b*c - a*d)^2*(c + d*x^2)^2) + (b*x)/(2*a*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) + (d*(4*b*c - a*d)*(b*c + 3*a*d)*x)/(8*a*c^2*(b*c - a*d)^3*(c + d*x^2)) + (b^(5/2)*(b*c - 7*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(3/2)*(b*c - a*d)^4) + (d^(3/2)*(35*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*atan((sqrt(d)*x)/sqrt(c)))/(8*c^(5/2)*(b*c - a*d)^4)]
    @test_int [1/(x^1*(a + b*x^2)^2*(c + d*x^2)^3), x, 3, b^3/(2*a*(b*c - a*d)^3*(a + b*x^2)) + d^2/(4*c*(b*c - a*d)^2*(c + d*x^2)^2) + (d^2*(3*b*c - a*d))/(2*c^2*(b*c - a*d)^3*(c + d*x^2)) + log(x)/(a^2*c^3) - (b^3*(b*c - 4*a*d)*log(a + b*x^2))/(2*a^2*(b*c - a*d)^4) - (d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*log(c + d*x^2))/(2*c^3*(b*c - a*d)^4)]
    @test_int [1/(x^2*(a + b*x^2)^2*(c + d*x^2)^3), x, 7, -((3*(2*b*c - a*d)*(2*b^2*c^2 - 3*a*b*c*d + 5*a^2*d^2))/(8*a^2*c^3*(b*c - a*d)^3*x)) + (d*(2*b*c + a*d))/(4*a*c*(b*c - a*d)^2*x*(c + d*x^2)^2) + b/(2*a*(b*c - a*d)*x*(a + b*x^2)*(c + d*x^2)^2) + (d*(4*b^2*c^2 + 13*a*b*c*d - 5*a^2*d^2))/(8*a*c^2*(b*c - a*d)^3*x*(c + d*x^2)) - (3*b^(7/2)*(b*c - 3*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(5/2)*(b*c - a*d)^4) - (3*d^(5/2)*(21*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*atan((sqrt(d)*x)/sqrt(c)))/(8*c^(7/2)*(b*c - a*d)^4)]
    @test_int [1/(x^3*(a + b*x^2)^2*(c + d*x^2)^3), x, 3, -1/(2*a^2*c^3*x^2) - b^4/(2*a^2*(b*c - a*d)^3*(a + b*x^2)) - d^3/(4*c^2*(b*c - a*d)^2*(c + d*x^2)^2) - (d^3*(2*b*c - a*d))/(c^3*(b*c - a*d)^3*(c + d*x^2)) - ((2*b*c + 3*a*d)*log(x))/(a^3*c^4) + (b^4*(2*b*c - 5*a*d)*log(a + b*x^2))/(2*a^3*(b*c - a*d)^4) + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*log(c + d*x^2))/(2*c^4*(b*c - a*d)^4)]
    @test_int [1/(x^4*(a + b*x^2)^2*(c + d*x^2)^3), x, 8, -((20*b^3*c^3 - 24*a*b^2*c^2*d + 75*a^2*b*c*d^2 - 35*a^3*d^3)/(24*a^2*c^3*(b*c - a*d)^3*x^3)) + (20*b^4*c^4 - 24*a*b^3*c^3*d - 24*a^2*b^2*c^2*d^2 + 75*a^3*b*c*d^3 - 35*a^4*d^4)/(8*a^3*c^4*(b*c - a*d)^3*x) + (d*(2*b*c + a*d))/(4*a*c*(b*c - a*d)^2*x^3*(c + d*x^2)^2) + b/(2*a*(b*c - a*d)*x^3*(a + b*x^2)*(c + d*x^2)^2) + (d*(4*b^2*c^2 + 15*a*b*c*d - 7*a^2*d^2))/(8*a*c^2*(b*c - a*d)^3*x^3*(c + d*x^2)) + (b^(9/2)*(5*b*c - 11*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(7/2)*(b*c - a*d)^4) + (d^(7/2)*(99*b^2*c^2 - 110*a*b*c*d + 35*a^2*d^2)*atan((sqrt(d)*x)/sqrt(c)))/(8*c^(9/2)*(b*c - a*d)^4)]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^2)^p / (c+d*x^2)^3=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^2)^p*(c+d*x^2)^q*with*m*symbolic=#

    #= ::Subsubsection::Closed:: =#
    #=q>0=#

    @test_int [x^m*(a + b*x^2)^3*(A + B*x^2), x, 2, (a^3*A*x^(1 + m))/(1 + m) + (a^2*(3*A*b + a*B)*x^(3 + m))/(3 + m) + (3*a*b*(A*b + a*B)*x^(5 + m))/(5 + m) + (b^2*(A*b + 3*a*B)*x^(7 + m))/(7 + m) + (b^3*B*x^(9 + m))/(9 + m)]
    @test_int [x^m*(a + b*x^2)^2*(A + B*x^2), x, 2, (a^2*A*x^(1 + m))/(1 + m) + (a*(2*A*b + a*B)*x^(3 + m))/(3 + m) + (b*(A*b + 2*a*B)*x^(5 + m))/(5 + m) + (b^2*B*x^(7 + m))/(7 + m)]
    @test_int [x^m*(a + b*x^2)^1*(A + B*x^2), x, 2, (a*A*x^(1 + m))/(1 + m) + ((A*b + a*B)*x^(3 + m))/(3 + m) + (b*B*x^(5 + m))/(5 + m)]
    @test_int [x^m*(A + B*x^2)/(a + b*x^2)^1, x, 2, (B*x^(1 + m))/(b*(1 + m)) + ((A*b - a*B)*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(a*b*(1 + m))]
    @test_int [x^m*(A + B*x^2)/(a + b*x^2)^2, x, 2, ((A*b - a*B)*x^(1 + m))/(2*a*b*(a + b*x^2)) + ((a*B*(1 + m) + A*(b - b*m))*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(2*a^2*b*(1 + m))]
    @test_int [x^m*(A + B*x^2)/(a + b*x^2)^3, x, 2, ((A*b - a*B)*x^(1 + m))/(4*a*b*(a + b*x^2)^2) + ((A*b*(3 - m) + a*B*(1 + m))*x^(1 + m)*HypergeometricFunctions._₂F₁(2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(4*a^3*b*(1 + m))]

    @test_int [x^m*(a + b*x^2)^2*(c + d*x^2)^3, x, 2, (a^2*c^3*x^(1 + m))/(1 + m) + (a*c^2*(2*b*c + 3*a*d)*x^(3 + m))/(3 + m) + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(5 + m))/(5 + m) + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(7 + m))/(7 + m) + (b*d^2*(3*b*c + 2*a*d)*x^(9 + m))/(9 + m) + (b^2*d^3*x^(11 + m))/(11 + m)]
    @test_int [x^m*(a + b*x^2)^2*(c + d*x^2)^2, x, 2, (a^2*c^2*x^(1 + m))/(1 + m) + (2*a*c*(b*c + a*d)*x^(3 + m))/(3 + m) + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(5 + m))/(5 + m) + (2*b*d*(b*c + a*d)*x^(7 + m))/(7 + m) + (b^2*d^2*x^(9 + m))/(9 + m)]
    @test_int [x^m*(a + b*x^2)^2*(c + d*x^2)^1, x, 2, (a^2*c*x^(1 + m))/(1 + m) + (a*(2*b*c + a*d)*x^(3 + m))/(3 + m) + (b*(b*c + 2*a*d)*x^(5 + m))/(5 + m) + (b^2*d*x^(7 + m))/(7 + m)]
    @test_int [x^m*(a + b*x^2)^2/(c + d*x^2)^1, x, 3, -((b*(b*c - 2*a*d)*x^(1 + m))/(d^2*(1 + m))) + (b^2*x^(3 + m))/(d*(3 + m)) + ((b*c - a*d)^2*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)))/(c*d^2*(1 + m))]
    @test_int [x^m*(a + b*x^2)^2/(c + d*x^2)^2, x, 3, (b^2*x^(1 + m))/(d^2*(1 + m)) + ((b*c - a*d)^2*x^(1 + m))/(2*c*d^2*(c + d*x^2)) - ((b*c - a*d)*(a*d*(1 - m) + b*c*(3 + m))*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)))/(2*c^2*d^2*(1 + m))]
    @test_int [x^m*(a + b*x^2)^2/(c + d*x^2)^3, x, 3, ((b*c - a*d)^2*x^(1 + m))/(4*c*d^2*(c + d*x^2)^2) - ((b*c - a*d)*(a*d*(3 - m) + b*c*(5 + m))*x^(1 + m))/(8*c^2*d^2*(c + d*x^2)) + ((2*a*b*c*d*(1 - m^2) + a^2*d^2*(3 - 4*m + m^2) + b^2*c^2*(3 + 4*m + m^2))*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)))/(8*c^3*d^2*(1 + m))]

    #= ::Subsubsection::Closed:: =#
    #=q<0=#

    @test_int [x^m*(c + d*x^2)^3/(a + b*x^2), x, 3, (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^(1 + m))/(b^3*(1 + m)) + (d^2*(3*b*c - a*d)*x^(3 + m))/(b^2*(3 + m)) + (d^3*x^(5 + m))/(b*(5 + m)) + ((b*c - a*d)^3*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(a*b^3*(1 + m))]
    @test_int [x^m*(c + d*x^2)^2/(a + b*x^2), x, 3, (d*(2*b*c - a*d)*x^(1 + m))/(b^2*(1 + m)) + (d^2*x^(3 + m))/(b*(3 + m)) + ((b*c - a*d)^2*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(a*b^2*(1 + m))]
    @test_int [x^m*(c + d*x^2)^1/(a + b*x^2), x, 2, (d*x^(1 + m))/(b*(1 + m)) + ((b*c - a*d)*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(a*b*(1 + m))]
    @test_int [x^m/((a + b*x^2)^1*(c + d*x^2)), x, 3, (b*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(a*(b*c - a*d)*(1 + m)) - (d*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)))/(c*(b*c - a*d)*(1 + m))]
    @test_int [x^m/((a + b*x^2)^2*(c + d*x^2)), x, 5, (b*x^(1 + m))/(2*a*(b*c - a*d)*(a + b*x^2)) + (b*(b*c*(1 - m) - a*d*(3 - m))*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(2*a^2*(b*c - a*d)^2*(1 + m)) + (d^2*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)))/(c*(b*c - a*d)^2*(1 + m))]
    @test_int [x^m/((a + b*x^2)^3*(c + d*x^2)), x, 6, (b*x^(1 + m))/(4*a*(b*c - a*d)*(a + b*x^2)^2) + (b*(b*c*(3 - m) - a*d*(7 - m))*x^(1 + m))/(8*a^2*(b*c - a*d)^2*(a + b*x^2)) + (b*(a^2*d^2*(15 - 8*m + m^2) - 2*a*b*c*d*(5 - 6*m + m^2) + b^2*c^2*(3 - 4*m + m^2))*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(8*a^3*(b*c - a*d)^3*(1 + m)) - (d^3*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)))/(c*(b*c - a*d)^3*(1 + m))]

    @test_int [x^m*(c + d*x^2)^3/(a + b*x^2)^2, x, 4, -((d*(2*b^2*c^2*(1 + m) - 3*a*b*c*d*(3 + m) + a^2*d^2*(5 + m))*x^(1 + m))/(2*a*b^3*(1 + m))) - (d^2*(b*c*(3 + m) - a*d*(5 + m))*x^(3 + m))/(2*a*b^2*(3 + m)) + ((b*c - a*d)*x^(1 + m)*(c + d*x^2)^2)/(2*a*b*(a + b*x^2)) + ((b*c - a*d)^2*(a*d*(5 + m) + b*(c - c*m))*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(2*a^2*b^3*(1 + m))]
    @test_int [x^m*(c + d*x^2)^2/(a + b*x^2)^2, x, 3, (d^2*x^(1 + m))/(b^2*(1 + m)) + ((b*c - a*d)^2*x^(1 + m))/(2*a*b^2*(a + b*x^2)) + ((b*c - a*d)*(a*d*(3 + m) + b*(c - c*m))*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(2*a^2*b^2*(1 + m))]
    @test_int [x^m*(c + d*x^2)^1/(a + b*x^2)^2, x, 2, ((b*c - a*d)*x^(1 + m))/(2*a*b*(a + b*x^2)) + ((a*d*(1 + m) + b*(c - c*m))*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(2*a^2*b*(1 + m))]
    @test_int [x^m/((a + b*x^2)^2*(c + d*x^2)^1), x, 5, (b*x^(1 + m))/(2*a*(b*c - a*d)*(a + b*x^2)) + (b*(b*c*(1 - m) - a*d*(3 - m))*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(2*a^2*(b*c - a*d)^2*(1 + m)) + (d^2*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)))/(c*(b*c - a*d)^2*(1 + m))]
    @test_int [x^m/((a + b*x^2)^2*(c + d*x^2)^2), x, 6, (d*(b*c + a*d)*x^(1 + m))/(2*a*c*(b*c - a*d)^2*(c + d*x^2)) + (b*x^(1 + m))/(2*a*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)) - (b^2*(a*d*(5 - m) - b*(c - c*m))*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(2*a^2*(b*c - a*d)^3*(1 + m)) - (d^2*(a*d*(1 - m) - b*c*(5 - m))*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)))/(2*c^2*(b*c - a*d)^3*(1 + m))]
    @test_int [x^m/((a + b*x^2)^2*(c + d*x^2)^3), x, 7, (d*(2*b*c + a*d)*x^(1 + m))/(4*a*c*(b*c - a*d)^2*(c + d*x^2)^2) + (b*x^(1 + m))/(2*a*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) + (d*(4*b^2*c^2 - a^2*d^2*(3 - m) + a*b*c*d*(11 - m))*x^(1 + m))/(8*a*c^2*(b*c - a*d)^3*(c + d*x^2)) - (b^3*(a*d*(7 - m) - b*(c - c*m))*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(2*a^2*(b*c - a*d)^4*(1 + m)) + (d^2*(b^2*c^2*(35 - 12*m + m^2) - 2*a*b*c*d*(7 - 8*m + m^2) + a^2*d^2*(3 - 4*m + m^2))*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)))/(8*c^3*(b*c - a*d)^4*(1 + m))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(e*x)^(m/2)*(a+b*x^2)^p*(c+d*x^2)^q=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^2)^p*(c+d*x^2)^1=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^(7/2)*(a + b*x^2)*(A + B*x^2), x, 2, (2*a*A*x^(9/2))/9 + (2*(A*b + a*B)*x^(13/2))/13 + (2*b*B*x^(17/2))/17]
    @test_int [x^(5/2)*(a + b*x^2)*(A + B*x^2), x, 2, (2*a*A*x^(7/2))/7 + (2*(A*b + a*B)*x^(11/2))/11 + (2*b*B*x^(15/2))/15]
    @test_int [x^(3/2)*(a + b*x^2)*(A + B*x^2), x, 2, (2*a*A*x^(5/2))/5 + (2*(A*b + a*B)*x^(9/2))/9 + (2*b*B*x^(13/2))/13]
    @test_int [sqrt(x)*(a + b*x^2)*(A + B*x^2), x, 2, (2*a*A*x^(3/2))/3 + (2*(A*b + a*B)*x^(7/2))/7 + (2*b*B*x^(11/2))/11]
    @test_int [((a + b*x^2)*(A + B*x^2))/sqrt(x), x, 2, 2*a*A*sqrt(x) + (2*(A*b + a*B)*x^(5/2))/5 + (2*b*B*x^(9/2))/9]
    @test_int [((a + b*x^2)*(A + B*x^2))/x^(3/2), x, 2, (-2*a*A)/sqrt(x) + (2*(A*b + a*B)*x^(3/2))/3 + (2*b*B*x^(7/2))/7]
    @test_int [((a + b*x^2)*(A + B*x^2))/x^(5/2), x, 2, (-2*a*A)/(3*x^(3/2)) + 2*(A*b + a*B)*sqrt(x) + (2*b*B*x^(5/2))/5]
    @test_int [((a + b*x^2)*(A + B*x^2))/x^(7/2), x, 2, (-2*a*A)/(5*x^(5/2)) - (2*(A*b + a*B))/sqrt(x) + (2*b*B*x^(3/2))/3]

    @test_int [x^(7/2)*(a + b*x^2)^2*(A + B*x^2), x, 2, (2*a^2*A*x^(9/2))/9 + (2*a*(2*A*b + a*B)*x^(13/2))/13 + (2*b*(A*b + 2*a*B)*x^(17/2))/17 + (2*b^2*B*x^(21/2))/21]
    @test_int [x^(5/2)*(a + b*x^2)^2*(A + B*x^2), x, 2, (2*a^2*A*x^(7/2))/7 + (2*a*(2*A*b + a*B)*x^(11/2))/11 + (2*b*(A*b + 2*a*B)*x^(15/2))/15 + (2*b^2*B*x^(19/2))/19]
    @test_int [x^(3/2)*(a + b*x^2)^2*(A + B*x^2), x, 2, (2*a^2*A*x^(5/2))/5 + (2*a*(2*A*b + a*B)*x^(9/2))/9 + (2*b*(A*b + 2*a*B)*x^(13/2))/13 + (2*b^2*B*x^(17/2))/17]
    @test_int [sqrt(x)*(a + b*x^2)^2*(A + B*x^2), x, 2, (2*a^2*A*x^(3/2))/3 + (2*a*(2*A*b + a*B)*x^(7/2))/7 + (2*b*(A*b + 2*a*B)*x^(11/2))/11 + (2*b^2*B*x^(15/2))/15]
    @test_int [((a + b*x^2)^2*(A + B*x^2))/sqrt(x), x, 2, 2*a^2*A*sqrt(x) + (2*a*(2*A*b + a*B)*x^(5/2))/5 + (2*b*(A*b + 2*a*B)*x^(9/2))/9 + (2*b^2*B*x^(13/2))/13]
    @test_int [((a + b*x^2)^2*(A + B*x^2))/x^(3/2), x, 2, (-2*a^2*A)/sqrt(x) + (2*a*(2*A*b + a*B)*x^(3/2))/3 + (2*b*(A*b + 2*a*B)*x^(7/2))/7 + (2*b^2*B*x^(11/2))/11]
    @test_int [((a + b*x^2)^2*(A + B*x^2))/x^(5/2), x, 2, (-2*a^2*A)/(3*x^(3/2)) + 2*a*(2*A*b + a*B)*sqrt(x) + (2*b*(A*b + 2*a*B)*x^(5/2))/5 + (2*b^2*B*x^(9/2))/9]
    @test_int [((a + b*x^2)^2*(A + B*x^2))/x^(7/2), x, 2, (-2*a^2*A)/(5*x^(5/2)) - (2*a*(2*A*b + a*B))/sqrt(x) + (2*b*(A*b + 2*a*B)*x^(3/2))/3 + (2*b^2*B*x^(7/2))/7]

    @test_int [x^(7/2)*(a + b*x^2)^3*(A + B*x^2), x, 2, (2*a^3*A*x^(9/2))/9 + (2*a^2*(3*A*b + a*B)*x^(13/2))/13 + (6*a*b*(A*b + a*B)*x^(17/2))/17 + (2*b^2*(A*b + 3*a*B)*x^(21/2))/21 + (2*b^3*B*x^(25/2))/25]
    @test_int [x^(5/2)*(a + b*x^2)^3*(A + B*x^2), x, 2, (2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(11/2))/11 + (2*a*b*(A*b + a*B)*x^(15/2))/5 + (2*b^2*(A*b + 3*a*B)*x^(19/2))/19 + (2*b^3*B*x^(23/2))/23]
    @test_int [x^(3/2)*(a + b*x^2)^3*(A + B*x^2), x, 2, (2*a^3*A*x^(5/2))/5 + (2*a^2*(3*A*b + a*B)*x^(9/2))/9 + (6*a*b*(A*b + a*B)*x^(13/2))/13 + (2*b^2*(A*b + 3*a*B)*x^(17/2))/17 + (2*b^3*B*x^(21/2))/21]
    @test_int [sqrt(x)*(a + b*x^2)^3*(A + B*x^2), x, 2, (2*a^3*A*x^(3/2))/3 + (2*a^2*(3*A*b + a*B)*x^(7/2))/7 + (6*a*b*(A*b + a*B)*x^(11/2))/11 + (2*b^2*(A*b + 3*a*B)*x^(15/2))/15 + (2*b^3*B*x^(19/2))/19]
    @test_int [((a + b*x^2)^3*(A + B*x^2))/sqrt(x), x, 2, 2*a^3*A*sqrt(x) + (2*a^2*(3*A*b + a*B)*x^(5/2))/5 + (2*a*b*(A*b + a*B)*x^(9/2))/3 + (2*b^2*(A*b + 3*a*B)*x^(13/2))/13 + (2*b^3*B*x^(17/2))/17]
    @test_int [((a + b*x^2)^3*(A + B*x^2))/x^(3/2), x, 2, (-2*a^3*A)/sqrt(x) + (2*a^2*(3*A*b + a*B)*x^(3/2))/3 + (6*a*b*(A*b + a*B)*x^(7/2))/7 + (2*b^2*(A*b + 3*a*B)*x^(11/2))/11 + (2*b^3*B*x^(15/2))/15]
    @test_int [((a + b*x^2)^3*(A + B*x^2))/x^(5/2), x, 2, (-2*a^3*A)/(3*x^(3/2)) + 2*a^2*(3*A*b + a*B)*sqrt(x) + (6*a*b*(A*b + a*B)*x^(5/2))/5 + (2*b^2*(A*b + 3*a*B)*x^(9/2))/9 + (2*b^3*B*x^(13/2))/13]
    @test_int [((a + b*x^2)^3*(A + B*x^2))/x^(7/2), x, 2, (-2*a^3*A)/(5*x^(5/2)) - (2*a^2*(3*A*b + a*B))/sqrt(x) + 2*a*b*(A*b + a*B)*x^(3/2) + (2*b^2*(A*b + 3*a*B)*x^(7/2))/7 + (2*b^3*B*x^(11/2))/11]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(x^(7/2)*(A + B*x^2))/(a + b*x^2), x, 13, -((2*a*(A*b - a*B)*sqrt(x))/b^3) + (2*(A*b - a*B)*x^(5/2))/(5*b^2) + (2*B*x^(9/2))/(9*b) - (a^(5/4)*(A*b - a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(13/4)) + (a^(5/4)*(A*b - a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(13/4)) - (a^(5/4)*(A*b - a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(13/4)) + (a^(5/4)*(A*b - a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(13/4))]
    @test_int [(x^(5/2)*(A + B*x^2))/(a + b*x^2), x, 12, (2*(A*b - a*B)*x^(3/2))/(3*b^2) + (2*B*x^(7/2))/(7*b) + (a^(3/4)*(A*b - a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(11/4)) - (a^(3/4)*(A*b - a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(11/4)) - (a^(3/4)*(A*b - a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(11/4)) + (a^(3/4)*(A*b - a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(11/4))]
    @test_int [(x^(3/2)*(A + B*x^2))/(a + b*x^2), x, 12, (2*(A*b - a*B)*sqrt(x))/b^2 + (2*B*x^(5/2))/(5*b) + (a^(1/4)*(A*b - a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(9/4)) - (a^(1/4)*(A*b - a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(9/4)) + (a^(1/4)*(A*b - a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(9/4)) - (a^(1/4)*(A*b - a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(9/4))]
    @test_int [(sqrt(x)*(A + B*x^2))/(a + b*x^2), x, 11, (2*B*x^(3/2))/(3*b) - ((A*b - a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(1/4)*b^(7/4)) + ((A*b - a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(1/4)*b^(7/4)) + ((A*b - a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(1/4)*b^(7/4)) - ((A*b - a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(1/4)*b^(7/4))]
    @test_int [(A + B*x^2)/(sqrt(x)*(a + b*x^2)), x, 11, (2*B*sqrt(x))/b - ((A*b - a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(3/4)*b^(5/4)) + ((A*b - a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(3/4)*b^(5/4)) - ((A*b - a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(3/4)*b^(5/4)) + ((A*b - a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(3/4)*b^(5/4))]
    @test_int [(A + B*x^2)/(x^(3/2)*(a + b*x^2)), x, 11, -((2*A)/(a*sqrt(x))) + ((A*b - a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(5/4)*b^(3/4)) - ((A*b - a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(5/4)*b^(3/4)) - ((A*b - a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(5/4)*b^(3/4)) + ((A*b - a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(5/4)*b^(3/4))]
    @test_int [(A + B*x^2)/(x^(5/2)*(a + b*x^2)), x, 11, -((2*A)/(3*a*x^(3/2))) + ((A*b - a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(7/4)*b^(1/4)) - ((A*b - a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(7/4)*b^(1/4)) + ((A*b - a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(7/4)*b^(1/4)) - ((A*b - a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(7/4)*b^(1/4))]
    @test_int [(A + B*x^2)/(x^(7/2)*(a + b*x^2)), x, 12, -((2*A)/(5*a*x^(5/2))) + (2*(A*b - a*B))/(a^2*sqrt(x)) - (b^(1/4)*(A*b - a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(9/4)) + (b^(1/4)*(A*b - a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(9/4)) + (b^(1/4)*(A*b - a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(9/4)) - (b^(1/4)*(A*b - a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(9/4))]

    @test_int [(x^(7/2)*(A + B*x^2))/(a + b*x^2)^2, x, 13, ((5*A*b - 9*a*B)*sqrt(x))/(2*b^3) - ((5*A*b - 9*a*B)*x^(5/2))/(10*a*b^2) + ((A*b - a*B)*x^(9/2))/(2*a*b*(a + b*x^2)) + (a^(1/4)*(5*A*b - 9*a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*b^(13/4)) - (a^(1/4)*(5*A*b - 9*a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*b^(13/4)) + (a^(1/4)*(5*A*b - 9*a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*b^(13/4)) - (a^(1/4)*(5*A*b - 9*a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*b^(13/4))]
    @test_int [(x^(5/2)*(A + B*x^2))/(a + b*x^2)^2, x, 12, -(((3*A*b - 7*a*B)*x^(3/2))/(6*a*b^2)) + ((A*b - a*B)*x^(7/2))/(2*a*b*(a + b*x^2)) - ((3*A*b - 7*a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(1/4)*b^(11/4)) + ((3*A*b - 7*a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(1/4)*b^(11/4)) + ((3*A*b - 7*a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(1/4)*b^(11/4)) - ((3*A*b - 7*a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(1/4)*b^(11/4))]
    @test_int [(x^(3/2)*(A + B*x^2))/(a + b*x^2)^2, x, 12, -(((A*b - 5*a*B)*sqrt(x))/(2*a*b^2)) + ((A*b - a*B)*x^(5/2))/(2*a*b*(a + b*x^2)) - ((A*b - 5*a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(3/4)*b^(9/4)) + ((A*b - 5*a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(3/4)*b^(9/4)) - ((A*b - 5*a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(3/4)*b^(9/4)) + ((A*b - 5*a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(3/4)*b^(9/4))]
    @test_int [(sqrt(x)*(A + B*x^2))/(a + b*x^2)^2, x, 11, ((A*b - a*B)*x^(3/2))/(2*a*b*(a + b*x^2)) - ((A*b + 3*a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(5/4)*b^(7/4)) + ((A*b + 3*a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(5/4)*b^(7/4)) + ((A*b + 3*a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(5/4)*b^(7/4)) - ((A*b + 3*a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(5/4)*b^(7/4))]
    @test_int [(A + B*x^2)/(sqrt(x)*(a + b*x^2)^2), x, 11, ((A*b - a*B)*sqrt(x))/(2*a*b*(a + b*x^2)) - ((3*A*b + a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(7/4)*b^(5/4)) + ((3*A*b + a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(7/4)*b^(5/4)) - ((3*A*b + a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(7/4)*b^(5/4)) + ((3*A*b + a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(7/4)*b^(5/4))]
    @test_int [(A + B*x^2)/(x^(3/2)*(a + b*x^2)^2), x, 12, -((5*A*b - a*B)/(2*a^2*b*sqrt(x))) + (A*b - a*B)/(2*a*b*sqrt(x)*(a + b*x^2)) + ((5*A*b - a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(9/4)*b^(3/4)) - ((5*A*b - a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(9/4)*b^(3/4)) - ((5*A*b - a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(9/4)*b^(3/4)) + ((5*A*b - a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(9/4)*b^(3/4))]
    @test_int [(A + B*x^2)/(x^(5/2)*(a + b*x^2)^2), x, 12, -((7*A*b - 3*a*B)/(6*a^2*b*x^(3/2))) + (A*b - a*B)/(2*a*b*x^(3/2)*(a + b*x^2)) + ((7*A*b - 3*a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(11/4)*b^(1/4)) - ((7*A*b - 3*a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(11/4)*b^(1/4)) + ((7*A*b - 3*a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(11/4)*b^(1/4)) - ((7*A*b - 3*a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(11/4)*b^(1/4))]
    @test_int [(A + B*x^2)/(x^(7/2)*(a + b*x^2)^2), x, 13, -((9*A*b - 5*a*B)/(10*a^2*b*x^(5/2))) + (9*A*b - 5*a*B)/(2*a^3*sqrt(x)) + (A*b - a*B)/(2*a*b*x^(5/2)*(a + b*x^2)) - (b^(1/4)*(9*A*b - 5*a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(13/4)) + (b^(1/4)*(9*A*b - 5*a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(13/4)) + (b^(1/4)*(9*A*b - 5*a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(13/4)) - (b^(1/4)*(9*A*b - 5*a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(13/4))]

    @test_int [(x^(7/2)*(A + B*x^2))/(a + b*x^2)^3, x, 13, -((5*(A*b - 9*a*B)*sqrt(x))/(16*a*b^3)) + ((A*b - a*B)*x^(9/2))/(4*a*b*(a + b*x^2)^2) + ((A*b - 9*a*B)*x^(5/2))/(16*a*b^2*(a + b*x^2)) - (5*(A*b - 9*a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(3/4)*b^(13/4)) + (5*(A*b - 9*a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(3/4)*b^(13/4)) - (5*(A*b - 9*a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(3/4)*b^(13/4)) + (5*(A*b - 9*a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(3/4)*b^(13/4))]
    @test_int [(x^(5/2)*(A + B*x^2))/(a + b*x^2)^3, x, 12, ((A*b - a*B)*x^(7/2))/(4*a*b*(a + b*x^2)^2) - ((A*b + 7*a*B)*x^(3/2))/(16*a*b^2*(a + b*x^2)) - (3*(A*b + 7*a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(5/4)*b^(11/4)) + (3*(A*b + 7*a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(5/4)*b^(11/4)) + (3*(A*b + 7*a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(5/4)*b^(11/4)) - (3*(A*b + 7*a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(5/4)*b^(11/4))]
    @test_int [(x^(3/2)*(A + B*x^2))/(a + b*x^2)^3, x, 12, ((A*b - a*B)*x^(5/2))/(4*a*b*(a + b*x^2)^2) - ((3*A*b + 5*a*B)*sqrt(x))/(16*a*b^2*(a + b*x^2)) - ((3*A*b + 5*a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(7/4)*b^(9/4)) + ((3*A*b + 5*a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(7/4)*b^(9/4)) - ((3*A*b + 5*a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(7/4)*b^(9/4)) + ((3*A*b + 5*a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(7/4)*b^(9/4))]
    @test_int [(sqrt(x)*(A + B*x^2))/(a + b*x^2)^3, x, 12, ((A*b - a*B)*x^(3/2))/(4*a*b*(a + b*x^2)^2) + ((5*A*b + 3*a*B)*x^(3/2))/(16*a^2*b*(a + b*x^2)) - ((5*A*b + 3*a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(9/4)*b^(7/4)) + ((5*A*b + 3*a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(9/4)*b^(7/4)) + ((5*A*b + 3*a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(9/4)*b^(7/4)) - ((5*A*b + 3*a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(9/4)*b^(7/4))]
    @test_int [(A + B*x^2)/(sqrt(x)*(a + b*x^2)^3), x, 12, ((A*b - a*B)*sqrt(x))/(4*a*b*(a + b*x^2)^2) + ((7*A*b + a*B)*sqrt(x))/(16*a^2*b*(a + b*x^2)) - (3*(7*A*b + a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(11/4)*b^(5/4)) + (3*(7*A*b + a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(11/4)*b^(5/4)) - (3*(7*A*b + a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(11/4)*b^(5/4)) + (3*(7*A*b + a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(11/4)*b^(5/4))]
    @test_int [(A + B*x^2)/(x^(3/2)*(a + b*x^2)^3), x, 13, -((5*(9*A*b - a*B))/(16*a^3*b*sqrt(x))) + (A*b - a*B)/(4*a*b*sqrt(x)*(a + b*x^2)^2) + (9*A*b - a*B)/(16*a^2*b*sqrt(x)*(a + b*x^2)) + (5*(9*A*b - a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(13/4)*b^(3/4)) - (5*(9*A*b - a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(13/4)*b^(3/4)) - (5*(9*A*b - a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(13/4)*b^(3/4)) + (5*(9*A*b - a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(13/4)*b^(3/4))]
    @test_int [(A + B*x^2)/(x^(5/2)*(a + b*x^2)^3), x, 13, -((7*(11*A*b - 3*a*B))/(48*a^3*b*x^(3/2))) + (A*b - a*B)/(4*a*b*x^(3/2)*(a + b*x^2)^2) + (11*A*b - 3*a*B)/(16*a^2*b*x^(3/2)*(a + b*x^2)) + (7*(11*A*b - 3*a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(15/4)*b^(1/4)) - (7*(11*A*b - 3*a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(15/4)*b^(1/4)) + (7*(11*A*b - 3*a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(15/4)*b^(1/4)) - (7*(11*A*b - 3*a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(15/4)*b^(1/4))]
    @test_int [(A + B*x^2)/(x^(7/2)*(a + b*x^2)^3), x, 14, -((9*(13*A*b - 5*a*B))/(80*a^3*b*x^(5/2))) + (9*(13*A*b - 5*a*B))/(16*a^4*sqrt(x)) + (A*b - a*B)/(4*a*b*x^(5/2)*(a + b*x^2)^2) + (13*A*b - 5*a*B)/(16*a^2*b*x^(5/2)*(a + b*x^2)) - (9*b^(1/4)*(13*A*b - 5*a*B)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(17/4)) + (9*b^(1/4)*(13*A*b - 5*a*B)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(17/4)) + (9*b^(1/4)*(13*A*b - 5*a*B)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(17/4)) - (9*b^(1/4)*(13*A*b - 5*a*B)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(17/4))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^2)^p*(c+d*x^2)^2=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^(7/2)*(a + b*x^2)^2*(c + d*x^2), x, 2, (2*a^2*c*x^(9/2))/9 + (2*a*(2*b*c + a*d)*x^(13/2))/13 + (2*b*(b*c + 2*a*d)*x^(17/2))/17 + (2*b^2*d*x^(21/2))/21]
    @test_int [x^(5/2)*(a + b*x^2)^2*(c + d*x^2), x, 2, (2*a^2*c*x^(7/2))/7 + (2*a*(2*b*c + a*d)*x^(11/2))/11 + (2*b*(b*c + 2*a*d)*x^(15/2))/15 + (2*b^2*d*x^(19/2))/19]
    @test_int [x^(3/2)*(a + b*x^2)^2*(c + d*x^2), x, 2, (2*a^2*c*x^(5/2))/5 + (2*a*(2*b*c + a*d)*x^(9/2))/9 + (2*b*(b*c + 2*a*d)*x^(13/2))/13 + (2*b^2*d*x^(17/2))/17]
    @test_int [sqrt(x)*(a + b*x^2)^2*(c + d*x^2), x, 2, (2*a^2*c*x^(3/2))/3 + (2*a*(2*b*c + a*d)*x^(7/2))/7 + (2*b*(b*c + 2*a*d)*x^(11/2))/11 + (2*b^2*d*x^(15/2))/15]
    @test_int [((a + b*x^2)^2*(c + d*x^2))/sqrt(x), x, 2, 2*a^2*c*sqrt(x) + (2*a*(2*b*c + a*d)*x^(5/2))/5 + (2*b*(b*c + 2*a*d)*x^(9/2))/9 + (2*b^2*d*x^(13/2))/13]
    @test_int [((a + b*x^2)^2*(c + d*x^2))/x^(3/2), x, 2, (-2*a^2*c)/sqrt(x) + (2*a*(2*b*c + a*d)*x^(3/2))/3 + (2*b*(b*c + 2*a*d)*x^(7/2))/7 + (2*b^2*d*x^(11/2))/11]
    @test_int [((a + b*x^2)^2*(c + d*x^2))/x^(5/2), x, 2, (-2*a^2*c)/(3*x^(3/2)) + 2*a*(2*b*c + a*d)*sqrt(x) + (2*b*(b*c + 2*a*d)*x^(5/2))/5 + (2*b^2*d*x^(9/2))/9]
    @test_int [((a + b*x^2)^2*(c + d*x^2))/x^(7/2), x, 2, (-2*a^2*c)/(5*x^(5/2)) - (2*a*(2*b*c + a*d))/sqrt(x) + (2*b*(b*c + 2*a*d)*x^(3/2))/3 + (2*b^2*d*x^(7/2))/7]

    @test_int [x^(7/2)*(a + b*x^2)^2*(c + d*x^2)^2, x, 2, (2/9)*a^2*c^2*x^(9/2) + (4/13)*a*c*(b*c + a*d)*x^(13/2) + (2/17)*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(17/2) + (4/21)*b*d*(b*c + a*d)*x^(21/2) + (2/25)*b^2*d^2*x^(25/2)]
    @test_int [x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^2, x, 2, (2/7)*a^2*c^2*x^(7/2) + (4/11)*a*c*(b*c + a*d)*x^(11/2) + (2/15)*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(15/2) + (4/19)*b*d*(b*c + a*d)*x^(19/2) + (2/23)*b^2*d^2*x^(23/2)]
    @test_int [x^(3/2)*(a + b*x^2)^2*(c + d*x^2)^2, x, 2, (2/5)*a^2*c^2*x^(5/2) + (4/9)*a*c*(b*c + a*d)*x^(9/2) + (2/13)*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(13/2) + (4/17)*b*d*(b*c + a*d)*x^(17/2) + (2/21)*b^2*d^2*x^(21/2)]
    @test_int [sqrt(x)*(a + b*x^2)^2*(c + d*x^2)^2, x, 2, (2/3)*a^2*c^2*x^(3/2) + (4/7)*a*c*(b*c + a*d)*x^(7/2) + (2/11)*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(11/2) + (4/15)*b*d*(b*c + a*d)*x^(15/2) + (2/19)*b^2*d^2*x^(19/2)]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^2)/sqrt(x), x, 2, 2*a^2*c^2*sqrt(x) + (4/5)*a*c*(b*c + a*d)*x^(5/2) + (2/9)*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(9/2) + (4/13)*b*d*(b*c + a*d)*x^(13/2) + (2/17)*b^2*d^2*x^(17/2)]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^2)/x^(3/2), x, 2, -((2*a^2*c^2)/sqrt(x)) + (4/3)*a*c*(b*c + a*d)*x^(3/2) + (2/7)*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(7/2) + (4/11)*b*d*(b*c + a*d)*x^(11/2) + (2/15)*b^2*d^2*x^(15/2)]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^2)/x^(5/2), x, 2, -((2*a^2*c^2)/(3*x^(3/2))) + 4*a*c*(b*c + a*d)*sqrt(x) + (2/5)*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(5/2) + (4/9)*b*d*(b*c + a*d)*x^(9/2) + (2/13)*b^2*d^2*x^(13/2)]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^2)/x^(7/2), x, 2, -((2*a^2*c^2)/(5*x^(5/2))) - (4*a*c*(b*c + a*d))/sqrt(x) + (2/3)*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(3/2) + (4/7)*b*d*(b*c + a*d)*x^(7/2) + (2/11)*b^2*d^2*x^(11/2)]

    @test_int [x^(7/2)*(a + b*x^2)^2*(c + d*x^2)^3, x, 2, (2/9)*a^2*c^3*x^(9/2) + (2/13)*a*c^2*(2*b*c + 3*a*d)*x^(13/2) + (2/17)*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(17/2) + (2/21)*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(21/2) + (2/25)*b*d^2*(3*b*c + 2*a*d)*x^(25/2) + (2/29)*b^2*d^3*x^(29/2)]
    @test_int [x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^3, x, 2, (2/7)*a^2*c^3*x^(7/2) + (2/11)*a*c^2*(2*b*c + 3*a*d)*x^(11/2) + (2/15)*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(15/2) + (2/19)*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(19/2) + (2/23)*b*d^2*(3*b*c + 2*a*d)*x^(23/2) + (2/27)*b^2*d^3*x^(27/2)]
    @test_int [x^(3/2)*(a + b*x^2)^2*(c + d*x^2)^3, x, 2, (2/5)*a^2*c^3*x^(5/2) + (2/9)*a*c^2*(2*b*c + 3*a*d)*x^(9/2) + (2/13)*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(13/2) + (2/17)*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(17/2) + (2/21)*b*d^2*(3*b*c + 2*a*d)*x^(21/2) + (2/25)*b^2*d^3*x^(25/2)]
    @test_int [sqrt(x)*(a + b*x^2)^2*(c + d*x^2)^3, x, 2, (2/3)*a^2*c^3*x^(3/2) + (2/7)*a*c^2*(2*b*c + 3*a*d)*x^(7/2) + (2/11)*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(11/2) + (2/15)*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(15/2) + (2/19)*b*d^2*(3*b*c + 2*a*d)*x^(19/2) + (2/23)*b^2*d^3*x^(23/2)]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^3)/sqrt(x), x, 2, 2*a^2*c^3*sqrt(x) + (2/5)*a*c^2*(2*b*c + 3*a*d)*x^(5/2) + (2/9)*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(9/2) + (2/13)*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(13/2) + (2/17)*b*d^2*(3*b*c + 2*a*d)*x^(17/2) + (2/21)*b^2*d^3*x^(21/2)]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^3)/x^(3/2), x, 2, -((2*a^2*c^3)/sqrt(x)) + (2/3)*a*c^2*(2*b*c + 3*a*d)*x^(3/2) + (2/7)*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(7/2) + (2/11)*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(11/2) + (2/15)*b*d^2*(3*b*c + 2*a*d)*x^(15/2) + (2/19)*b^2*d^3*x^(19/2)]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^3)/x^(5/2), x, 2, -((2*a^2*c^3)/(3*x^(3/2))) + 2*a*c^2*(2*b*c + 3*a*d)*sqrt(x) + (2/5)*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(5/2) + (2/9)*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(9/2) + (2/13)*b*d^2*(3*b*c + 2*a*d)*x^(13/2) + (2/17)*b^2*d^3*x^(17/2)]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^3)/x^(7/2), x, 2, -((2*a^2*c^3)/(5*x^(5/2))) - (2*a*c^2*(2*b*c + 3*a*d))/sqrt(x) + (2/3)*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(3/2) + (2/7)*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(7/2) + (2/11)*b*d^2*(3*b*c + 2*a*d)*x^(11/2) + (2/15)*b^2*d^3*x^(15/2)]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(x^(7/2)*(a + b*x^2)^2)/(c + d*x^2), x, 14, -((2*c*(b*c - a*d)^2*sqrt(x))/d^4) + (2*(b*c - a*d)^2*x^(5/2))/(5*d^3) - (2*b*(b*c - 2*a*d)*x^(9/2))/(9*d^2) + (2*b^2*x^(13/2))/(13*d) - (c^(5/4)*(b*c - a*d)^2*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*d^(17/4)) + (c^(5/4)*(b*c - a*d)^2*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*d^(17/4)) - (c^(5/4)*(b*c - a*d)^2*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*d^(17/4)) + (c^(5/4)*(b*c - a*d)^2*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*d^(17/4))]
    @test_int [(x^(5/2)*(a + b*x^2)^2)/(c + d*x^2), x, 13, (2*(b*c - a*d)^2*x^(3/2))/(3*d^3) - (2*b*(b*c - 2*a*d)*x^(7/2))/(7*d^2) + (2*b^2*x^(11/2))/(11*d) + (c^(3/4)*(b*c - a*d)^2*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*d^(15/4)) - (c^(3/4)*(b*c - a*d)^2*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*d^(15/4)) - (c^(3/4)*(b*c - a*d)^2*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*d^(15/4)) + (c^(3/4)*(b*c - a*d)^2*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*d^(15/4))]
    @test_int [(x^(3/2)*(a + b*x^2)^2)/(c + d*x^2), x, 13, (2*(b*c - a*d)^2*sqrt(x))/d^3 - (2*b*(b*c - 2*a*d)*x^(5/2))/(5*d^2) + (2*b^2*x^(9/2))/(9*d) + (c^(1/4)*(b*c - a*d)^2*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*d^(13/4)) - (c^(1/4)*(b*c - a*d)^2*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*d^(13/4)) + (c^(1/4)*(b*c - a*d)^2*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*d^(13/4)) - (c^(1/4)*(b*c - a*d)^2*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*d^(13/4))]
    @test_int [(sqrt(x)*(a + b*x^2)^2)/(c + d*x^2), x, 12, -((2*b*(b*c - 2*a*d)*x^(3/2))/(3*d^2)) + (2*b^2*x^(7/2))/(7*d) - ((b*c - a*d)^2*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(1/4)*d^(11/4)) + ((b*c - a*d)^2*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(1/4)*d^(11/4)) + ((b*c - a*d)^2*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(1/4)*d^(11/4)) - ((b*c - a*d)^2*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(1/4)*d^(11/4))]
    @test_int [(a + b*x^2)^2/(sqrt(x)*(c + d*x^2)), x, 12, -((2*b*(b*c - 2*a*d)*sqrt(x))/d^2) + (2*b^2*x^(5/2))/(5*d) - ((b*c - a*d)^2*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(3/4)*d^(9/4)) + ((b*c - a*d)^2*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(3/4)*d^(9/4)) - ((b*c - a*d)^2*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(3/4)*d^(9/4)) + ((b*c - a*d)^2*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(3/4)*d^(9/4))]
    @test_int [(a + b*x^2)^2/(x^(3/2)*(c + d*x^2)), x, 12, -((2*a^2)/(c*sqrt(x))) + (2*b^2*x^(3/2))/(3*d) + ((b*c - a*d)^2*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(5/4)*d^(7/4)) - ((b*c - a*d)^2*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(5/4)*d^(7/4)) - ((b*c - a*d)^2*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(5/4)*d^(7/4)) + ((b*c - a*d)^2*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(5/4)*d^(7/4))]
    @test_int [(a + b*x^2)^2/(x^(5/2)*(c + d*x^2)), x, 12, -((2*a^2)/(3*c*x^(3/2))) + (2*b^2*sqrt(x))/d + ((b*c - a*d)^2*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(7/4)*d^(5/4)) - ((b*c - a*d)^2*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(7/4)*d^(5/4)) + ((b*c - a*d)^2*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(7/4)*d^(5/4)) - ((b*c - a*d)^2*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(7/4)*d^(5/4))]
    @test_int [(a + b*x^2)^2/(x^(7/2)*(c + d*x^2)), x, 12, -((2*a^2)/(5*c*x^(5/2))) - (2*a*(2*b*c - a*d))/(c^2*sqrt(x)) - ((b*c - a*d)^2*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(9/4)*d^(3/4)) + ((b*c - a*d)^2*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(9/4)*d^(3/4)) + ((b*c - a*d)^2*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(9/4)*d^(3/4)) - ((b*c - a*d)^2*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(9/4)*d^(3/4))]
    @test_int [(a + b*x^2)^2/(x^(9/2)*(c + d*x^2)), x, 12, -((2*a^2)/(7*c*x^(7/2))) - (2*a*(2*b*c - a*d))/(3*c^2*x^(3/2)) - ((b*c - a*d)^2*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(11/4)*d^(1/4)) + ((b*c - a*d)^2*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(11/4)*d^(1/4)) - ((b*c - a*d)^2*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(11/4)*d^(1/4)) + ((b*c - a*d)^2*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(11/4)*d^(1/4))]
    @test_int [(c + d*x^2)^2/(x^(11/2)*(a + b*x^2)), x, 13, -((2*c^2)/(9*a*x^(9/2))) + (2*c*(b*c - 2*a*d))/(5*a^2*x^(5/2)) - (2*(b*c - a*d)^2)/(a^3*sqrt(x)) + (b^(1/4)*(b*c - a*d)^2*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(13/4)) - (b^(1/4)*(b*c - a*d)^2*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(13/4)) - (b^(1/4)*(b*c - a*d)^2*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(13/4)) + (b^(1/4)*(b*c - a*d)^2*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(13/4))]

    @test_int [(x^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^2, x, 14, ((13*b*c - 5*a*d)*(b*c - a*d)*sqrt(x))/(2*d^4) - ((13*b*c - 5*a*d)*(b*c - a*d)*x^(5/2))/(10*c*d^3) + (2*b^2*x^(9/2))/(9*d^2) + ((b*c - a*d)^2*x^(9/2))/(2*c*d^2*(c + d*x^2)) + (c^(1/4)*(13*b*c - 5*a*d)*(b*c - a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*d^(17/4)) - (c^(1/4)*(13*b*c - 5*a*d)*(b*c - a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*d^(17/4)) + (c^(1/4)*(13*b*c - 5*a*d)*(b*c - a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*d^(17/4)) - (c^(1/4)*(13*b*c - 5*a*d)*(b*c - a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*d^(17/4))]
    @test_int [(x^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^2, x, 13, -(((11*b*c - 3*a*d)*(b*c - a*d)*x^(3/2))/(6*c*d^3)) + (2*b^2*x^(7/2))/(7*d^2) + ((b*c - a*d)^2*x^(7/2))/(2*c*d^2*(c + d*x^2)) - ((11*b*c - 3*a*d)*(b*c - a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(1/4)*d^(15/4)) + ((11*b*c - 3*a*d)*(b*c - a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(1/4)*d^(15/4)) + ((11*b*c - 3*a*d)*(b*c - a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(1/4)*d^(15/4)) - ((11*b*c - 3*a*d)*(b*c - a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(1/4)*d^(15/4))]
    @test_int [(x^(3/2)*(a + b*x^2)^2)/(c + d*x^2)^2, x, 13, -(((b*c - a*d)*(9*b*c - a*d)*sqrt(x))/(2*c*d^3)) + (2*b^2*x^(5/2))/(5*d^2) + ((b*c - a*d)^2*x^(5/2))/(2*c*d^2*(c + d*x^2)) - ((b*c - a*d)*(9*b*c - a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(3/4)*d^(13/4)) + ((b*c - a*d)*(9*b*c - a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(3/4)*d^(13/4)) - ((b*c - a*d)*(9*b*c - a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(3/4)*d^(13/4)) + ((b*c - a*d)*(9*b*c - a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(3/4)*d^(13/4))]
    @test_int [(sqrt(x)*(a + b*x^2)^2)/(c + d*x^2)^2, x, 12, (2*b^2*x^(3/2))/(3*d^2) + ((b*c - a*d)^2*x^(3/2))/(2*c*d^2*(c + d*x^2)) + ((b*c - a*d)*(7*b*c + a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(5/4)*d^(11/4)) - ((b*c - a*d)*(7*b*c + a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(5/4)*d^(11/4)) - ((b*c - a*d)*(7*b*c + a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(5/4)*d^(11/4)) + ((b*c - a*d)*(7*b*c + a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(5/4)*d^(11/4))]
    @test_int [(a + b*x^2)^2/(sqrt(x)*(c + d*x^2)^2), x, 12, (2*b^2*sqrt(x))/d^2 + ((b*c - a*d)^2*sqrt(x))/(2*c*d^2*(c + d*x^2)) + ((b*c - a*d)*(5*b*c + 3*a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(7/4)*d^(9/4)) - ((b*c - a*d)*(5*b*c + 3*a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(7/4)*d^(9/4)) + ((b*c - a*d)*(5*b*c + 3*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(7/4)*d^(9/4)) - ((b*c - a*d)*(5*b*c + 3*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(7/4)*d^(9/4))]
    @test_int [(a + b*x^2)^2/(x^(3/2)*(c + d*x^2)^2), x, 12, -((2*a^2)/(c*sqrt(x)*(c + d*x^2))) - ((b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*x^(3/2))/(2*c^2*d*(c + d*x^2)) - ((b*c - a*d)*(3*b*c + 5*a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(9/4)*d^(7/4)) + ((b*c - a*d)*(3*b*c + 5*a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(9/4)*d^(7/4)) + ((b*c - a*d)*(3*b*c + 5*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(9/4)*d^(7/4)) - ((b*c - a*d)*(3*b*c + 5*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(9/4)*d^(7/4))]
    @test_int [(a + b*x^2)^2/(x^(5/2)*(c + d*x^2)^2), x, 12, -((2*a^2)/(3*c*x^(3/2)*(c + d*x^2))) - ((3*b^2*c^2 - 6*a*b*c*d + 7*a^2*d^2)*sqrt(x))/(6*c^2*d*(c + d*x^2)) - ((b*c - a*d)*(b*c + 7*a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(11/4)*d^(5/4)) + ((b*c - a*d)*(b*c + 7*a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(11/4)*d^(5/4)) - ((b*c - a*d)*(b*c + 7*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(11/4)*d^(5/4)) + ((b*c - a*d)*(b*c + 7*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(11/4)*d^(5/4))]
    @test_int [(a + b*x^2)^2/(x^(7/2)*(c + d*x^2)^2), x, 13, ((b*c - 9*a*d)*(b*c - a*d))/(2*c^3*d*sqrt(x)) - (2*a^2)/(5*c*x^(5/2)*(c + d*x^2)) - (5*b^2*c^2 - 10*a*b*c*d + 9*a^2*d^2)/(10*c^2*d*sqrt(x)*(c + d*x^2)) - ((b*c - 9*a*d)*(b*c - a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(13/4)*d^(3/4)) + ((b*c - 9*a*d)*(b*c - a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(13/4)*d^(3/4)) + ((b*c - 9*a*d)*(b*c - a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(13/4)*d^(3/4)) - ((b*c - 9*a*d)*(b*c - a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(13/4)*d^(3/4))]

    @test_int [(x^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^3, x, 14, -(((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*sqrt(x))/(16*c*d^4)) + ((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*x^(5/2))/(80*c^2*d^3) + ((b*c - a*d)^2*x^(9/2))/(4*c*d^2*(c + d*x^2)^2) - ((b*c - a*d)*(17*b*c - a*d)*x^(9/2))/(16*c^2*d^2*(c + d*x^2)) - ((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(3/4)*d^(17/4)) + ((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(3/4)*d^(17/4)) - ((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(3/4)*d^(17/4)) + ((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(3/4)*d^(17/4))]
    @test_int [(x^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^3, x, 13, -(((42*a*b - (77*b^2*c)/d + (3*a^2*d)/c)*x^(3/2))/(48*c*d^2)) + ((b*c - a*d)^2*x^(7/2))/(4*c*d^2*(c + d*x^2)^2) - ((b*c - a*d)*(15*b*c + a*d)*x^(7/2))/(16*c^2*d^2*(c + d*x^2)) + ((77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(5/4)*d^(15/4)) - ((77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(5/4)*d^(15/4)) - ((77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(5/4)*d^(15/4)) + ((77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(5/4)*d^(15/4))]
    @test_int [(x^(3/2)*(a + b*x^2)^2)/(c + d*x^2)^3, x, 13, -(((10*a*b - (45*b^2*c)/d + (3*a^2*d)/c)*sqrt(x))/(16*c*d^2)) + ((b*c - a*d)^2*x^(5/2))/(4*c*d^2*(c + d*x^2)^2) - ((b*c - a*d)*(13*b*c + 3*a*d)*x^(5/2))/(16*c^2*d^2*(c + d*x^2)) + ((45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(7/4)*d^(13/4)) - ((45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(7/4)*d^(13/4)) + ((45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(7/4)*d^(13/4)) - ((45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(7/4)*d^(13/4))]
    @test_int [(sqrt(x)*(a + b*x^2)^2)/(c + d*x^2)^3, x, 12, ((b*c - a*d)^2*x^(3/2))/(4*c*d^2*(c + d*x^2)^2) - ((b*c - a*d)*(11*b*c + 5*a*d)*x^(3/2))/(16*c^2*d^2*(c + d*x^2)) - ((21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(9/4)*d^(11/4)) + ((21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(9/4)*d^(11/4)) + ((21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(9/4)*d^(11/4)) - ((21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(9/4)*d^(11/4))]
    @test_int [(a + b*x^2)^2/(sqrt(x)*(c + d*x^2)^3), x, 12, ((b*c - a*d)^2*sqrt(x))/(4*c*d^2*(c + d*x^2)^2) - ((b*c - a*d)*(9*b*c + 7*a*d)*sqrt(x))/(16*c^2*d^2*(c + d*x^2)) - ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(11/4)*d^(9/4)) + ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(11/4)*d^(9/4)) - ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(11/4)*d^(9/4)) + ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(11/4)*d^(9/4))]
    @test_int [(a + b*x^2)^2/(x^(3/2)*(c + d*x^2)^3), x, 13, -((2*a^2)/(c*sqrt(x)*(c + d*x^2)^2)) - ((b^2*c^2 - 2*a*b*c*d + 9*a^2*d^2)*x^(3/2))/(4*c^2*d*(c + d*x^2)^2) + ((3*b^2*c^2 + 5*a*d*(2*b*c - 9*a*d))*x^(3/2))/(16*c^3*d*(c + d*x^2)) - ((3*b^2*c^2 + 5*a*d*(2*b*c - 9*a*d))*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(13/4)*d^(7/4)) + ((3*b^2*c^2 + 5*a*d*(2*b*c - 9*a*d))*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(13/4)*d^(7/4)) + ((3*b^2*c^2 + 5*a*d*(2*b*c - 9*a*d))*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(13/4)*d^(7/4)) - ((3*b^2*c^2 + 5*a*d*(2*b*c - 9*a*d))*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(13/4)*d^(7/4))]
    @test_int [(a + b*x^2)^2/(x^(5/2)*(c + d*x^2)^3), x, 13, -((2*a^2)/(3*c*x^(3/2)*(c + d*x^2)^2)) - ((3*b^2*c^2 - 6*a*b*c*d + 11*a^2*d^2)*sqrt(x))/(12*c^2*d*(c + d*x^2)^2) + ((3*b^2*c^2 + 7*a*d*(6*b*c - 11*a*d))*sqrt(x))/(48*c^3*d*(c + d*x^2)) - ((3*b^2*c^2 + 7*a*d*(6*b*c - 11*a*d))*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(15/4)*d^(5/4)) + ((3*b^2*c^2 + 7*a*d*(6*b*c - 11*a*d))*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(15/4)*d^(5/4)) - ((3*b^2*c^2 + 7*a*d*(6*b*c - 11*a*d))*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(15/4)*d^(5/4)) + ((3*b^2*c^2 + 7*a*d*(6*b*c - 11*a*d))*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(15/4)*d^(5/4))]
    @test_int [(a + b*x^2)^2/(x^(7/2)*(c + d*x^2)^3), x, 14, (5*b^2*c^2 - 9*a*d*(10*b*c - 13*a*d))/(16*c^4*d*sqrt(x)) - (2*a^2)/(5*c*x^(5/2)*(c + d*x^2)^2) - (5*b^2*c^2 - 10*a*b*c*d + 13*a^2*d^2)/(20*c^2*d*sqrt(x)*(c + d*x^2)^2) - (5*b^2*c^2 - 9*a*d*(10*b*c - 13*a*d))/(80*c^3*d*sqrt(x)*(c + d*x^2)) - ((5*b^2*c^2 - 9*a*d*(10*b*c - 13*a*d))*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(17/4)*d^(3/4)) + ((5*b^2*c^2 - 9*a*d*(10*b*c - 13*a*d))*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(17/4)*d^(3/4)) + ((5*b^2*c^2 - 9*a*d*(10*b*c - 13*a*d))*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(17/4)*d^(3/4)) - ((5*b^2*c^2 - 9*a*d*(10*b*c - 13*a*d))*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(17/4)*d^(3/4))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^2)^p*(c+d*x^2)^3=#

    #= ::Subsubsection:: =#
    #=p>0=#

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(x^(5/2)*(c + d*x^2)^3)/(a + b*x^2), x, 13, (2*(b*c - a*d)^3*x^(3/2))/(3*b^4) + (2*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^(7/2))/(7*b^3) + (2*d^2*(3*b*c - a*d)*x^(11/2))/(11*b^2) + (2*d^3*x^(15/2))/(15*b) + (a^(3/4)*(b*c - a*d)^3*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(19/4)) - (a^(3/4)*(b*c - a*d)^3*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(19/4)) - (a^(3/4)*(b*c - a*d)^3*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(19/4)) + (a^(3/4)*(b*c - a*d)^3*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(19/4))]
    @test_int [(x^(3/2)*(c + d*x^2)^3)/(a + b*x^2), x, 13, (2*(b*c - a*d)^3*sqrt(x))/b^4 + (2*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^(5/2))/(5*b^3) + (2*d^2*(3*b*c - a*d)*x^(9/2))/(9*b^2) + (2*d^3*x^(13/2))/(13*b) + (a^(1/4)*(b*c - a*d)^3*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(17/4)) - (a^(1/4)*(b*c - a*d)^3*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(17/4)) + (a^(1/4)*(b*c - a*d)^3*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(17/4)) - (a^(1/4)*(b*c - a*d)^3*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(17/4))]
    @test_int [(sqrt(x)*(c + d*x^2)^3)/(a + b*x^2), x, 12, (2*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^(3/2))/(3*b^3) + (2*d^2*(3*b*c - a*d)*x^(7/2))/(7*b^2) + (2*d^3*x^(11/2))/(11*b) - ((b*c - a*d)^3*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(1/4)*b^(15/4)) + ((b*c - a*d)^3*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(1/4)*b^(15/4)) + ((b*c - a*d)^3*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(1/4)*b^(15/4)) - ((b*c - a*d)^3*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(1/4)*b^(15/4))]
    @test_int [(c + d*x^2)^3/(sqrt(x)*(a + b*x^2)), x, 12, (2*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*sqrt(x))/b^3 + (2*d^2*(3*b*c - a*d)*x^(5/2))/(5*b^2) + (2*d^3*x^(9/2))/(9*b) - ((b*c - a*d)^3*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(3/4)*b^(13/4)) + ((b*c - a*d)^3*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(3/4)*b^(13/4)) - ((b*c - a*d)^3*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(3/4)*b^(13/4)) + ((b*c - a*d)^3*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(3/4)*b^(13/4))]
    @test_int [(c + d*x^2)^3/(x^(3/2)*(a + b*x^2)), x, 12, -((2*c^3)/(a*sqrt(x))) + (2*d^2*(3*b*c - a*d)*x^(3/2))/(3*b^2) + (2*d^3*x^(7/2))/(7*b) + ((b*c - a*d)^3*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(5/4)*b^(11/4)) - ((b*c - a*d)^3*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(5/4)*b^(11/4)) - ((b*c - a*d)^3*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(5/4)*b^(11/4)) + ((b*c - a*d)^3*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(5/4)*b^(11/4))]
    @test_int [(c + d*x^2)^3/(x^(5/2)*(a + b*x^2)), x, 12, -((2*c^3)/(3*a*x^(3/2))) + (2*d^2*(3*b*c - a*d)*sqrt(x))/b^2 + (2*d^3*x^(5/2))/(5*b) + ((b*c - a*d)^3*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(7/4)*b^(9/4)) - ((b*c - a*d)^3*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(7/4)*b^(9/4)) + ((b*c - a*d)^3*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(7/4)*b^(9/4)) - ((b*c - a*d)^3*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(7/4)*b^(9/4))]
    @test_int [(c + d*x^2)^3/(x^(7/2)*(a + b*x^2)), x, 12, -((2*c^3)/(5*a*x^(5/2))) + (2*c^2*(b*c - 3*a*d))/(a^2*sqrt(x)) + (2*d^3*x^(3/2))/(3*b) - ((b*c - a*d)^3*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(9/4)*b^(7/4)) + ((b*c - a*d)^3*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(9/4)*b^(7/4)) + ((b*c - a*d)^3*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(9/4)*b^(7/4)) - ((b*c - a*d)^3*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(9/4)*b^(7/4))]
    @test_int [(c + d*x^2)^3/(x^(9/2)*(a + b*x^2)), x, 12, -((2*c^3)/(7*a*x^(7/2))) + (2*c^2*(b*c - 3*a*d))/(3*a^2*x^(3/2)) + (2*d^3*sqrt(x))/b - ((b*c - a*d)^3*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(11/4)*b^(5/4)) + ((b*c - a*d)^3*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(11/4)*b^(5/4)) - ((b*c - a*d)^3*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(11/4)*b^(5/4)) + ((b*c - a*d)^3*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(11/4)*b^(5/4))]
    @test_int [(c + d*x^2)^3/(x^(11/2)*(a + b*x^2)), x, 12, -((2*c^3)/(9*a*x^(9/2))) + (2*c^2*(b*c - 3*a*d))/(5*a^2*x^(5/2)) - (2*c*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2))/(a^3*sqrt(x)) + ((b*c - a*d)^3*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(13/4)*b^(3/4)) - ((b*c - a*d)^3*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(13/4)*b^(3/4)) - ((b*c - a*d)^3*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(13/4)*b^(3/4)) + ((b*c - a*d)^3*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(13/4)*b^(3/4))]
    @test_int [(c + d*x^2)^3/(x^(13/2)*(a + b*x^2)), x, 12, -((2*c^3)/(11*a*x^(11/2))) + (2*c^2*(b*c - 3*a*d))/(7*a^2*x^(7/2)) - (2*c*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2))/(3*a^3*x^(3/2)) + ((b*c - a*d)^3*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(15/4)*b^(1/4)) - ((b*c - a*d)^3*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(15/4)*b^(1/4)) + ((b*c - a*d)^3*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(15/4)*b^(1/4)) - ((b*c - a*d)^3*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(15/4)*b^(1/4))]
    @test_int [(c + d*x^2)^3/(x^(15/2)*(a + b*x^2)), x, 12, -((2*c^3)/(13*a*x^(13/2))) + (2*c^2*(b*c - 3*a*d))/(9*a^2*x^(9/2)) - (2*c*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2))/(5*a^3*x^(5/2)) + (2*(b*c - a*d)^3)/(a^4*sqrt(x)) - (b^(1/4)*(b*c - a*d)^3*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(17/4)) + (b^(1/4)*(b*c - a*d)^3*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(17/4)) + (b^(1/4)*(b*c - a*d)^3*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(17/4)) - (b^(1/4)*(b*c - a*d)^3*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(17/4))]

    @test_int [(x^(7/2)*(c + d*x^2)^3)/(a + b*x^2)^2, x, 13, ((5*b*c - 17*a*d)*(b*c - a*d)^2*sqrt(x))/(2*b^5) + (d*(27*b^2*c^2 - 39*a*b*c*d + 17*a^2*d^2)*x^(5/2))/(10*b^4) + (d^2*(39*b*c - 17*a*d)*x^(9/2))/(18*b^3) + (17*d^3*x^(13/2))/(26*b^2) - (x^(5/2)*(c + d*x^2)^3)/(2*b*(a + b*x^2)) + (a^(1/4)*(5*b*c - 17*a*d)*(b*c - a*d)^2*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*b^(21/4)) - (a^(1/4)*(5*b*c - 17*a*d)*(b*c - a*d)^2*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*b^(21/4)) + (a^(1/4)*(5*b*c - 17*a*d)*(b*c - a*d)^2*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*b^(21/4)) - (a^(1/4)*(5*b*c - 17*a*d)*(b*c - a*d)^2*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*b^(21/4))]
    @test_int [(x^(5/2)*(c + d*x^2)^3)/(a + b*x^2)^2, x, 13, (d*(7*b^2*c^2 - 11*a*b*c*d + 5*a^2*d^2)*x^(3/2))/(2*b^4) + (3*d^2*(11*b*c - 5*a*d)*x^(7/2))/(14*b^3) + (15*d^3*x^(11/2))/(22*b^2) - (x^(3/2)*(c + d*x^2)^3)/(2*b*(a + b*x^2)) - (3*(b*c - 5*a*d)*(b*c - a*d)^2*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(1/4)*b^(19/4)) + (3*(b*c - 5*a*d)*(b*c - a*d)^2*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(1/4)*b^(19/4)) + (3*(b*c - 5*a*d)*(b*c - a*d)^2*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(1/4)*b^(19/4)) - (3*(b*c - 5*a*d)*(b*c - a*d)^2*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(1/4)*b^(19/4))]
    @test_int [(x^(3/2)*(c + d*x^2)^3)/(a + b*x^2)^2, x, 14, (d*(497*b^2*c^2 - 1098*a*b*c*d + 585*a^2*d^2)*sqrt(x))/(90*b^4) + (d*(113*b*c - 117*a*d)*sqrt(x)*(c + d*x^2))/(90*b^3) + (13*d*sqrt(x)*(c + d*x^2)^2)/(18*b^2) - (sqrt(x)*(c + d*x^2)^3)/(2*b*(a + b*x^2)) - ((b*c - 13*a*d)*(b*c - a*d)^2*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(3/4)*b^(17/4)) + ((b*c - 13*a*d)*(b*c - a*d)^2*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(3/4)*b^(17/4)) - ((b*c - 13*a*d)*(b*c - a*d)^2*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(3/4)*b^(17/4)) + ((b*c - 13*a*d)*(b*c - a*d)^2*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(3/4)*b^(17/4))]
    @test_int [(sqrt(x)*(c + d*x^2)^3)/(a + b*x^2)^2, x, 13, -((d*(6*b^2*c^2 - 21*a*b*c*d + 11*a^2*d^2)*x^(3/2))/(6*a*b^3)) - (d^2*(7*b*c - 11*a*d)*x^(7/2))/(14*a*b^2) + ((b*c - a*d)*x^(3/2)*(c + d*x^2)^2)/(2*a*b*(a + b*x^2)) - ((b*c - a*d)^2*(b*c + 11*a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(5/4)*b^(15/4)) + ((b*c - a*d)^2*(b*c + 11*a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(5/4)*b^(15/4)) + ((b*c - a*d)^2*(b*c + 11*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(5/4)*b^(15/4)) - ((b*c - a*d)^2*(b*c + 11*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(5/4)*b^(15/4))]
    @test_int [(c + d*x^2)^3/(sqrt(x)*(a + b*x^2)^2), x, 13, (2*d^2*(3*b*c - 2*a*d)*sqrt(x))/b^3 + (2*d^3*x^(5/2))/(5*b^2) + ((b*c - a*d)^3*sqrt(x))/(2*a*b^3*(a + b*x^2)) - (3*(b*c - a*d)^2*(b*c + 3*a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(7/4)*b^(13/4)) + (3*(b*c - a*d)^2*(b*c + 3*a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(7/4)*b^(13/4)) - (3*(b*c - a*d)^2*(b*c + 3*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(7/4)*b^(13/4)) + (3*(b*c - a*d)^2*(b*c + 3*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(7/4)*b^(13/4))]
    @test_int [(c + d*x^2)^3/(x^(3/2)*(a + b*x^2)^2), x, 13, -((c^2*(5*b*c - a*d))/(2*a^2*b*sqrt(x))) - (d^2*(3*b*c - 7*a*d)*x^(3/2))/(6*a*b^2) + ((b*c - a*d)*(c + d*x^2)^2)/(2*a*b*sqrt(x)*(a + b*x^2)) + ((b*c - a*d)^2*(5*b*c + 7*a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(9/4)*b^(11/4)) - ((b*c - a*d)^2*(5*b*c + 7*a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(9/4)*b^(11/4)) - ((b*c - a*d)^2*(5*b*c + 7*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(9/4)*b^(11/4)) + ((b*c - a*d)^2*(5*b*c + 7*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(9/4)*b^(11/4))]
    @test_int [(c + d*x^2)^3/(x^(5/2)*(a + b*x^2)^2), x, 13, -((c^2*(7*b*c - 3*a*d))/(6*a^2*b*x^(3/2))) - (d^2*(b*c - 5*a*d)*sqrt(x))/(2*a*b^2) + ((b*c - a*d)*(c + d*x^2)^2)/(2*a*b*x^(3/2)*(a + b*x^2)) + ((b*c - a*d)^2*(7*b*c + 5*a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(11/4)*b^(9/4)) - ((b*c - a*d)^2*(7*b*c + 5*a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(11/4)*b^(9/4)) + ((b*c - a*d)^2*(7*b*c + 5*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(11/4)*b^(9/4)) - ((b*c - a*d)^2*(7*b*c + 5*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(11/4)*b^(9/4))]
    @test_int [(c + d*x^2)^3/(x^(7/2)*(a + b*x^2)^2), x, 13, -((c^2*(9*b*c - 5*a*d))/(10*a^2*b*x^(5/2))) + (c*(9*b^2*c^2 - 15*a*b*c*d + 2*a^2*d^2))/(2*a^3*b*sqrt(x)) + ((b*c - a*d)*(c + d*x^2)^2)/(2*a*b*x^(5/2)*(a + b*x^2)) - (3*(b*c - a*d)^2*(3*b*c + a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(13/4)*b^(7/4)) + (3*(b*c - a*d)^2*(3*b*c + a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(13/4)*b^(7/4)) + (3*(b*c - a*d)^2*(3*b*c + a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(13/4)*b^(7/4)) - (3*(b*c - a*d)^2*(3*b*c + a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(13/4)*b^(7/4))]
    @test_int [(c + d*x^2)^3/(x^(9/2)*(a + b*x^2)^2), x, 13, -((c^2*(11*b*c - 7*a*d))/(14*a^2*b*x^(7/2))) + (c*(11*b^2*c^2 - 21*a*b*c*d + 6*a^2*d^2))/(6*a^3*b*x^(3/2)) + ((b*c - a*d)*(c + d*x^2)^2)/(2*a*b*x^(7/2)*(a + b*x^2)) - ((b*c - a*d)^2*(11*b*c + a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(15/4)*b^(5/4)) + ((b*c - a*d)^2*(11*b*c + a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(15/4)*b^(5/4)) - ((b*c - a*d)^2*(11*b*c + a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(15/4)*b^(5/4)) + ((b*c - a*d)^2*(11*b*c + a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(15/4)*b^(5/4))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^2)^p / (c+d*x^2)^1=#

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^(9/2)/((a + b*x^2)*(c + d*x^2)), x, 22, (2*x^(3/2))/(3*b*d) - (a^(7/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(7/4)*(b*c - a*d)) + (a^(7/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(7/4)*(b*c - a*d)) + (c^(7/4)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*d^(7/4)*(b*c - a*d)) - (c^(7/4)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*d^(7/4)*(b*c - a*d)) + (a^(7/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(7/4)*(b*c - a*d)) - (a^(7/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(7/4)*(b*c - a*d)) - (c^(7/4)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*d^(7/4)*(b*c - a*d)) + (c^(7/4)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*d^(7/4)*(b*c - a*d))]
    @test_int [x^(7/2)/((a + b*x^2)*(c + d*x^2)), x, 21, (2*sqrt(x))/(b*d) - (a^(5/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(5/4)*(b*c - a*d)) + (a^(5/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(5/4)*(b*c - a*d)) + (c^(5/4)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*d^(5/4)*(b*c - a*d)) - (c^(5/4)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*d^(5/4)*(b*c - a*d)) - (a^(5/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(5/4)*(b*c - a*d)) + (a^(5/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(5/4)*(b*c - a*d)) + (c^(5/4)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*d^(5/4)*(b*c - a*d)) - (c^(5/4)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*d^(5/4)*(b*c - a*d))]
    @test_int [x^(5/2)/((a + b*x^2)*(c + d*x^2)), x, 20, (a^(3/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(3/4)*(b*c - a*d)) - (a^(3/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(3/4)*(b*c - a*d)) - (c^(3/4)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*d^(3/4)*(b*c - a*d)) + (c^(3/4)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*d^(3/4)*(b*c - a*d)) - (a^(3/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(3/4)*(b*c - a*d)) + (a^(3/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(3/4)*(b*c - a*d)) + (c^(3/4)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*d^(3/4)*(b*c - a*d)) - (c^(3/4)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*d^(3/4)*(b*c - a*d))]
    @test_int [x^(3/2)/((a + b*x^2)*(c + d*x^2)), x, 20, (a^(1/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(1/4)*(b*c - a*d)) - (a^(1/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(1/4)*(b*c - a*d)) - (c^(1/4)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*d^(1/4)*(b*c - a*d)) + (c^(1/4)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*d^(1/4)*(b*c - a*d)) + (a^(1/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(1/4)*(b*c - a*d)) - (a^(1/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(1/4)*(b*c - a*d)) - (c^(1/4)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*d^(1/4)*(b*c - a*d)) + (c^(1/4)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*d^(1/4)*(b*c - a*d))]
    @test_int [sqrt(x)/((a + b*x^2)*(c + d*x^2)), x, 20, -((b^(1/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(1/4)*(b*c - a*d))) + (b^(1/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(1/4)*(b*c - a*d)) + (d^(1/4)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(1/4)*(b*c - a*d)) - (d^(1/4)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(1/4)*(b*c - a*d)) + (b^(1/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(1/4)*(b*c - a*d)) - (b^(1/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(1/4)*(b*c - a*d)) - (d^(1/4)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(1/4)*(b*c - a*d)) + (d^(1/4)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(1/4)*(b*c - a*d))]
    @test_int [1/(sqrt(x)*(a + b*x^2)*(c + d*x^2)), x, 20, -((b^(3/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(3/4)*(b*c - a*d))) + (b^(3/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(3/4)*(b*c - a*d)) + (d^(3/4)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(3/4)*(b*c - a*d)) - (d^(3/4)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(3/4)*(b*c - a*d)) - (b^(3/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(3/4)*(b*c - a*d)) + (b^(3/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(3/4)*(b*c - a*d)) + (d^(3/4)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(3/4)*(b*c - a*d)) - (d^(3/4)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(3/4)*(b*c - a*d))]
    @test_int [1/(x^(3/2)*(a + b*x^2)*(c + d*x^2)), x, 22, -(2/(a*c*sqrt(x))) + (b^(5/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(5/4)*(b*c - a*d)) - (b^(5/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(5/4)*(b*c - a*d)) - (d^(5/4)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(5/4)*(b*c - a*d)) + (d^(5/4)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(5/4)*(b*c - a*d)) - (b^(5/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(5/4)*(b*c - a*d)) + (b^(5/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(5/4)*(b*c - a*d)) + (d^(5/4)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(5/4)*(b*c - a*d)) - (d^(5/4)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(5/4)*(b*c - a*d))]
    @test_int [1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)), x, 21, -(2/(3*a*c*x^(3/2))) + (b^(7/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(7/4)*(b*c - a*d)) - (b^(7/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(7/4)*(b*c - a*d)) - (d^(7/4)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(7/4)*(b*c - a*d)) + (d^(7/4)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(7/4)*(b*c - a*d)) + (b^(7/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(7/4)*(b*c - a*d)) - (b^(7/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(7/4)*(b*c - a*d)) - (d^(7/4)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(7/4)*(b*c - a*d)) + (d^(7/4)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(7/4)*(b*c - a*d))]
    @test_int [1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)), x, 23, -(2/(5*a*c*x^(5/2))) + (2*(b*c + a*d))/(a^2*c^2*sqrt(x)) - (b^(9/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(9/4)*(b*c - a*d)) + (b^(9/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(9/4)*(b*c - a*d)) + (d^(9/4)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(9/4)*(b*c - a*d)) - (d^(9/4)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(sqrt(2)*c^(9/4)*(b*c - a*d)) + (b^(9/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(9/4)*(b*c - a*d)) - (b^(9/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(9/4)*(b*c - a*d)) - (d^(9/4)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(9/4)*(b*c - a*d)) + (d^(9/4)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(2*sqrt(2)*c^(9/4)*(b*c - a*d))]

    @test_int [x^(11/2)/((a + b*x^2)*(c + d*x^2)^2), x, 22, ((5*b*c - 4*a*d)*sqrt(x))/(2*b*d^2*(b*c - a*d)) - (c*x^(5/2))/(2*d*(b*c - a*d)*(c + d*x^2)) + (a^(9/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(5/4)*(b*c - a*d)^2) - (a^(9/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(5/4)*(b*c - a*d)^2) + (c^(5/4)*(5*b*c - 9*a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*d^(9/4)*(b*c - a*d)^2) - (c^(5/4)*(5*b*c - 9*a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*d^(9/4)*(b*c - a*d)^2) + (a^(9/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(5/4)*(b*c - a*d)^2) - (a^(9/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(5/4)*(b*c - a*d)^2) + (c^(5/4)*(5*b*c - 9*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*d^(9/4)*(b*c - a*d)^2) - (c^(5/4)*(5*b*c - 9*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*d^(9/4)*(b*c - a*d)^2)]
    @test_int [x^(9/2)/((a + b*x^2)*(c + d*x^2)^2), x, 22, -((c*x^(3/2))/(2*d*(b*c - a*d)*(c + d*x^2))) - (a^(7/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(3/4)*(b*c - a*d)^2) + (a^(7/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(3/4)*(b*c - a*d)^2) - (c^(3/4)*(3*b*c - 7*a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*d^(7/4)*(b*c - a*d)^2) + (c^(3/4)*(3*b*c - 7*a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*d^(7/4)*(b*c - a*d)^2) + (a^(7/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(3/4)*(b*c - a*d)^2) - (a^(7/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(3/4)*(b*c - a*d)^2) + (c^(3/4)*(3*b*c - 7*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*d^(7/4)*(b*c - a*d)^2) - (c^(3/4)*(3*b*c - 7*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*d^(7/4)*(b*c - a*d)^2)]
    @test_int [x^(7/2)/((a + b*x^2)*(c + d*x^2)^2), x, 21, -((c*sqrt(x))/(2*d*(b*c - a*d)*(c + d*x^2))) - (a^(5/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(1/4)*(b*c - a*d)^2) + (a^(5/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(1/4)*(b*c - a*d)^2) - (c^(1/4)*(b*c - 5*a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*d^(5/4)*(b*c - a*d)^2) + (c^(1/4)*(b*c - 5*a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*d^(5/4)*(b*c - a*d)^2) - (a^(5/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(1/4)*(b*c - a*d)^2) + (a^(5/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(1/4)*(b*c - a*d)^2) - (c^(1/4)*(b*c - 5*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*d^(5/4)*(b*c - a*d)^2) + (c^(1/4)*(b*c - 5*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*d^(5/4)*(b*c - a*d)^2)]
    @test_int [x^(5/2)/((a + b*x^2)*(c + d*x^2)^2), x, 22, x^(3/2)/(2*(b*c - a*d)*(c + d*x^2)) + (a^(3/4)*b^(1/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*(b*c - a*d)^2) - (a^(3/4)*b^(1/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*(b*c - a*d)^2) - ((b*c + 3*a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(1/4)*d^(3/4)*(b*c - a*d)^2) + ((b*c + 3*a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(1/4)*d^(3/4)*(b*c - a*d)^2) - (a^(3/4)*b^(1/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*(b*c - a*d)^2) + (a^(3/4)*b^(1/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*(b*c - a*d)^2) + ((b*c + 3*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(1/4)*d^(3/4)*(b*c - a*d)^2) - ((b*c + 3*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(1/4)*d^(3/4)*(b*c - a*d)^2)]
    @test_int [x^(3/2)/((a + b*x^2)*(c + d*x^2)^2), x, 21, sqrt(x)/(2*(b*c - a*d)*(c + d*x^2)) + (a^(1/4)*b^(3/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*(b*c - a*d)^2) - (a^(1/4)*b^(3/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*(b*c - a*d)^2) - ((3*b*c + a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(3/4)*d^(1/4)*(b*c - a*d)^2) + ((3*b*c + a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(3/4)*d^(1/4)*(b*c - a*d)^2) + (a^(1/4)*b^(3/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*(b*c - a*d)^2) - (a^(1/4)*b^(3/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*(b*c - a*d)^2) - ((3*b*c + a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(3/4)*d^(1/4)*(b*c - a*d)^2) + ((3*b*c + a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(3/4)*d^(1/4)*(b*c - a*d)^2)]
    @test_int [sqrt(x)/((a + b*x^2)*(c + d*x^2)^2), x, 22, -((d*x^(3/2))/(2*c*(b*c - a*d)*(c + d*x^2))) - (b^(5/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(1/4)*(b*c - a*d)^2) + (b^(5/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(1/4)*(b*c - a*d)^2) + (d^(1/4)*(5*b*c - a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(5/4)*(b*c - a*d)^2) - (d^(1/4)*(5*b*c - a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(5/4)*(b*c - a*d)^2) + (b^(5/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(1/4)*(b*c - a*d)^2) - (b^(5/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(1/4)*(b*c - a*d)^2) - (d^(1/4)*(5*b*c - a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(5/4)*(b*c - a*d)^2) + (d^(1/4)*(5*b*c - a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(5/4)*(b*c - a*d)^2)]
    @test_int [1/(sqrt(x)*(a + b*x^2)*(c + d*x^2)^2), x, 21, -((d*sqrt(x))/(2*c*(b*c - a*d)*(c + d*x^2))) - (b^(7/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(3/4)*(b*c - a*d)^2) + (b^(7/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(3/4)*(b*c - a*d)^2) + (d^(3/4)*(7*b*c - 3*a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(7/4)*(b*c - a*d)^2) - (d^(3/4)*(7*b*c - 3*a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(7/4)*(b*c - a*d)^2) - (b^(7/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(3/4)*(b*c - a*d)^2) + (b^(7/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(3/4)*(b*c - a*d)^2) + (d^(3/4)*(7*b*c - 3*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(7/4)*(b*c - a*d)^2) - (d^(3/4)*(7*b*c - 3*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(7/4)*(b*c - a*d)^2)]
    @test_int [1/(x^(3/2)*(a + b*x^2)*(c + d*x^2)^2), x, 23, -((4*b*c - 5*a*d)/(2*a*c^2*(b*c - a*d)*sqrt(x))) - d/(2*c*(b*c - a*d)*sqrt(x)*(c + d*x^2)) + (b^(9/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(5/4)*(b*c - a*d)^2) - (b^(9/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(5/4)*(b*c - a*d)^2) - (d^(5/4)*(9*b*c - 5*a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(9/4)*(b*c - a*d)^2) + (d^(5/4)*(9*b*c - 5*a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(9/4)*(b*c - a*d)^2) - (b^(9/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(5/4)*(b*c - a*d)^2) + (b^(9/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(5/4)*(b*c - a*d)^2) + (d^(5/4)*(9*b*c - 5*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(9/4)*(b*c - a*d)^2) - (d^(5/4)*(9*b*c - 5*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(9/4)*(b*c - a*d)^2)]
    @test_int [1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)^2), x, 22, -((4*b*c - 7*a*d)/(6*a*c^2*(b*c - a*d)*x^(3/2))) - d/(2*c*(b*c - a*d)*x^(3/2)*(c + d*x^2)) + (b^(11/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(7/4)*(b*c - a*d)^2) - (b^(11/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(7/4)*(b*c - a*d)^2) - (d^(7/4)*(11*b*c - 7*a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(11/4)*(b*c - a*d)^2) + (d^(7/4)*(11*b*c - 7*a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(11/4)*(b*c - a*d)^2) + (b^(11/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(7/4)*(b*c - a*d)^2) - (b^(11/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(7/4)*(b*c - a*d)^2) - (d^(7/4)*(11*b*c - 7*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(11/4)*(b*c - a*d)^2) + (d^(7/4)*(11*b*c - 7*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(11/4)*(b*c - a*d)^2)]
    @test_int [1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)^2), x, 24, -((4*b*c - 9*a*d)/(10*a*c^2*(b*c - a*d)*x^(5/2))) + (4*b^2*c^2 + 4*a*b*c*d - 9*a^2*d^2)/(2*a^2*c^3*(b*c - a*d)*sqrt(x)) - d/(2*c*(b*c - a*d)*x^(5/2)*(c + d*x^2)) - (b^(13/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(9/4)*(b*c - a*d)^2) + (b^(13/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(9/4)*(b*c - a*d)^2) + (d^(9/4)*(13*b*c - 9*a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(13/4)*(b*c - a*d)^2) - (d^(9/4)*(13*b*c - 9*a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(13/4)*(b*c - a*d)^2) + (b^(13/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(9/4)*(b*c - a*d)^2) - (b^(13/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(9/4)*(b*c - a*d)^2) - (d^(9/4)*(13*b*c - 9*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(13/4)*(b*c - a*d)^2) + (d^(9/4)*(13*b*c - 9*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(13/4)*(b*c - a*d)^2)]

    @test_int [x^(7/2)/((a + b*x^2)*(c + d*x^2)^3), x, 22, -((c*sqrt(x))/(4*d*(b*c - a*d)*(c + d*x^2)^2)) + ((b*c - 9*a*d)*sqrt(x))/(16*d*(b*c - a*d)^2*(c + d*x^2)) - (a^(5/4)*b^(3/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*(b*c - a*d)^3) + (a^(5/4)*b^(3/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*(b*c - a*d)^3) - ((3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(3/4)*d^(5/4)*(b*c - a*d)^3) + ((3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(3/4)*d^(5/4)*(b*c - a*d)^3) - (a^(5/4)*b^(3/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*(b*c - a*d)^3) + (a^(5/4)*b^(3/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*(b*c - a*d)^3) - ((3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(3/4)*d^(5/4)*(b*c - a*d)^3) + ((3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(3/4)*d^(5/4)*(b*c - a*d)^3)]
    @test_int [x^(5/2)/((a + b*x^2)*(c + d*x^2)^3), x, 23, x^(3/2)/(4*(b*c - a*d)*(c + d*x^2)^2) + ((5*b*c + 3*a*d)*x^(3/2))/(16*c*(b*c - a*d)^2*(c + d*x^2)) + (a^(3/4)*b^(5/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*(b*c - a*d)^3) - (a^(3/4)*b^(5/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*(b*c - a*d)^3) - ((5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(5/4)*d^(3/4)*(b*c - a*d)^3) + ((5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(5/4)*d^(3/4)*(b*c - a*d)^3) - (a^(3/4)*b^(5/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*(b*c - a*d)^3) + (a^(3/4)*b^(5/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*(b*c - a*d)^3) + ((5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(5/4)*d^(3/4)*(b*c - a*d)^3) - ((5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(5/4)*d^(3/4)*(b*c - a*d)^3)]
    @test_int [x^(3/2)/((a + b*x^2)*(c + d*x^2)^3), x, 22, sqrt(x)/(4*(b*c - a*d)*(c + d*x^2)^2) + ((7*b*c + a*d)*sqrt(x))/(16*c*(b*c - a*d)^2*(c + d*x^2)) + (a^(1/4)*b^(7/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*(b*c - a*d)^3) - (a^(1/4)*b^(7/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*(b*c - a*d)^3) - ((21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(7/4)*d^(1/4)*(b*c - a*d)^3) + ((21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(7/4)*d^(1/4)*(b*c - a*d)^3) + (a^(1/4)*b^(7/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*(b*c - a*d)^3) - (a^(1/4)*b^(7/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*(b*c - a*d)^3) - ((21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(7/4)*d^(1/4)*(b*c - a*d)^3) + ((21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(7/4)*d^(1/4)*(b*c - a*d)^3)]
    @test_int [sqrt(x)/((a + b*x^2)*(c + d*x^2)^3), x, 23, -((d*x^(3/2))/(4*c*(b*c - a*d)*(c + d*x^2)^2)) - (d*(13*b*c - 5*a*d)*x^(3/2))/(16*c^2*(b*c - a*d)^2*(c + d*x^2)) - (b^(9/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(1/4)*(b*c - a*d)^3) + (b^(9/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(1/4)*(b*c - a*d)^3) + (d^(1/4)*(45*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(9/4)*(b*c - a*d)^3) - (d^(1/4)*(45*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(9/4)*(b*c - a*d)^3) + (b^(9/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(1/4)*(b*c - a*d)^3) - (b^(9/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(1/4)*(b*c - a*d)^3) - (d^(1/4)*(45*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(9/4)*(b*c - a*d)^3) + (d^(1/4)*(45*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(9/4)*(b*c - a*d)^3)]
    @test_int [1/(sqrt(x)*(a + b*x^2)*(c + d*x^2)^3), x, 22, -((d*sqrt(x))/(4*c*(b*c - a*d)*(c + d*x^2)^2)) - (d*(15*b*c - 7*a*d)*sqrt(x))/(16*c^2*(b*c - a*d)^2*(c + d*x^2)) - (b^(11/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(3/4)*(b*c - a*d)^3) + (b^(11/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(3/4)*(b*c - a*d)^3) + (d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(11/4)*(b*c - a*d)^3) - (d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(11/4)*(b*c - a*d)^3) - (b^(11/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(3/4)*(b*c - a*d)^3) + (b^(11/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(3/4)*(b*c - a*d)^3) + (d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(11/4)*(b*c - a*d)^3) - (d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(11/4)*(b*c - a*d)^3)]
    @test_int [1/(x^(3/2)*(a + b*x^2)*(c + d*x^2)^3), x, 24, -((32*b^2*c^2 - 85*a*b*c*d + 45*a^2*d^2)/(16*a*c^3*(b*c - a*d)^2*sqrt(x))) - d/(4*c*(b*c - a*d)*sqrt(x)*(c + d*x^2)^2) - (d*(17*b*c - 9*a*d))/(16*c^2*(b*c - a*d)^2*sqrt(x)*(c + d*x^2)) + (b^(13/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(5/4)*(b*c - a*d)^3) - (b^(13/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(5/4)*(b*c - a*d)^3) - (d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(13/4)*(b*c - a*d)^3) + (d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(13/4)*(b*c - a*d)^3) - (b^(13/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(5/4)*(b*c - a*d)^3) + (b^(13/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(5/4)*(b*c - a*d)^3) + (d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(13/4)*(b*c - a*d)^3) - (d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(13/4)*(b*c - a*d)^3)]
    @test_int [1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)^3), x, 23, -((32*b^2*c^2 - 133*a*b*c*d + 77*a^2*d^2)/(48*a*c^3*(b*c - a*d)^2*x^(3/2))) - d/(4*c*(b*c - a*d)*x^(3/2)*(c + d*x^2)^2) - (d*(19*b*c - 11*a*d))/(16*c^2*(b*c - a*d)^2*x^(3/2)*(c + d*x^2)) + (b^(15/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(7/4)*(b*c - a*d)^3) - (b^(15/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(7/4)*(b*c - a*d)^3) - (d^(7/4)*(165*b^2*c^2 - 210*a*b*c*d + 77*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(15/4)*(b*c - a*d)^3) + (d^(7/4)*(165*b^2*c^2 - 210*a*b*c*d + 77*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(15/4)*(b*c - a*d)^3) + (b^(15/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(7/4)*(b*c - a*d)^3) - (b^(15/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(7/4)*(b*c - a*d)^3) - (d^(7/4)*(165*b^2*c^2 - 210*a*b*c*d + 77*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(15/4)*(b*c - a*d)^3) + (d^(7/4)*(165*b^2*c^2 - 210*a*b*c*d + 77*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(15/4)*(b*c - a*d)^3)]
    @test_int [1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)^3), x, 25, -((32*b^2*c^2 - 189*a*b*c*d + 117*a^2*d^2)/(80*a*c^3*(b*c - a*d)^2*x^(5/2))) + (32*b^3*c^3 + 32*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 117*a^3*d^3)/(16*a^2*c^4*(b*c - a*d)^2*sqrt(x)) - d/(4*c*(b*c - a*d)*x^(5/2)*(c + d*x^2)^2) - (d*(21*b*c - 13*a*d))/(16*c^2*(b*c - a*d)^2*x^(5/2)*(c + d*x^2)) - (b^(17/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(9/4)*(b*c - a*d)^3) + (b^(17/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(9/4)*(b*c - a*d)^3) + (d^(9/4)*(221*b^2*c^2 - 306*a*b*c*d + 117*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(17/4)*(b*c - a*d)^3) - (d^(9/4)*(221*b^2*c^2 - 306*a*b*c*d + 117*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(17/4)*(b*c - a*d)^3) + (b^(17/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(9/4)*(b*c - a*d)^3) - (b^(17/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(9/4)*(b*c - a*d)^3) - (d^(9/4)*(221*b^2*c^2 - 306*a*b*c*d + 117*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(17/4)*(b*c - a*d)^3) + (d^(9/4)*(221*b^2*c^2 - 306*a*b*c*d + 117*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(17/4)*(b*c - a*d)^3)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^2)^p / (c+d*x^2)^2=#

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^(7/2)/((a + b*x^2)^2*(c + d*x^2)^2), x, 22, ((b*c + a*d)*sqrt(x))/(2*b*(b*c - a*d)^2*(c + d*x^2)) + (a*sqrt(x))/(2*b*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)) + (a^(1/4)*(5*b*c + 3*a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*b^(1/4)*(b*c - a*d)^3) - (a^(1/4)*(5*b*c + 3*a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*b^(1/4)*(b*c - a*d)^3) - (c^(1/4)*(3*b*c + 5*a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*d^(1/4)*(b*c - a*d)^3) + (c^(1/4)*(3*b*c + 5*a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*d^(1/4)*(b*c - a*d)^3) + (a^(1/4)*(5*b*c + 3*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*b^(1/4)*(b*c - a*d)^3) - (a^(1/4)*(5*b*c + 3*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*b^(1/4)*(b*c - a*d)^3) - (c^(1/4)*(3*b*c + 5*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*d^(1/4)*(b*c - a*d)^3) + (c^(1/4)*(3*b*c + 5*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*d^(1/4)*(b*c - a*d)^3)]
    @test_int [x^(5/2)/((a + b*x^2)^2*(c + d*x^2)^2), x, 23, -((d*x^(3/2))/((b*c - a*d)^2*(c + d*x^2))) - x^(3/2)/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)) - (b^(1/4)*(3*b*c + 5*a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(1/4)*(b*c - a*d)^3) + (b^(1/4)*(3*b*c + 5*a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(1/4)*(b*c - a*d)^3) + (d^(1/4)*(5*b*c + 3*a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(1/4)*(b*c - a*d)^3) - (d^(1/4)*(5*b*c + 3*a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(1/4)*(b*c - a*d)^3) + (b^(1/4)*(3*b*c + 5*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(1/4)*(b*c - a*d)^3) - (b^(1/4)*(3*b*c + 5*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(1/4)*(b*c - a*d)^3) - (d^(1/4)*(5*b*c + 3*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(1/4)*(b*c - a*d)^3) + (d^(1/4)*(5*b*c + 3*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(1/4)*(b*c - a*d)^3)]
    @test_int [x^(3/2)/((a + b*x^2)^2*(c + d*x^2)^2), x, 22, -((d*sqrt(x))/((b*c - a*d)^2*(c + d*x^2))) - sqrt(x)/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)) - (b^(3/4)*(b*c + 7*a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(3/4)*(b*c - a*d)^3) + (b^(3/4)*(b*c + 7*a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(3/4)*(b*c - a*d)^3) + (d^(3/4)*(7*b*c + a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(3/4)*(b*c - a*d)^3) - (d^(3/4)*(7*b*c + a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(3/4)*(b*c - a*d)^3) - (b^(3/4)*(b*c + 7*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(3/4)*(b*c - a*d)^3) + (b^(3/4)*(b*c + 7*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(3/4)*(b*c - a*d)^3) + (d^(3/4)*(7*b*c + a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(3/4)*(b*c - a*d)^3) - (d^(3/4)*(7*b*c + a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(3/4)*(b*c - a*d)^3)]
    @test_int [sqrt(x)/((a + b*x^2)^2*(c + d*x^2)^2), x, 23, (d*(b*c + a*d)*x^(3/2))/(2*a*c*(b*c - a*d)^2*(c + d*x^2)) + (b*x^(3/2))/(2*a*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)) - (b^(5/4)*(b*c - 9*a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(5/4)*(b*c - a*d)^3) + (b^(5/4)*(b*c - 9*a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(5/4)*(b*c - a*d)^3) - (d^(5/4)*(9*b*c - a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(5/4)*(b*c - a*d)^3) + (d^(5/4)*(9*b*c - a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(5/4)*(b*c - a*d)^3) + (b^(5/4)*(b*c - 9*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(5/4)*(b*c - a*d)^3) - (b^(5/4)*(b*c - 9*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(5/4)*(b*c - a*d)^3) + (d^(5/4)*(9*b*c - a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(5/4)*(b*c - a*d)^3) - (d^(5/4)*(9*b*c - a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(5/4)*(b*c - a*d)^3)]
    @test_int [1/(sqrt(x)*(a + b*x^2)^2*(c + d*x^2)^2), x, 22, (d*(b*c + a*d)*sqrt(x))/(2*a*c*(b*c - a*d)^2*(c + d*x^2)) + (b*sqrt(x))/(2*a*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)) - (b^(7/4)*(3*b*c - 11*a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(7/4)*(b*c - a*d)^3) + (b^(7/4)*(3*b*c - 11*a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(7/4)*(b*c - a*d)^3) - (d^(7/4)*(11*b*c - 3*a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(7/4)*(b*c - a*d)^3) + (d^(7/4)*(11*b*c - 3*a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(7/4)*(b*c - a*d)^3) - (b^(7/4)*(3*b*c - 11*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(7/4)*(b*c - a*d)^3) + (b^(7/4)*(3*b*c - 11*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(7/4)*(b*c - a*d)^3) - (d^(7/4)*(11*b*c - 3*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(7/4)*(b*c - a*d)^3) + (d^(7/4)*(11*b*c - 3*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(7/4)*(b*c - a*d)^3)]
    @test_int [1/(x^(3/2)*(a + b*x^2)^2*(c + d*x^2)^2), x, 24, -(5*b^2*c^2 - 8*a*b*c*d + 5*a^2*d^2)/(2*a^2*c^2*(b*c - a*d)^2*sqrt(x)) + (d*(b*c + a*d))/(2*a*c*(b*c - a*d)^2*sqrt(x)*(c + d*x^2)) + b/(2*a*(b*c - a*d)*sqrt(x)*(a + b*x^2)*(c + d*x^2)) + (b^(9/4)*(5*b*c - 13*a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(9/4)*(b*c - a*d)^3) - (b^(9/4)*(5*b*c - 13*a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(9/4)*(b*c - a*d)^3) + (d^(9/4)*(13*b*c - 5*a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(9/4)*(b*c - a*d)^3) - (d^(9/4)*(13*b*c - 5*a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(9/4)*(b*c - a*d)^3) - (b^(9/4)*(5*b*c - 13*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(9/4)*(b*c - a*d)^3) + (b^(9/4)*(5*b*c - 13*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(9/4)*(b*c - a*d)^3) - (d^(9/4)*(13*b*c - 5*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(9/4)*(b*c - a*d)^3) + (d^(9/4)*(13*b*c - 5*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(9/4)*(b*c - a*d)^3)]
    @test_int [1/(x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^2), x, 23, -(7*b^2*c^2 - 8*a*b*c*d + 7*a^2*d^2)/(6*a^2*c^2*(b*c - a*d)^2*x^(3/2)) + (d*(b*c + a*d))/(2*a*c*(b*c - a*d)^2*x^(3/2)*(c + d*x^2)) + b/(2*a*(b*c - a*d)*x^(3/2)*(a + b*x^2)*(c + d*x^2)) + (b^(11/4)*(7*b*c - 15*a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(11/4)*(b*c - a*d)^3) - (b^(11/4)*(7*b*c - 15*a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(11/4)*(b*c - a*d)^3) + (d^(11/4)*(15*b*c - 7*a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(11/4)*(b*c - a*d)^3) - (d^(11/4)*(15*b*c - 7*a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(11/4)*(b*c - a*d)^3) + (b^(11/4)*(7*b*c - 15*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(11/4)*(b*c - a*d)^3) - (b^(11/4)*(7*b*c - 15*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(11/4)*(b*c - a*d)^3) + (d^(11/4)*(15*b*c - 7*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(11/4)*(b*c - a*d)^3) - (d^(11/4)*(15*b*c - 7*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(11/4)*(b*c - a*d)^3)]
    @test_int [1/(x^(7/2)*(a + b*x^2)^2*(c + d*x^2)^2), x, 25, -(9*b^2*c^2 - 8*a*b*c*d + 9*a^2*d^2)/(10*a^2*c^2*(b*c - a*d)^2*x^(5/2)) + ((b*c + a*d)*(9*b^2*c^2 - 17*a*b*c*d + 9*a^2*d^2))/(2*a^3*c^3*(b*c - a*d)^2*sqrt(x)) + (d*(b*c + a*d))/(2*a*c*(b*c - a*d)^2*x^(5/2)*(c + d*x^2)) + b/(2*a*(b*c - a*d)*x^(5/2)*(a + b*x^2)*(c + d*x^2)) - (b^(13/4)*(9*b*c - 17*a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(13/4)*(b*c - a*d)^3) + (b^(13/4)*(9*b*c - 17*a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(13/4)*(b*c - a*d)^3) - (d^(13/4)*(17*b*c - 9*a*d)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(13/4)*(b*c - a*d)^3) + (d^(13/4)*(17*b*c - 9*a*d)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(4*sqrt(2)*c^(13/4)*(b*c - a*d)^3) + (b^(13/4)*(9*b*c - 17*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(13/4)*(b*c - a*d)^3) - (b^(13/4)*(9*b*c - 17*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(13/4)*(b*c - a*d)^3) + (d^(13/4)*(17*b*c - 9*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(13/4)*(b*c - a*d)^3) - (d^(13/4)*(17*b*c - 9*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(8*sqrt(2)*c^(13/4)*(b*c - a*d)^3)]

    @test_int [x^(7/2)/((a + b*x^2)^2*(c + d*x^2)^3), x, 23, ((b*c + 2*a*d)*sqrt(x))/(4*b*(b*c - a*d)^2*(c + d*x^2)^2) + (a*sqrt(x))/(2*b*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) + ((7*b*c + 17*a*d)*sqrt(x))/(16*(b*c - a*d)^3*(c + d*x^2)) + (a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*(b*c - a*d)^4) - (a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*(b*c - a*d)^4) - ((21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(3/4)*d^(1/4)*(b*c - a*d)^4) + ((21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(3/4)*d^(1/4)*(b*c - a*d)^4) + (a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*(b*c - a*d)^4) - (a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*(b*c - a*d)^4) - ((21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(3/4)*d^(1/4)*(b*c - a*d)^4) + ((21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(3/4)*d^(1/4)*(b*c - a*d)^4)]
    @test_int [x^(5/2)/((a + b*x^2)^2*(c + d*x^2)^3), x, 24, (-3*d*x^(3/2))/(4*(b*c - a*d)^2*(c + d*x^2)^2) - x^(3/2)/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) - (3*d*(7*b*c + a*d)*x^(3/2))/(16*c*(b*c - a*d)^3*(c + d*x^2)) - (3*b^(5/4)*(b*c + 3*a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(1/4)*(b*c - a*d)^4) + (3*b^(5/4)*(b*c + 3*a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(1/4)*(b*c - a*d)^4) + (3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(5/4)*(b*c - a*d)^4) - (3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(5/4)*(b*c - a*d)^4) + (3*b^(5/4)*(b*c + 3*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(1/4)*(b*c - a*d)^4) - (3*b^(5/4)*(b*c + 3*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(1/4)*(b*c - a*d)^4) - (3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(5/4)*(b*c - a*d)^4) + (3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(5/4)*(b*c - a*d)^4)]
    @test_int [x^(3/2)/((a + b*x^2)^2*(c + d*x^2)^3), x, 23, (-3*d*sqrt(x))/(4*(b*c - a*d)^2*(c + d*x^2)^2) - sqrt(x)/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) - (d*(23*b*c + a*d)*sqrt(x))/(16*c*(b*c - a*d)^3*(c + d*x^2)) - (b^(7/4)*(b*c + 11*a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(3/4)*(b*c - a*d)^4) + (b^(7/4)*(b*c + 11*a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(3/4)*(b*c - a*d)^4) + (d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(7/4)*(b*c - a*d)^4) - (d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(7/4)*(b*c - a*d)^4) - (b^(7/4)*(b*c + 11*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(3/4)*(b*c - a*d)^4) + (b^(7/4)*(b*c + 11*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(3/4)*(b*c - a*d)^4) + (d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(7/4)*(b*c - a*d)^4) - (d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(7/4)*(b*c - a*d)^4)]
    @test_int [sqrt(x)/((a + b*x^2)^2*(c + d*x^2)^3), x, 24, (d*(2*b*c + a*d)*x^(3/2))/(4*a*c*(b*c - a*d)^2*(c + d*x^2)^2) + (b*x^(3/2))/(2*a*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) + (d*(8*b^2*c^2 + 21*a*b*c*d - 5*a^2*d^2)*x^(3/2))/(16*a*c^2*(b*c - a*d)^3*(c + d*x^2)) - (b^(9/4)*(b*c - 13*a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(5/4)*(b*c - a*d)^4) + (b^(9/4)*(b*c - 13*a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(5/4)*(b*c - a*d)^4) - (d^(5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(9/4)*(b*c - a*d)^4) + (d^(5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(9/4)*(b*c - a*d)^4) + (b^(9/4)*(b*c - 13*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(5/4)*(b*c - a*d)^4) - (b^(9/4)*(b*c - 13*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(5/4)*(b*c - a*d)^4) + (d^(5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(9/4)*(b*c - a*d)^4) - (d^(5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(9/4)*(b*c - a*d)^4)]
    @test_int [1/(sqrt(x)*(a + b*x^2)^2*(c + d*x^2)^3), x, 23, (d*(2*b*c + a*d)*sqrt(x))/(4*a*c*(b*c - a*d)^2*(c + d*x^2)^2) + (b*sqrt(x))/(2*a*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) + (d*(8*b^2*c^2 + 23*a*b*c*d - 7*a^2*d^2)*sqrt(x))/(16*a*c^2*(b*c - a*d)^3*(c + d*x^2)) - (3*b^(11/4)*(b*c - 5*a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(7/4)*(b*c - a*d)^4) + (3*b^(11/4)*(b*c - 5*a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(7/4)*(b*c - a*d)^4) - (3*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(11/4)*(b*c - a*d)^4) + (3*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(11/4)*(b*c - a*d)^4) - (3*b^(11/4)*(b*c - 5*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(7/4)*(b*c - a*d)^4) + (3*b^(11/4)*(b*c - 5*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(7/4)*(b*c - a*d)^4) - (3*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(11/4)*(b*c - a*d)^4) + (3*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(11/4)*(b*c - a*d)^4)]
    @test_int [1/(x^(3/2)*(a + b*x^2)^2*(c + d*x^2)^3), x, 25, -(40*b^3*c^3 - 96*a*b^2*c^2*d + 125*a^2*b*c*d^2 - 45*a^3*d^3)/(16*a^2*c^3*(b*c - a*d)^3*sqrt(x)) + (d*(2*b*c + a*d))/(4*a*c*(b*c - a*d)^2*sqrt(x)*(c + d*x^2)^2) + b/(2*a*(b*c - a*d)*sqrt(x)*(a + b*x^2)*(c + d*x^2)^2) + (d*(8*b^2*c^2 + 25*a*b*c*d - 9*a^2*d^2))/(16*a*c^2*(b*c - a*d)^3*sqrt(x)*(c + d*x^2)) + (b^(13/4)*(5*b*c - 17*a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(9/4)*(b*c - a*d)^4) - (b^(13/4)*(5*b*c - 17*a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(9/4)*(b*c - a*d)^4) + (d^(9/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(13/4)*(b*c - a*d)^4) - (d^(9/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(13/4)*(b*c - a*d)^4) - (b^(13/4)*(5*b*c - 17*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(9/4)*(b*c - a*d)^4) + (b^(13/4)*(5*b*c - 17*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(9/4)*(b*c - a*d)^4) - (d^(9/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(13/4)*(b*c - a*d)^4) + (d^(9/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(13/4)*(b*c - a*d)^4)]
    @test_int [1/(x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^3), x, 24, -(56*b^3*c^3 - 96*a*b^2*c^2*d + 189*a^2*b*c*d^2 - 77*a^3*d^3)/(48*a^2*c^3*(b*c - a*d)^3*x^(3/2)) + (d*(2*b*c + a*d))/(4*a*c*(b*c - a*d)^2*x^(3/2)*(c + d*x^2)^2) + b/(2*a*(b*c - a*d)*x^(3/2)*(a + b*x^2)*(c + d*x^2)^2) + (d*(8*b^2*c^2 + 27*a*b*c*d - 11*a^2*d^2))/(16*a*c^2*(b*c - a*d)^3*x^(3/2)*(c + d*x^2)) + (b^(15/4)*(7*b*c - 19*a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(11/4)*(b*c - a*d)^4) - (b^(15/4)*(7*b*c - 19*a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(11/4)*(b*c - a*d)^4) + (d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(15/4)*(b*c - a*d)^4) - (d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(15/4)*(b*c - a*d)^4) + (b^(15/4)*(7*b*c - 19*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(11/4)*(b*c - a*d)^4) - (b^(15/4)*(7*b*c - 19*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(11/4)*(b*c - a*d)^4) + (d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(15/4)*(b*c - a*d)^4) - (d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(15/4)*(b*c - a*d)^4)]
    @test_int [1/(x^(7/2)*(a + b*x^2)^2*(c + d*x^2)^3), x, 26, (-3*(24*b^3*c^3 - 32*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 39*a^3*d^3))/(80*a^2*c^3*(b*c - a*d)^3*x^(5/2)) + (3*(24*b^4*c^4 - 32*a*b^3*c^3*d - 32*a^2*b^2*c^2*d^2 + 87*a^3*b*c*d^3 - 39*a^4*d^4))/(16*a^3*c^4*(b*c - a*d)^3*sqrt(x)) + (d*(2*b*c + a*d))/(4*a*c*(b*c - a*d)^2*x^(5/2)*(c + d*x^2)^2) + b/(2*a*(b*c - a*d)*x^(5/2)*(a + b*x^2)*(c + d*x^2)^2) + (d*(8*b^2*c^2 + 29*a*b*c*d - 13*a^2*d^2))/(16*a*c^2*(b*c - a*d)^3*x^(5/2)*(c + d*x^2)) - (3*b^(17/4)*(3*b*c - 7*a*d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(13/4)*(b*c - a*d)^4) + (3*b^(17/4)*(3*b*c - 7*a*d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(13/4)*(b*c - a*d)^4) - (3*d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(17/4)*(b*c - a*d)^4) + (3*d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*sqrt(x))/c^(1/4)))/(32*sqrt(2)*c^(17/4)*(b*c - a*d)^4) + (3*b^(17/4)*(3*b*c - 7*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(13/4)*(b*c - a*d)^4) - (3*b^(17/4)*(3*b*c - 7*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(13/4)*(b*c - a*d)^4) + (3*d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(17/4)*(b*c - a*d)^4) - (3*d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x))/(64*sqrt(2)*c^(17/4)*(b*c - a*d)^4)]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(e*x)^m*(a+b*x^2)^(p/2)*(c+d*x^2)^q=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^2)^(p/2)*(c+d*x^2)^1=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^5*sqrt(a + b*x^2)*(A + B*x^2), x, 3, (a^2*(A*b - a*B)*(a + b*x^2)^(3/2))/(3*b^4) - (a*(2*A*b - 3*a*B)*(a + b*x^2)^(5/2))/(5*b^4) + ((A*b - 3*a*B)*(a + b*x^2)^(7/2))/(7*b^4) + (B*(a + b*x^2)^(9/2))/(9*b^4)]
    @test_int [x^4*sqrt(a + b*x^2)*(A + B*x^2), x, 6, -((a^2*(8*A*b - 5*a*B)*x*sqrt(a + b*x^2))/(128*b^3)) + (a*(8*A*b - 5*a*B)*x^3*sqrt(a + b*x^2))/(192*b^2) + ((8*A*b - 5*a*B)*x^5*sqrt(a + b*x^2))/(48*b) + (B*x^5*(a + b*x^2)^(3/2))/(8*b) + (a^3*(8*A*b - 5*a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(128*b^(7/2))]
    @test_int [x^3*sqrt(a + b*x^2)*(A + B*x^2), x, 3, -((a*(A*b - a*B)*(a + b*x^2)^(3/2))/(3*b^3)) + ((A*b - 2*a*B)*(a + b*x^2)^(5/2))/(5*b^3) + (B*(a + b*x^2)^(7/2))/(7*b^3)]
    @test_int [x^2*sqrt(a + b*x^2)*(A + B*x^2), x, 5, (a*(2*A*b - a*B)*x*sqrt(a + b*x^2))/(16*b^2) + ((2*A*b - a*B)*x^3*sqrt(a + b*x^2))/(8*b) + (B*x^3*(a + b*x^2)^(3/2))/(6*b) - (a^2*(2*A*b - a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(16*b^(5/2))]
    @test_int [x*sqrt(a + b*x^2)*(A + B*x^2), x, 3, ((A*b - a*B)*(a + b*x^2)^(3/2))/(3*b^2) + (B*(a + b*x^2)^(5/2))/(5*b^2)]
    @test_int [sqrt(a + b*x^2)*(A + B*x^2), x, 4, ((4*A*b - a*B)*x*sqrt(a + b*x^2))/(8*b) + (B*x*(a + b*x^2)^(3/2))/(4*b) + (a*(4*A*b - a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(8*b^(3/2))]
    @test_int [(sqrt(a + b*x^2)*(A + B*x^2))/x, x, 5, A*sqrt(a + b*x^2) + (B*(a + b*x^2)^(3/2))/(3*b) - sqrt(a)*A*atanh(sqrt(a + b*x^2)/sqrt(a))]
    @test_int [(sqrt(a + b*x^2)*(A + B*x^2))/x^2, x, 4, ((2*A*b + a*B)*x*sqrt(a + b*x^2))/(2*a) - (A*(a + b*x^2)^(3/2))/(a*x) + ((2*A*b + a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(2*sqrt(b))]
    @test_int [(sqrt(a + b*x^2)*(A + B*x^2))/x^3, x, 5, ((A*b + 2*a*B)*sqrt(a + b*x^2))/(2*a) - (A*(a + b*x^2)^(3/2))/(2*a*x^2) - ((A*b + 2*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(2*sqrt(a))]
    @test_int [(sqrt(a + b*x^2)*(A + B*x^2))/x^4, x, 4, -((B*sqrt(a + b*x^2))/x) - (A*(a + b*x^2)^(3/2))/(3*a*x^3) + sqrt(b)*B*atanh((sqrt(b)*x)/sqrt(a + b*x^2))]
    @test_int [(sqrt(a + b*x^2)*(A + B*x^2))/x^5, x, 5, ((A*b - 4*a*B)*sqrt(a + b*x^2))/(8*a*x^2) - (A*(a + b*x^2)^(3/2))/(4*a*x^4) + (b*(A*b - 4*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(8*a^(3/2))]
    @test_int [(sqrt(a + b*x^2)*(A + B*x^2))/x^6, x, 2, -(A*(a + b*x^2)^(3/2))/(5*a*x^5) + ((2*A*b - 5*a*B)*(a + b*x^2)^(3/2))/(15*a^2*x^3)]
    @test_int [(sqrt(a + b*x^2)*(A + B*x^2))/x^7, x, 6, ((A*b - 2*a*B)*sqrt(a + b*x^2))/(8*a*x^4) + (b*(A*b - 2*a*B)*sqrt(a + b*x^2))/(16*a^2*x^2) - (A*(a + b*x^2)^(3/2))/(6*a*x^6) - (b^2*(A*b - 2*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(16*a^(5/2))]
    @test_int [(sqrt(a + b*x^2)*(A + B*x^2))/x^8, x, 3, -((A*(a + b*x^2)^(3/2))/(7*a*x^7)) + ((4*A*b - 7*a*B)*(a + b*x^2)^(3/2))/(35*a^2*x^5) - (2*b*(4*A*b - 7*a*B)*(a + b*x^2)^(3/2))/(105*a^3*x^3)]
    @test_int [(sqrt(a + b*x^2)*(A + B*x^2))/x^9, x, 7, ((5*A*b - 8*a*B)*sqrt(a + b*x^2))/(48*a*x^6) + (b*(5*A*b - 8*a*B)*sqrt(a + b*x^2))/(192*a^2*x^4) - (b^2*(5*A*b - 8*a*B)*sqrt(a + b*x^2))/(128*a^3*x^2) - (A*(a + b*x^2)^(3/2))/(8*a*x^8) + (b^3*(5*A*b - 8*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(128*a^(7/2))]
    @test_int [(sqrt(a + b*x^2)*(A + B*x^2))/x^10, x, 4, -((A*(a + b*x^2)^(3/2))/(9*a*x^9)) + ((2*A*b - 3*a*B)*(a + b*x^2)^(3/2))/(21*a^2*x^7) - (4*b*(2*A*b - 3*a*B)*(a + b*x^2)^(3/2))/(105*a^3*x^5) + (8*b^2*(2*A*b - 3*a*B)*(a + b*x^2)^(3/2))/(315*a^4*x^3)]
    @test_int [(sqrt(a + b*x^2)*(A + B*x^2))/x^11, x, 8, ((7*A*b - 10*a*B)*sqrt(a + b*x^2))/(80*a*x^8) + (b*(7*A*b - 10*a*B)*sqrt(a + b*x^2))/(480*a^2*x^6) - (b^2*(7*A*b - 10*a*B)*sqrt(a + b*x^2))/(384*a^3*x^4) + (b^3*(7*A*b - 10*a*B)*sqrt(a + b*x^2))/(256*a^4*x^2) - (A*(a + b*x^2)^(3/2))/(10*a*x^10) - (b^4*(7*A*b - 10*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(256*a^(9/2))]

    @test_int [x^5*(A + B*x^2)*(a + b*x^2)^(3/2), x, 3, (a^2*(A*b - a*B)*(a + b*x^2)^(5/2))/(5*b^4) - (a*(2*A*b - 3*a*B)*(a + b*x^2)^(7/2))/(7*b^4) + ((A*b - 3*a*B)*(a + b*x^2)^(9/2))/(9*b^4) + (B*(a + b*x^2)^(11/2))/(11*b^4)]
    @test_int [x^4*(A + B*x^2)*(a + b*x^2)^(3/2), x, 7, -((3*a^3*(2*A*b - a*B)*x*sqrt(a + b*x^2))/(256*b^3)) + (a^2*(2*A*b - a*B)*x^3*sqrt(a + b*x^2))/(128*b^2) + (a*(2*A*b - a*B)*x^5*sqrt(a + b*x^2))/(32*b) + ((2*A*b - a*B)*x^5*(a + b*x^2)^(3/2))/(16*b) + (B*x^5*(a + b*x^2)^(5/2))/(10*b) + (3*a^4*(2*A*b - a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(256*b^(7/2))]
    @test_int [x^3*(A + B*x^2)*(a + b*x^2)^(3/2), x, 3, -((a*(A*b - a*B)*(a + b*x^2)^(5/2))/(5*b^3)) + ((A*b - 2*a*B)*(a + b*x^2)^(7/2))/(7*b^3) + (B*(a + b*x^2)^(9/2))/(9*b^3)]
    @test_int [x^2*(A + B*x^2)*(a + b*x^2)^(3/2), x, 6, (a^2*(8*A*b - 3*a*B)*x*sqrt(a + b*x^2))/(128*b^2) + (a*(8*A*b - 3*a*B)*x^3*sqrt(a + b*x^2))/(64*b) + ((8*A*b - 3*a*B)*x^3*(a + b*x^2)^(3/2))/(48*b) + (B*x^3*(a + b*x^2)^(5/2))/(8*b) - (a^3*(8*A*b - 3*a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(128*b^(5/2))]
    @test_int [x^1*(A + B*x^2)*(a + b*x^2)^(3/2), x, 3, ((A*b - a*B)*(a + b*x^2)^(5/2))/(5*b^2) + (B*(a + b*x^2)^(7/2))/(7*b^2)]
    @test_int [x^0*(A + B*x^2)*(a + b*x^2)^(3/2), x, 5, (a*(6*A*b - a*B)*x*sqrt(a + b*x^2))/(16*b) + ((6*A*b - a*B)*x*(a + b*x^2)^(3/2))/(24*b) + (B*x*(a + b*x^2)^(5/2))/(6*b) + (a^2*(6*A*b - a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(16*b^(3/2))]
    @test_int [(A + B*x^2)*(a + b*x^2)^(3/2)/x^1, x, 6, a*A*sqrt(a + b*x^2) + (1/3)*A*(a + b*x^2)^(3/2) + (B*(a + b*x^2)^(5/2))/(5*b) - a^(3/2)*A*atanh(sqrt(a + b*x^2)/sqrt(a))]
    @test_int [(A + B*x^2)*(a + b*x^2)^(3/2)/x^2, x, 5, (3/8)*(4*A*b + a*B)*x*sqrt(a + b*x^2) + ((4*A*b + a*B)*x*(a + b*x^2)^(3/2))/(4*a) - (A*(a + b*x^2)^(5/2))/(a*x) + (3*a*(4*A*b + a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(8*sqrt(b))]
    @test_int [(A + B*x^2)*(a + b*x^2)^(3/2)/x^3, x, 6, (1/2)*(3*A*b + 2*a*B)*sqrt(a + b*x^2) + ((3*A*b + 2*a*B)*(a + b*x^2)^(3/2))/(6*a) - (A*(a + b*x^2)^(5/2))/(2*a*x^2) - (1/2)*sqrt(a)*(3*A*b + 2*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a))]
    @test_int [(A + B*x^2)*(a + b*x^2)^(3/2)/x^4, x, 5, (b*(2*A*b + 3*a*B)*x*sqrt(a + b*x^2))/(2*a) - ((2*A*b + 3*a*B)*(a + b*x^2)^(3/2))/(3*a*x) - (A*(a + b*x^2)^(5/2))/(3*a*x^3) + (1/2)*sqrt(b)*(2*A*b + 3*a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2))]
    @test_int [(A + B*x^2)*(a + b*x^2)^(3/2)/x^5, x, 6, (3*b*(A*b + 4*a*B)*sqrt(a + b*x^2))/(8*a) - ((A*b + 4*a*B)*(a + b*x^2)^(3/2))/(8*a*x^2) - (A*(a + b*x^2)^(5/2))/(4*a*x^4) - (3*b*(A*b + 4*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(8*sqrt(a))]
    @test_int [(A + B*x^2)*(a + b*x^2)^(3/2)/x^6, x, 5, -((b*B*sqrt(a + b*x^2))/x) - (B*(a + b*x^2)^(3/2))/(3*x^3) - (A*(a + b*x^2)^(5/2))/(5*a*x^5) + b^(3/2)*B*atanh((sqrt(b)*x)/sqrt(a + b*x^2))]
    @test_int [(A + B*x^2)*(a + b*x^2)^(3/2)/x^7, x, 6, (b*(A*b - 6*a*B)*sqrt(a + b*x^2))/(16*a*x^2) + ((A*b - 6*a*B)*(a + b*x^2)^(3/2))/(24*a*x^4) - (A*(a + b*x^2)^(5/2))/(6*a*x^6) + (b^2*(A*b - 6*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(16*a^(3/2))]
    @test_int [(A + B*x^2)*(a + b*x^2)^(3/2)/x^8, x, 2, -((A*(a + b*x^2)^(5/2))/(7*a*x^7)) + ((2*A*b - 7*a*B)*(a + b*x^2)^(5/2))/(35*a^2*x^5)]
    @test_int [(A + B*x^2)*(a + b*x^2)^(3/2)/x^9, x, 7, (b*(3*A*b - 8*a*B)*sqrt(a + b*x^2))/(64*a*x^4) + (b^2*(3*A*b - 8*a*B)*sqrt(a + b*x^2))/(128*a^2*x^2) + ((3*A*b - 8*a*B)*(a + b*x^2)^(3/2))/(48*a*x^6) - (A*(a + b*x^2)^(5/2))/(8*a*x^8) - (b^3*(3*A*b - 8*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(128*a^(5/2))]
    @test_int [(A + B*x^2)*(a + b*x^2)^(3/2)/x^10, x, 3, -((A*(a + b*x^2)^(5/2))/(9*a*x^9)) + ((4*A*b - 9*a*B)*(a + b*x^2)^(5/2))/(63*a^2*x^7) - (2*b*(4*A*b - 9*a*B)*(a + b*x^2)^(5/2))/(315*a^3*x^5)]
    @test_int [(A + B*x^2)*(a + b*x^2)^(3/2)/x^11, x, 8, (b*(A*b - 2*a*B)*sqrt(a + b*x^2))/(32*a*x^6) + (b^2*(A*b - 2*a*B)*sqrt(a + b*x^2))/(128*a^2*x^4) - (3*b^3*(A*b - 2*a*B)*sqrt(a + b*x^2))/(256*a^3*x^2) + ((A*b - 2*a*B)*(a + b*x^2)^(3/2))/(16*a*x^8) - (A*(a + b*x^2)^(5/2))/(10*a*x^10) + (3*b^4*(A*b - 2*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(256*a^(7/2))]

    @test_int [x^5*(a + b*x^2)^(5/2)*(A + B*x^2), x, 3, (a^2*(A*b - a*B)*(a + b*x^2)^(7/2))/(7*b^4) - (a*(2*A*b - 3*a*B)*(a + b*x^2)^(9/2))/(9*b^4) + ((A*b - 3*a*B)*(a + b*x^2)^(11/2))/(11*b^4) + (B*(a + b*x^2)^(13/2))/(13*b^4)]
    @test_int [x^4*(a + b*x^2)^(5/2)*(A + B*x^2), x, 8, -(a^4*(12*A*b - 5*a*B)*x*sqrt(a + b*x^2))/(1024*b^3) + (a^3*(12*A*b - 5*a*B)*x^3*sqrt(a + b*x^2))/(1536*b^2) + (a^2*(12*A*b - 5*a*B)*x^5*sqrt(a + b*x^2))/(384*b) + (a*(12*A*b - 5*a*B)*x^5*(a + b*x^2)^(3/2))/(192*b) + ((12*A*b - 5*a*B)*x^5*(a + b*x^2)^(5/2))/(120*b) + (B*x^5*(a + b*x^2)^(7/2))/(12*b) + (a^5*(12*A*b - 5*a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(1024*b^(7/2))]
    @test_int [x^3*(a + b*x^2)^(5/2)*(A + B*x^2), x, 3, -(a*(A*b - a*B)*(a + b*x^2)^(7/2))/(7*b^3) + ((A*b - 2*a*B)*(a + b*x^2)^(9/2))/(9*b^3) + (B*(a + b*x^2)^(11/2))/(11*b^3)]
    @test_int [x^2*(a + b*x^2)^(5/2)*(A + B*x^2), x, 7, (a^3*(10*A*b - 3*a*B)*x*sqrt(a + b*x^2))/(256*b^2) + (a^2*(10*A*b - 3*a*B)*x^3*sqrt(a + b*x^2))/(128*b) + (a*(10*A*b - 3*a*B)*x^3*(a + b*x^2)^(3/2))/(96*b) + ((10*A*b - 3*a*B)*x^3*(a + b*x^2)^(5/2))/(80*b) + (B*x^3*(a + b*x^2)^(7/2))/(10*b) - (a^4*(10*A*b - 3*a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(256*b^(5/2))]
    @test_int [x*(a + b*x^2)^(5/2)*(A + B*x^2), x, 3, ((A*b - a*B)*(a + b*x^2)^(7/2))/(7*b^2) + (B*(a + b*x^2)^(9/2))/(9*b^2)]
    @test_int [(a + b*x^2)^(5/2)*(A + B*x^2), x, 6, (5*a^2*(8*A*b - a*B)*x*sqrt(a + b*x^2))/(128*b) + (5*a*(8*A*b - a*B)*x*(a + b*x^2)^(3/2))/(192*b) + ((8*A*b - a*B)*x*(a + b*x^2)^(5/2))/(48*b) + (B*x*(a + b*x^2)^(7/2))/(8*b) + (5*a^3*(8*A*b - a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(128*b^(3/2))]
    @test_int [((a + b*x^2)^(5/2)*(A + B*x^2))/x, x, 7, a^2*A*sqrt(a + b*x^2) + (a*A*(a + b*x^2)^(3/2))/3 + (A*(a + b*x^2)^(5/2))/5 + (B*(a + b*x^2)^(7/2))/(7*b) - a^(5/2)*A*atanh(sqrt(a + b*x^2)/sqrt(a))]
    @test_int [((a + b*x^2)^(5/2)*(A + B*x^2))/x^2, x, 6, (5*a*(6*A*b + a*B)*x*sqrt(a + b*x^2))/16 + (5*(6*A*b + a*B)*x*(a + b*x^2)^(3/2))/24 + ((6*A*b + a*B)*x*(a + b*x^2)^(5/2))/(6*a) - (A*(a + b*x^2)^(7/2))/(a*x) + (5*a^2*(6*A*b + a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(16*sqrt(b))]
    @test_int [((a + b*x^2)^(5/2)*(A + B*x^2))/x^3, x, 7, (a*(5*A*b + 2*a*B)*sqrt(a + b*x^2))/2 + ((5*A*b + 2*a*B)*(a + b*x^2)^(3/2))/6 + ((5*A*b + 2*a*B)*(a + b*x^2)^(5/2))/(10*a) - (A*(a + b*x^2)^(7/2))/(2*a*x^2) - (a^(3/2)*(5*A*b + 2*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/2]
    @test_int [((a + b*x^2)^(5/2)*(A + B*x^2))/x^4, x, 6, (5*b*(4*A*b + 3*a*B)*x*sqrt(a + b*x^2))/8 + (5*b*(4*A*b + 3*a*B)*x*(a + b*x^2)^(3/2))/(12*a) - ((4*A*b + 3*a*B)*(a + b*x^2)^(5/2))/(3*a*x) - (A*(a + b*x^2)^(7/2))/(3*a*x^3) + (5*a*sqrt(b)*(4*A*b + 3*a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/8]
    @test_int [((a + b*x^2)^(5/2)*(A + B*x^2))/x^5, x, 7, (5*b*(3*A*b + 4*a*B)*sqrt(a + b*x^2))/8 + (5*b*(3*A*b + 4*a*B)*(a + b*x^2)^(3/2))/(24*a) - ((3*A*b + 4*a*B)*(a + b*x^2)^(5/2))/(8*a*x^2) - (A*(a + b*x^2)^(7/2))/(4*a*x^4) - (5*sqrt(a)*b*(3*A*b + 4*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/8]
    @test_int [((a + b*x^2)^(5/2)*(A + B*x^2))/x^6, x, 6, (b^2*(2*A*b + 5*a*B)*x*sqrt(a + b*x^2))/(2*a) - (b*(2*A*b + 5*a*B)*(a + b*x^2)^(3/2))/(3*a*x) - ((2*A*b + 5*a*B)*(a + b*x^2)^(5/2))/(15*a*x^3) - (A*(a + b*x^2)^(7/2))/(5*a*x^5) + (b^(3/2)*(2*A*b + 5*a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/2]
    @test_int [((a + b*x^2)^(5/2)*(A + B*x^2))/x^7, x, 7, (5*b^2*(A*b + 6*a*B)*sqrt(a + b*x^2))/(16*a) - (5*b*(A*b + 6*a*B)*(a + b*x^2)^(3/2))/(48*a*x^2) - ((A*b + 6*a*B)*(a + b*x^2)^(5/2))/(24*a*x^4) - (A*(a + b*x^2)^(7/2))/(6*a*x^6) - (5*b^2*(A*b + 6*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(16*sqrt(a))]
    @test_int [((a + b*x^2)^(5/2)*(A + B*x^2))/x^8, x, 6, -((b^2*B*sqrt(a + b*x^2))/x) - (b*B*(a + b*x^2)^(3/2))/(3*x^3) - (B*(a + b*x^2)^(5/2))/(5*x^5) - (A*(a + b*x^2)^(7/2))/(7*a*x^7) + b^(5/2)*B*atanh((sqrt(b)*x)/sqrt(a + b*x^2))]
    @test_int [((a + b*x^2)^(5/2)*(A + B*x^2))/x^9, x, 7, (5*b^2*(A*b - 8*a*B)*sqrt(a + b*x^2))/(128*a*x^2) + (5*b*(A*b - 8*a*B)*(a + b*x^2)^(3/2))/(192*a*x^4) + ((A*b - 8*a*B)*(a + b*x^2)^(5/2))/(48*a*x^6) - (A*(a + b*x^2)^(7/2))/(8*a*x^8) + (5*b^3*(A*b - 8*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(128*a^(3/2))]
    @test_int [((a + b*x^2)^(5/2)*(A + B*x^2))/x^10, x, 2, -(A*(a + b*x^2)^(7/2))/(9*a*x^9) + ((2*A*b - 9*a*B)*(a + b*x^2)^(7/2))/(63*a^2*x^7)]
    @test_int [((a + b*x^2)^(5/2)*(A + B*x^2))/x^11, x, 8, (b^2*(3*A*b - 10*a*B)*sqrt(a + b*x^2))/(128*a*x^4) + (b^3*(3*A*b - 10*a*B)*sqrt(a + b*x^2))/(256*a^2*x^2) + (b*(3*A*b - 10*a*B)*(a + b*x^2)^(3/2))/(96*a*x^6) + ((3*A*b - 10*a*B)*(a + b*x^2)^(5/2))/(80*a*x^8) - (A*(a + b*x^2)^(7/2))/(10*a*x^10) - (b^4*(3*A*b - 10*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(256*a^(5/2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(x^5*(A + B*x^2))/sqrt(a + b*x^2), x, 3, (a^2*(A*b - a*B)*sqrt(a + b*x^2))/b^4 - (a*(2*A*b - 3*a*B)*(a + b*x^2)^(3/2))/(3*b^4) + ((A*b - 3*a*B)*(a + b*x^2)^(5/2))/(5*b^4) + (B*(a + b*x^2)^(7/2))/(7*b^4)]
    @test_int [(x^4*(A + B*x^2))/sqrt(a + b*x^2), x, 5, -((a*(6*A*b - 5*a*B)*x*sqrt(a + b*x^2))/(16*b^3)) + ((6*A*b - 5*a*B)*x^3*sqrt(a + b*x^2))/(24*b^2) + (B*x^5*sqrt(a + b*x^2))/(6*b) + (a^2*(6*A*b - 5*a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(16*b^(7/2))]
    @test_int [(x^3*(A + B*x^2))/sqrt(a + b*x^2), x, 3, -((a*(A*b - a*B)*sqrt(a + b*x^2))/b^3) + ((A*b - 2*a*B)*(a + b*x^2)^(3/2))/(3*b^3) + (B*(a + b*x^2)^(5/2))/(5*b^3)]
    @test_int [(x^2*(A + B*x^2))/sqrt(a + b*x^2), x, 4, ((4*A*b - 3*a*B)*x*sqrt(a + b*x^2))/(8*b^2) + (B*x^3*sqrt(a + b*x^2))/(4*b) - (a*(4*A*b - 3*a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(8*b^(5/2))]
    @test_int [(x*(A + B*x^2))/sqrt(a + b*x^2), x, 3, ((A*b - a*B)*sqrt(a + b*x^2))/b^2 + (B*(a + b*x^2)^(3/2))/(3*b^2)]
    @test_int [(A + B*x^2)/sqrt(a + b*x^2), x, 3, (B*x*sqrt(a + b*x^2))/(2*b) + ((2*A*b - a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(2*b^(3/2))]
    @test_int [(A + B*x^2)/(x*sqrt(a + b*x^2)), x, 4, (B*sqrt(a + b*x^2))/b - (A*atanh(sqrt(a + b*x^2)/sqrt(a)))/sqrt(a)]
    @test_int [(A + B*x^2)/(x^2*sqrt(a + b*x^2)), x, 3, -((A*sqrt(a + b*x^2))/(a*x)) + (B*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/sqrt(b)]
    @test_int [(A + B*x^2)/(x^3*sqrt(a + b*x^2)), x, 4, -(A*sqrt(a + b*x^2))/(2*a*x^2) + ((A*b - 2*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(2*a^(3/2))]
    @test_int [(A + B*x^2)/(x^4*sqrt(a + b*x^2)), x, 2, -(A*sqrt(a + b*x^2))/(3*a*x^3) + ((2*A*b - 3*a*B)*sqrt(a + b*x^2))/(3*a^2*x)]
    @test_int [(A + B*x^2)/(x^5*sqrt(a + b*x^2)), x, 5, -(A*sqrt(a + b*x^2))/(4*a*x^4) + ((3*A*b - 4*a*B)*sqrt(a + b*x^2))/(8*a^2*x^2) - (b*(3*A*b - 4*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(8*a^(5/2))]
    @test_int [(A + B*x^2)/(x^6*sqrt(a + b*x^2)), x, 3, -(A*sqrt(a + b*x^2))/(5*a*x^5) + ((4*A*b - 5*a*B)*sqrt(a + b*x^2))/(15*a^2*x^3) - (2*b*(4*A*b - 5*a*B)*sqrt(a + b*x^2))/(15*a^3*x)]
    @test_int [(A + B*x^2)/(x^7*sqrt(a + b*x^2)), x, 6, -(A*sqrt(a + b*x^2))/(6*a*x^6) + ((5*A*b - 6*a*B)*sqrt(a + b*x^2))/(24*a^2*x^4) - (b*(5*A*b - 6*a*B)*sqrt(a + b*x^2))/(16*a^3*x^2) + (b^2*(5*A*b - 6*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(16*a^(7/2))]
    @test_int [(A + B*x^2)/(x^8*sqrt(a + b*x^2)), x, 4, -(A*sqrt(a + b*x^2))/(7*a*x^7) + ((6*A*b - 7*a*B)*sqrt(a + b*x^2))/(35*a^2*x^5) - (4*b*(6*A*b - 7*a*B)*sqrt(a + b*x^2))/(105*a^3*x^3) + (8*b^2*(6*A*b - 7*a*B)*sqrt(a + b*x^2))/(105*a^4*x)]

    @test_int [x^6*(A + B*x^2)/(a + b*x^2)^(3/2), x, 6, -(((6*A*b - 7*a*B)*x^5)/(6*b^2*sqrt(a + b*x^2))) + (B*x^7)/(6*b*sqrt(a + b*x^2)) - (5*a*(6*A*b - 7*a*B)*x*sqrt(a + b*x^2))/(16*b^4) + (5*(6*A*b - 7*a*B)*x^3*sqrt(a + b*x^2))/(24*b^3) + (5*a^2*(6*A*b - 7*a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(16*b^(9/2))]
    @test_int [x^5*(A + B*x^2)/(a + b*x^2)^(3/2), x, 3, -((a^2*(A*b - a*B))/(b^4*sqrt(a + b*x^2))) - (a*(2*A*b - 3*a*B)*sqrt(a + b*x^2))/b^4 + ((A*b - 3*a*B)*(a + b*x^2)^(3/2))/(3*b^4) + (B*(a + b*x^2)^(5/2))/(5*b^4)]
    @test_int [x^4*(A + B*x^2)/(a + b*x^2)^(3/2), x, 5, -(((4*A*b - 5*a*B)*x^3)/(4*b^2*sqrt(a + b*x^2))) + (B*x^5)/(4*b*sqrt(a + b*x^2)) + (3*(4*A*b - 5*a*B)*x*sqrt(a + b*x^2))/(8*b^3) - (3*a*(4*A*b - 5*a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(8*b^(7/2))]
    @test_int [x^3*(A + B*x^2)/(a + b*x^2)^(3/2), x, 3, (a*(A*b - a*B))/(b^3*sqrt(a + b*x^2)) + ((A*b - 2*a*B)*sqrt(a + b*x^2))/b^3 + (B*(a + b*x^2)^(3/2))/(3*b^3)]
    @test_int [x^2*(A + B*x^2)/(a + b*x^2)^(3/2), x, 4, -(((A*b - a*B)*x)/(b^2*sqrt(a + b*x^2))) + (B*x*sqrt(a + b*x^2))/(2*b^2) + ((2*A*b - 3*a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(2*b^(5/2))]
    @test_int [x^1*(A + B*x^2)/(a + b*x^2)^(3/2), x, 3, -((A*b - a*B)/(b^2*sqrt(a + b*x^2))) + (B*sqrt(a + b*x^2))/b^2]
    @test_int [x^0*(A + B*x^2)/(a + b*x^2)^(3/2), x, 3, ((A*b - a*B)*x)/(a*b*sqrt(a + b*x^2)) + (B*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/b^(3/2)]
    @test_int [(A + B*x^2)/(x^1*(a + b*x^2)^(3/2)), x, 4, (A*b - a*B)/(a*b*sqrt(a + b*x^2)) - (A*atanh(sqrt(a + b*x^2)/sqrt(a)))/a^(3/2)]
    @test_int [(A + B*x^2)/(x^2*(a + b*x^2)^(3/2)), x, 2, -(A/(a*x*sqrt(a + b*x^2))) - ((2*A*b - a*B)*x)/(a^2*sqrt(a + b*x^2))]
    @test_int [(A + B*x^2)/(x^3*(a + b*x^2)^(3/2)), x, 5, -((3*A*b - 2*a*B)/(2*a^2*sqrt(a + b*x^2))) - A/(2*a*x^2*sqrt(a + b*x^2)) + ((3*A*b - 2*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(2*a^(5/2))]
    @test_int [(A + B*x^2)/(x^4*(a + b*x^2)^(3/2)), x, 3, -(A/(3*a*x^3*sqrt(a + b*x^2))) + (4*A*b - 3*a*B)/(3*a^2*x*sqrt(a + b*x^2)) + (2*b*(4*A*b - 3*a*B)*x)/(3*a^3*sqrt(a + b*x^2))]
    @test_int [(A + B*x^2)/(x^5*(a + b*x^2)^(3/2)), x, 6, (3*b*(5*A*b - 4*a*B))/(8*a^3*sqrt(a + b*x^2)) - A/(4*a*x^4*sqrt(a + b*x^2)) + (5*A*b - 4*a*B)/(8*a^2*x^2*sqrt(a + b*x^2)) - (3*b*(5*A*b - 4*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(8*a^(7/2))]
    @test_int [(A + B*x^2)/(x^6*(a + b*x^2)^(3/2)), x, 4, -(A/(5*a*x^5*sqrt(a + b*x^2))) + (6*A*b - 5*a*B)/(15*a^2*x^3*sqrt(a + b*x^2)) - (4*b*(6*A*b - 5*a*B))/(15*a^3*x*sqrt(a + b*x^2)) - (8*b^2*(6*A*b - 5*a*B)*x)/(15*a^4*sqrt(a + b*x^2))]
    @test_int [(A + B*x^2)/(x^7*(a + b*x^2)^(3/2)), x, 7, -((5*b^2*(7*A*b - 6*a*B))/(16*a^4*sqrt(a + b*x^2))) - A/(6*a*x^6*sqrt(a + b*x^2)) + (7*A*b - 6*a*B)/(24*a^2*x^4*sqrt(a + b*x^2)) - (5*b*(7*A*b - 6*a*B))/(48*a^3*x^2*sqrt(a + b*x^2)) + (5*b^2*(7*A*b - 6*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(16*a^(9/2))]
    @test_int [(A + B*x^2)/(x^8*(a + b*x^2)^(3/2)), x, 5, -(A/(7*a*x^7*sqrt(a + b*x^2))) + (8*A*b - 7*a*B)/(35*a^2*x^5*sqrt(a + b*x^2)) - (2*b*(8*A*b - 7*a*B))/(35*a^3*x^3*sqrt(a + b*x^2)) + (8*b^2*(8*A*b - 7*a*B))/(35*a^4*x*sqrt(a + b*x^2)) + (16*b^3*(8*A*b - 7*a*B)*x)/(35*a^5*sqrt(a + b*x^2))]

    @test_int [(x^7*(A + B*x^2))/(a + b*x^2)^(5/2), x, 3, (a^3*(A*b - a*B))/(3*b^5*(a + b*x^2)^(3/2)) - (a^2*(3*A*b - 4*a*B))/(b^5*sqrt(a + b*x^2)) - (3*a*(A*b - 2*a*B)*sqrt(a + b*x^2))/b^5 + ((A*b - 4*a*B)*(a + b*x^2)^(3/2))/(3*b^5) + (B*(a + b*x^2)^(5/2))/(5*b^5)]
    @test_int [(x^6*(A + B*x^2))/(a + b*x^2)^(5/2), x, 6, -(((4*A*b - 7*a*B)*x^5)/(12*b^2*(a + b*x^2)^(3/2))) + (B*x^7)/(4*b*(a + b*x^2)^(3/2)) - (5*(4*A*b - 7*a*B)*x^3)/(12*b^3*sqrt(a + b*x^2)) + (5*(4*A*b - 7*a*B)*x*sqrt(a + b*x^2))/(8*b^4) - (5*a*(4*A*b - 7*a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(8*b^(9/2))]
    @test_int [(x^5*(A + B*x^2))/(a + b*x^2)^(5/2), x, 3, -(a^2*(A*b - a*B))/(3*b^4*(a + b*x^2)^(3/2)) + (a*(2*A*b - 3*a*B))/(b^4*sqrt(a + b*x^2)) + ((A*b - 3*a*B)*sqrt(a + b*x^2))/b^4 + (B*(a + b*x^2)^(3/2))/(3*b^4)]
    @test_int [(x^4*(A + B*x^2))/(a + b*x^2)^(5/2), x, 5, (a*(A*b - a*B)*x)/(3*b^3*(a + b*x^2)^(3/2)) - ((4*A*b - 7*a*B)*x)/(3*b^3*sqrt(a + b*x^2)) + (B*x*sqrt(a + b*x^2))/(2*b^3) + ((2*A*b - 5*a*B)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(2*b^(7/2))]
    @test_int [(x^3*(A + B*x^2))/(a + b*x^2)^(5/2), x, 3, (a*(A*b - a*B))/(3*b^3*(a + b*x^2)^(3/2)) - (A*b - 2*a*B)/(b^3*sqrt(a + b*x^2)) + (B*sqrt(a + b*x^2))/b^3]
    @test_int [(x^2*(A + B*x^2))/(a + b*x^2)^(5/2), x, 4, ((A*b - a*B)*x^3)/(3*a*b*(a + b*x^2)^(3/2)) - (B*x)/(b^2*sqrt(a + b*x^2)) + (B*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/b^(5/2)]
    @test_int [(x*(A + B*x^2))/(a + b*x^2)^(5/2), x, 3, -(A*b - a*B)/(3*b^2*(a + b*x^2)^(3/2)) - B/(b^2*sqrt(a + b*x^2))]
    @test_int [(A + B*x^2)/(a + b*x^2)^(5/2), x, 2, (2*A*x)/(3*a^2*sqrt(a + b*x^2)) + (x*(A + B*x^2))/(3*a*(a + b*x^2)^(3/2))]
    @test_int [(A + B*x^2)/(x*(a + b*x^2)^(5/2)), x, 5, (A*b - a*B)/(3*a*b*(a + b*x^2)^(3/2)) + A/(a^2*sqrt(a + b*x^2)) - (A*atanh(sqrt(a + b*x^2)/sqrt(a)))/a^(5/2)]
    @test_int [(A + B*x^2)/(x^2*(a + b*x^2)^(5/2)), x, 3, -(A/(a*x*(a + b*x^2)^(3/2))) - ((4*A*b - a*B)*x)/(3*a^2*(a + b*x^2)^(3/2)) - (2*(4*A*b - a*B)*x)/(3*a^3*sqrt(a + b*x^2))]
    @test_int [(A + B*x^2)/(x^3*(a + b*x^2)^(5/2)), x, 6, -(5*A*b - 2*a*B)/(6*a^2*(a + b*x^2)^(3/2)) - A/(2*a*x^2*(a + b*x^2)^(3/2)) - (5*A*b - 2*a*B)/(2*a^3*sqrt(a + b*x^2)) + ((5*A*b - 2*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(2*a^(7/2))]
    @test_int [(A + B*x^2)/(x^4*(a + b*x^2)^(5/2)), x, 4, -(A/(3*a*x^3*(a + b*x^2)^(3/2))) + (2*A*b - a*B)/(a^2*x*(a + b*x^2)^(3/2)) + (4*b*(2*A*b - a*B)*x)/(3*a^3*(a + b*x^2)^(3/2)) + (8*b*(2*A*b - a*B)*x)/(3*a^4*sqrt(a + b*x^2))]
    @test_int [(A + B*x^2)/(x^5*(a + b*x^2)^(5/2)), x, 7, (5*b*(7*A*b - 4*a*B))/(24*a^3*(a + b*x^2)^(3/2)) - A/(4*a*x^4*(a + b*x^2)^(3/2)) + (7*A*b - 4*a*B)/(8*a^2*x^2*(a + b*x^2)^(3/2)) + (5*b*(7*A*b - 4*a*B))/(8*a^4*sqrt(a + b*x^2)) - (5*b*(7*A*b - 4*a*B)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(8*a^(9/2))]
    @test_int [(A + B*x^2)/(x^6*(a + b*x^2)^(5/2)), x, 5, -(A/(5*a*x^5*(a + b*x^2)^(3/2))) + (8*A*b - 5*a*B)/(15*a^2*x^3*(a + b*x^2)^(3/2)) - (2*b*(8*A*b - 5*a*B))/(5*a^3*x*(a + b*x^2)^(3/2)) - (8*b^2*(8*A*b - 5*a*B)*x)/(15*a^4*(a + b*x^2)^(3/2)) - (16*b^2*(8*A*b - 5*a*B)*x)/(15*a^5*sqrt(a + b*x^2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^2)^(p/2)*(c+d*x^2)^2=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^5*(a + b*x^2)^2*sqrt(c + d*x^2), x, 3, (c^2*(b*c - a*d)^2*(c + d*x^2)^(3/2))/(3*d^5) - (2*c*(b*c - a*d)*(2*b*c - a*d)*(c + d*x^2)^(5/2))/(5*d^5) + ((6*b^2*c^2 - 6*a*b*c*d + a^2*d^2)*(c + d*x^2)^(7/2))/(7*d^5) - (2*b*(2*b*c - a*d)*(c + d*x^2)^(9/2))/(9*d^5) + (b^2*(c + d*x^2)^(11/2))/(11*d^5)]
    @test_int [x^3*(a + b*x^2)^2*sqrt(c + d*x^2), x, 3, -((c*(b*c - a*d)^2*(c + d*x^2)^(3/2))/(3*d^4)) + ((b*c - a*d)*(3*b*c - a*d)*(c + d*x^2)^(5/2))/(5*d^4) - (b*(3*b*c - 2*a*d)*(c + d*x^2)^(7/2))/(7*d^4) + (b^2*(c + d*x^2)^(9/2))/(9*d^4)]
    @test_int [x^1*(a + b*x^2)^2*sqrt(c + d*x^2), x, 3, ((b*c - a*d)^2*(c + d*x^2)^(3/2))/(3*d^3) - (2*b*(b*c - a*d)*(c + d*x^2)^(5/2))/(5*d^3) + (b^2*(c + d*x^2)^(7/2))/(7*d^3)]
    @test_int [((a + b*x^2)^2*sqrt(c + d*x^2))/x^1, x, 6, a^2*sqrt(c + d*x^2) - (b*(b*c - 2*a*d)*(c + d*x^2)^(3/2))/(3*d^2) + (b^2*(c + d*x^2)^(5/2))/(5*d^2) - a^2*sqrt(c)*atanh(sqrt(c + d*x^2)/sqrt(c))]
    @test_int [((a + b*x^2)^2*sqrt(c + d*x^2))/x^3, x, 6, (a*(4*b*c + a*d)*sqrt(c + d*x^2))/(2*c) + (b^2*(c + d*x^2)^(3/2))/(3*d) - (a^2*(c + d*x^2)^(3/2))/(2*c*x^2) - (a*(4*b*c + a*d)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(2*sqrt(c))]
    @test_int [((a + b*x^2)^2*sqrt(c + d*x^2))/x^5, x, 6, ((8*b^2*c^2 + a*d*(8*b*c - a*d))*sqrt(c + d*x^2))/(8*c^2) - (a^2*(c + d*x^2)^(3/2))/(4*c*x^4) - (a*(8*b*c - a*d)*(c + d*x^2)^(3/2))/(8*c^2*x^2) - ((8*b^2*c^2 + a*d*(8*b*c - a*d))*atanh(sqrt(c + d*x^2)/sqrt(c)))/(8*c^(3/2))]
    @test_int [((a + b*x^2)^2*sqrt(c + d*x^2))/x^7, x, 6, -(((8*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*sqrt(c + d*x^2))/(16*c^2*x^2)) - (a^2*(c + d*x^2)^(3/2))/(6*c*x^6) - (a*(4*b*c - a*d)*(c + d*x^2)^(3/2))/(8*c^2*x^4) - (d*(8*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(16*c^(5/2))]

    @test_int [x^2*(a + b*x^2)^2*sqrt(c + d*x^2), x, 6, (c*(16*a^2*d^2 + b*c*(5*b*c - 16*a*d))*x*sqrt(c + d*x^2))/(128*d^3) + ((16*a^2*d^2 + b*c*(5*b*c - 16*a*d))*x^3*sqrt(c + d*x^2))/(64*d^2) - (b*(5*b*c - 16*a*d)*x^3*(c + d*x^2)^(3/2))/(48*d^2) + (b^2*x^5*(c + d*x^2)^(3/2))/(8*d) - (c^2*(16*a^2*d^2 + b*c*(5*b*c - 16*a*d))*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(128*d^(7/2))]
    @test_int [(a + b*x^2)^2*sqrt(c + d*x^2), x, 5, ((b^2*c^2 - 4*a*b*c*d + 8*a^2*d^2)*x*sqrt(c + d*x^2))/(16*d^2) - (b*(3*b*c - 8*a*d)*x*(c + d*x^2)^(3/2))/(24*d^2) + (b*x*(a + b*x^2)*(c + d*x^2)^(3/2))/(6*d) + (c*(b^2*c^2 - 4*a*b*c*d + 8*a^2*d^2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(16*d^(5/2))]
    @test_int [((a + b*x^2)^2*sqrt(c + d*x^2))/x^2, x, 5, -(((b^2*c^2 - 8*a*d*(b*c + a*d))*x*sqrt(c + d*x^2))/(8*c*d)) - (a^2*(c + d*x^2)^(3/2))/(c*x) + (b^2*x*(c + d*x^2)^(3/2))/(4*d) - ((b^2*c^2 - 8*a*d*(b*c + a*d))*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(8*d^(3/2))]
    @test_int [((a + b*x^2)^2*sqrt(c + d*x^2))/x^4, x, 5, (b*(b*c + 4*a*d)*x*sqrt(c + d*x^2))/(2*c) - (a^2*(c + d*x^2)^(3/2))/(3*c*x^3) - (2*a*b*(c + d*x^2)^(3/2))/(c*x) + (b*(b*c + 4*a*d)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(2*sqrt(d))]
    @test_int [((a + b*x^2)^2*sqrt(c + d*x^2))/x^6, x, 5, -((b^2*sqrt(c + d*x^2))/x) - (a^2*(c + d*x^2)^(3/2))/(5*c*x^5) - (2*a*(5*b*c - a*d)*(c + d*x^2)^(3/2))/(15*c^2*x^3) + b^2*sqrt(d)*atanh((sqrt(d)*x)/sqrt(c + d*x^2))]
    @test_int [((a + b*x^2)^2*sqrt(c + d*x^2))/x^8, x, 3, -((a^2*(c + d*x^2)^(3/2))/(7*c*x^7)) - (2*a*(7*b*c - 2*a*d)*(c + d*x^2)^(3/2))/(35*c^2*x^5) - ((35*b^2*c^2 - 4*a*d*(7*b*c - 2*a*d))*(c + d*x^2)^(3/2))/(105*c^3*x^3)]
    @test_int [((a + b*x^2)^2*sqrt(c + d*x^2))/x^10, x, 4, -((a^2*(c + d*x^2)^(3/2))/(9*c*x^9)) - (2*a*(3*b*c - a*d)*(c + d*x^2)^(3/2))/(21*c^2*x^7) - ((21*b^2*c^2 - 8*a*d*(3*b*c - a*d))*(c + d*x^2)^(3/2))/(105*c^3*x^5) + (2*d*(21*b^2*c^2 - 8*a*d*(3*b*c - a*d))*(c + d*x^2)^(3/2))/(315*c^4*x^3)]
    @test_int [((a + b*x^2)^2*sqrt(c + d*x^2))/x^12, x, 5, -((a^2*(c + d*x^2)^(3/2))/(11*c*x^11)) - (2*a*(11*b*c - 4*a*d)*(c + d*x^2)^(3/2))/(99*c^2*x^9) - ((33*b^2*c^2 - 4*a*d*(11*b*c - 4*a*d))*(c + d*x^2)^(3/2))/(231*c^3*x^7) + (4*d*(33*b^2*c^2 - 4*a*d*(11*b*c - 4*a*d))*(c + d*x^2)^(3/2))/(1155*c^4*x^5) - (8*d^2*(33*b^2*c^2 - 4*a*d*(11*b*c - 4*a*d))*(c + d*x^2)^(3/2))/(3465*c^5*x^3)]

    @test_int [x^4*(a + b*x^2)^2*(c + d*x^2)^(3/2), x, 8, -((c^3*(24*a^2*d^2 + b*c*(7*b*c - 24*a*d))*x*sqrt(c + d*x^2))/(1024*d^4)) + (c^2*(24*a^2*d^2 + b*c*(7*b*c - 24*a*d))*x^3*sqrt(c + d*x^2))/(1536*d^3) + (c*(24*a^2*d^2 + b*c*(7*b*c - 24*a*d))*x^5*sqrt(c + d*x^2))/(384*d^2) + ((24*a^2*d^2 + b*c*(7*b*c - 24*a*d))*x^5*(c + d*x^2)^(3/2))/(192*d^2) - (b*(7*b*c - 24*a*d)*x^5*(c + d*x^2)^(5/2))/(120*d^2) + (b^2*x^7*(c + d*x^2)^(5/2))/(12*d) + (c^4*(24*a^2*d^2 + b*c*(7*b*c - 24*a*d))*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(1024*d^(9/2))]
    @test_int [x^3*(a + b*x^2)^2*(c + d*x^2)^(3/2), x, 3, -((c*(b*c - a*d)^2*(c + d*x^2)^(5/2))/(5*d^4)) + ((b*c - a*d)*(3*b*c - a*d)*(c + d*x^2)^(7/2))/(7*d^4) - (b*(3*b*c - 2*a*d)*(c + d*x^2)^(9/2))/(9*d^4) + (b^2*(c + d*x^2)^(11/2))/(11*d^4)]
    @test_int [x^2*(a + b*x^2)^2*(c + d*x^2)^(3/2), x, 7, (c^2*(16*a^2*d^2 + 3*b*c*(b*c - 4*a*d))*x*sqrt(c + d*x^2))/(256*d^3) + (c*(16*a^2*d^2 + 3*b*c*(b*c - 4*a*d))*x^3*sqrt(c + d*x^2))/(128*d^2) + ((16*a^2*d^2 + 3*b*c*(b*c - 4*a*d))*x^3*(c + d*x^2)^(3/2))/(96*d^2) - (b*(b*c - 4*a*d)*x^3*(c + d*x^2)^(5/2))/(16*d^2) + (b^2*x^5*(c + d*x^2)^(5/2))/(10*d) - (c^3*(16*a^2*d^2 + 3*b*c*(b*c - 4*a*d))*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(256*d^(7/2))]
    @test_int [x*(a + b*x^2)^2*(c + d*x^2)^(3/2), x, 3, ((b*c - a*d)^2*(c + d*x^2)^(5/2))/(5*d^3) - (2*b*(b*c - a*d)*(c + d*x^2)^(7/2))/(7*d^3) + (b^2*(c + d*x^2)^(9/2))/(9*d^3)]
    @test_int [(a + b*x^2)^2*(c + d*x^2)^(3/2), x, 6, (c*(3*b^2*c^2 - 16*a*b*c*d + 48*a^2*d^2)*x*sqrt(c + d*x^2))/(128*d^2) + ((3*b^2*c^2 - 16*a*b*c*d + 48*a^2*d^2)*x*(c + d*x^2)^(3/2))/(192*d^2) - (b*(3*b*c - 10*a*d)*x*(c + d*x^2)^(5/2))/(48*d^2) + (b*x*(a + b*x^2)*(c + d*x^2)^(5/2))/(8*d) + (c^2*(3*b^2*c^2 - 16*a*b*c*d + 48*a^2*d^2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(128*d^(5/2))]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^(3/2))/x, x, 7, a^2*c*sqrt(c + d*x^2) + (1/3)*a^2*(c + d*x^2)^(3/2) - (b*(b*c - 2*a*d)*(c + d*x^2)^(5/2))/(5*d^2) + (b^2*(c + d*x^2)^(7/2))/(7*d^2) - a^2*c^(3/2)*atanh(sqrt(c + d*x^2)/sqrt(c))]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^2, x, 6, -(((b^2*c^2 - 12*a*d*(b*c + 2*a*d))*x*sqrt(c + d*x^2))/(16*d)) - ((b^2*c^2 - 12*a*d*(b*c + 2*a*d))*x*(c + d*x^2)^(3/2))/(24*c*d) - (a^2*(c + d*x^2)^(5/2))/(c*x) + (b^2*x*(c + d*x^2)^(5/2))/(6*d) - (c*(b^2*c^2 - 12*a*d*(b*c + 2*a*d))*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(16*d^(3/2))]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^3, x, 7, (1/2)*a*(4*b*c + 3*a*d)*sqrt(c + d*x^2) + (a*(4*b*c + 3*a*d)*(c + d*x^2)^(3/2))/(6*c) + (b^2*(c + d*x^2)^(5/2))/(5*d) - (a^2*(c + d*x^2)^(5/2))/(2*c*x^2) - (1/2)*a*sqrt(c)*(4*b*c + 3*a*d)*atanh(sqrt(c + d*x^2)/sqrt(c))]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^4, x, 6, ((3*b^2*c^2 + 8*a*d*(3*b*c + a*d))*x*sqrt(c + d*x^2))/(8*c) + ((3*b^2*c^2 + 8*a*d*(3*b*c + a*d))*x*(c + d*x^2)^(3/2))/(12*c^2) - (a^2*(c + d*x^2)^(5/2))/(3*c*x^3) - (2*a*(3*b*c + a*d)*(c + d*x^2)^(5/2))/(3*c^2*x) + ((3*b^2*c^2 + 8*a*d*(3*b*c + a*d))*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(8*sqrt(d))]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^5, x, 7, ((8*b^2*c^2 + 3*a*d*(8*b*c + a*d))*sqrt(c + d*x^2))/(8*c) + ((8*b^2*c^2 + 3*a*d*(8*b*c + a*d))*(c + d*x^2)^(3/2))/(24*c^2) - (a^2*(c + d*x^2)^(5/2))/(4*c*x^4) - (a*(8*b*c + a*d)*(c + d*x^2)^(5/2))/(8*c^2*x^2) - ((8*b^2*c^2 + 3*a*d*(8*b*c + a*d))*atanh(sqrt(c + d*x^2)/sqrt(c)))/(8*sqrt(c))]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^6, x, 6, (b*d*(3*b*c + 4*a*d)*x*sqrt(c + d*x^2))/(2*c) - (b*(3*b*c + 4*a*d)*(c + d*x^2)^(3/2))/(3*c*x) - (a^2*(c + d*x^2)^(5/2))/(5*c*x^5) - (2*a*b*(c + d*x^2)^(5/2))/(3*c*x^3) + (1/2)*b*sqrt(d)*(3*b*c + 4*a*d)*atanh((sqrt(d)*x)/sqrt(c + d*x^2))]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^7, x, 7, (d*(24*b^2*c^2 + a*d*(12*b*c - a*d))*sqrt(c + d*x^2))/(16*c^2) - ((24*b^2*c^2 + a*d*(12*b*c - a*d))*(c + d*x^2)^(3/2))/(48*c^2*x^2) - (a^2*(c + d*x^2)^(5/2))/(6*c*x^6) - (a*(12*b*c - a*d)*(c + d*x^2)^(5/2))/(24*c^2*x^4) - (d*(24*b^2*c^2 + a*d*(12*b*c - a*d))*atanh(sqrt(c + d*x^2)/sqrt(c)))/(16*c^(3/2))]

    @test_int [x^3*(a + b*x^2)^2*(c + d*x^2)^(5/2), x, 3, -((c*(b*c - a*d)^2*(c + d*x^2)^(7/2))/(7*d^4)) + ((b*c - a*d)*(3*b*c - a*d)*(c + d*x^2)^(9/2))/(9*d^4) - (b*(3*b*c - 2*a*d)*(c + d*x^2)^(11/2))/(11*d^4) + (b^2*(c + d*x^2)^(13/2))/(13*d^4)]
    @test_int [x^2*(a + b*x^2)^2*(c + d*x^2)^(5/2), x, 8, (c^3*(40*a^2*d^2 + b*c*(5*b*c - 24*a*d))*x*sqrt(c + d*x^2))/(1024*d^3) + (c^2*(40*a^2*d^2 + b*c*(5*b*c - 24*a*d))*x^3*sqrt(c + d*x^2))/(512*d^2) + (c*(40*a^2*d^2 + b*c*(5*b*c - 24*a*d))*x^3*(c + d*x^2)^(3/2))/(384*d^2) + ((40*a^2*d^2 + b*c*(5*b*c - 24*a*d))*x^3*(c + d*x^2)^(5/2))/(320*d^2) - (b*(5*b*c - 24*a*d)*x^3*(c + d*x^2)^(7/2))/(120*d^2) + (b^2*x^5*(c + d*x^2)^(7/2))/(12*d) - (c^4*(40*a^2*d^2 + b*c*(5*b*c - 24*a*d))*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(1024*d^(7/2))]
    @test_int [x*(a + b*x^2)^2*(c + d*x^2)^(5/2), x, 3, ((b*c - a*d)^2*(c + d*x^2)^(7/2))/(7*d^3) - (2*b*(b*c - a*d)*(c + d*x^2)^(9/2))/(9*d^3) + (b^2*(c + d*x^2)^(11/2))/(11*d^3)]
    @test_int [(a + b*x^2)^2*(c + d*x^2)^(5/2), x, 7, (c^2*(3*b^2*c^2 - 20*a*b*c*d + 80*a^2*d^2)*x*sqrt(c + d*x^2))/(256*d^2) + (c*(3*b^2*c^2 - 20*a*b*c*d + 80*a^2*d^2)*x*(c + d*x^2)^(3/2))/(384*d^2) + ((3*b^2*c^2 - 20*a*b*c*d + 80*a^2*d^2)*x*(c + d*x^2)^(5/2))/(480*d^2) - (3*b*(b*c - 4*a*d)*x*(c + d*x^2)^(7/2))/(80*d^2) + (b*x*(a + b*x^2)*(c + d*x^2)^(7/2))/(10*d) + (c^3*(3*b^2*c^2 - 20*a*b*c*d + 80*a^2*d^2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(256*d^(5/2))]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^(5/2))/x, x, 8, a^2*c^2*sqrt(c + d*x^2) + (1/3)*a^2*c*(c + d*x^2)^(3/2) + (1/5)*a^2*(c + d*x^2)^(5/2) - (b*(b*c - 2*a*d)*(c + d*x^2)^(7/2))/(7*d^2) + (b^2*(c + d*x^2)^(9/2))/(9*d^2) - a^2*c^(5/2)*atanh(sqrt(c + d*x^2)/sqrt(c))]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^2, x, 7, -((5*c*(b^2*c^2 - 16*a*d*(b*c + 3*a*d))*x*sqrt(c + d*x^2))/(128*d)) - (5*(b^2*c^2 - 16*a*d*(b*c + 3*a*d))*x*(c + d*x^2)^(3/2))/(192*d) - ((b^2*c^2 - 16*a*d*(b*c + 3*a*d))*x*(c + d*x^2)^(5/2))/(48*c*d) - (a^2*(c + d*x^2)^(7/2))/(c*x) + (b^2*x*(c + d*x^2)^(7/2))/(8*d) - (5*c^2*(b^2*c^2 - 16*a*d*(b*c + 3*a*d))*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(128*d^(3/2))]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^3, x, 8, (1/2)*a*c*(4*b*c + 5*a*d)*sqrt(c + d*x^2) + (1/6)*a*(4*b*c + 5*a*d)*(c + d*x^2)^(3/2) + (a*(4*b*c + 5*a*d)*(c + d*x^2)^(5/2))/(10*c) + (b^2*(c + d*x^2)^(7/2))/(7*d) - (a^2*(c + d*x^2)^(7/2))/(2*c*x^2) - (1/2)*a*c^(3/2)*(4*b*c + 5*a*d)*atanh(sqrt(c + d*x^2)/sqrt(c))]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^4, x, 7, (5/16)*(b^2*c^2 + 4*a*d*(3*b*c + 2*a*d))*x*sqrt(c + d*x^2) + (5*(b^2*c^2 + 4*a*d*(3*b*c + 2*a*d))*x*(c + d*x^2)^(3/2))/(24*c) + ((b^2*c^2 + 4*a*d*(3*b*c + 2*a*d))*x*(c + d*x^2)^(5/2))/(6*c^2) - (a^2*(c + d*x^2)^(7/2))/(3*c*x^3) - (2*a*(3*b*c + 2*a*d)*(c + d*x^2)^(7/2))/(3*c^2*x) + (5*c*(b^2*c^2 + 4*a*d*(3*b*c + 2*a*d))*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(16*sqrt(d))]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^5, x, 8, (1/8)*(8*b^2*c^2 + 5*a*d*(8*b*c + 3*a*d))*sqrt(c + d*x^2) + ((8*b^2*c^2 + 5*a*d*(8*b*c + 3*a*d))*(c + d*x^2)^(3/2))/(24*c) + ((8*b^2*c^2 + 5*a*d*(8*b*c + 3*a*d))*(c + d*x^2)^(5/2))/(40*c^2) - (a^2*(c + d*x^2)^(7/2))/(4*c*x^4) - (a*(8*b*c + 3*a*d)*(c + d*x^2)^(7/2))/(8*c^2*x^2) - (1/8)*sqrt(c)*(8*b^2*c^2 + 5*a*d*(8*b*c + 3*a*d))*atanh(sqrt(c + d*x^2)/sqrt(c))]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^6, x, 7, (d*(15*b^2*c^2 + 8*a*d*(5*b*c + a*d))*x*sqrt(c + d*x^2))/(8*c) + (d*(15*b^2*c^2 + 8*a*d*(5*b*c + a*d))*x*(c + d*x^2)^(3/2))/(12*c^2) - ((15*b^2*c^2 + 8*a*d*(5*b*c + a*d))*(c + d*x^2)^(5/2))/(15*c^2*x) - (a^2*(c + d*x^2)^(7/2))/(5*c*x^5) - (2*a*(5*b*c + a*d)*(c + d*x^2)^(7/2))/(15*c^2*x^3) + (1/8)*sqrt(d)*(15*b^2*c^2 + 8*a*d*(5*b*c + a*d))*atanh((sqrt(d)*x)/sqrt(c + d*x^2))]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^7, x, 8, (5*d*(8*b^2*c^2 + a*d*(12*b*c + a*d))*sqrt(c + d*x^2))/(16*c) + (5*d*(8*b^2*c^2 + a*d*(12*b*c + a*d))*(c + d*x^2)^(3/2))/(48*c^2) - ((8*b^2*c^2 + a*d*(12*b*c + a*d))*(c + d*x^2)^(5/2))/(16*c^2*x^2) - (a^2*(c + d*x^2)^(7/2))/(6*c*x^6) - (a*(12*b*c + a*d)*(c + d*x^2)^(7/2))/(24*c^2*x^4) - (5*d*(8*b^2*c^2 + a*d*(12*b*c + a*d))*atanh(sqrt(c + d*x^2)/sqrt(c)))/(16*sqrt(c))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(x^4*(a + b*x^2)^2)/sqrt(c + d*x^2), x, 6, -((c*(48*a^2*d^2 + 5*b*c*(7*b*c - 16*a*d))*x*sqrt(c + d*x^2))/(128*d^4)) + ((48*a^2*d^2 + 5*b*c*(7*b*c - 16*a*d))*x^3*sqrt(c + d*x^2))/(192*d^3) - (b*(7*b*c - 16*a*d)*x^5*sqrt(c + d*x^2))/(48*d^2) + (b^2*x^7*sqrt(c + d*x^2))/(8*d) + (c^2*(48*a^2*d^2 + 5*b*c*(7*b*c - 16*a*d))*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(128*d^(9/2))]
    @test_int [(x^3*(a + b*x^2)^2)/sqrt(c + d*x^2), x, 3, -((c*(b*c - a*d)^2*sqrt(c + d*x^2))/d^4) + ((b*c - a*d)*(3*b*c - a*d)*(c + d*x^2)^(3/2))/(3*d^4) - (b*(3*b*c - 2*a*d)*(c + d*x^2)^(5/2))/(5*d^4) + (b^2*(c + d*x^2)^(7/2))/(7*d^4)]
    @test_int [(x^2*(a + b*x^2)^2)/sqrt(c + d*x^2), x, 5, ((8*a^2*d^2 + b*c*(5*b*c - 12*a*d))*x*sqrt(c + d*x^2))/(16*d^3) - (b*(5*b*c - 12*a*d)*x^3*sqrt(c + d*x^2))/(24*d^2) + (b^2*x^5*sqrt(c + d*x^2))/(6*d) - (c*(8*a^2*d^2 + b*c*(5*b*c - 12*a*d))*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(16*d^(7/2))]
    @test_int [(x*(a + b*x^2)^2)/sqrt(c + d*x^2), x, 3, ((b*c - a*d)^2*sqrt(c + d*x^2))/d^3 - (2*b*(b*c - a*d)*(c + d*x^2)^(3/2))/(3*d^3) + (b^2*(c + d*x^2)^(5/2))/(5*d^3)]
    @test_int [(a + b*x^2)^2/sqrt(c + d*x^2), x, 4, (-3*b*(b*c - 2*a*d)*x*sqrt(c + d*x^2))/(8*d^2) + (b*x*(a + b*x^2)*sqrt(c + d*x^2))/(4*d) + ((3*b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(8*d^(5/2))]
    @test_int [(a + b*x^2)^2/(x*sqrt(c + d*x^2)), x, 5, -((b*(b*c - 2*a*d)*sqrt(c + d*x^2))/d^2) + (b^2*(c + d*x^2)^(3/2))/(3*d^2) - (a^2*atanh(sqrt(c + d*x^2)/sqrt(c)))/sqrt(c)]
    @test_int [(a + b*x^2)^2/(x^2*sqrt(c + d*x^2)), x, 4, -((a^2*sqrt(c + d*x^2))/(c*x)) + (b^2*x*sqrt(c + d*x^2))/(2*d) - (b*(b*c - 4*a*d)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(2*d^(3/2))]
    @test_int [(a + b*x^2)^2/(x^3*sqrt(c + d*x^2)), x, 5, (b^2*sqrt(c + d*x^2))/d - (a^2*sqrt(c + d*x^2))/(2*c*x^2) - (a*(4*b*c - a*d)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(2*c^(3/2))]
    @test_int [(a + b*x^2)^2/(x^4*sqrt(c + d*x^2)), x, 4, -((a^2*sqrt(c + d*x^2))/(3*c*x^3)) - (2*a*(3*b*c - a*d)*sqrt(c + d*x^2))/(3*c^2*x) + (b^2*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/sqrt(d)]
    @test_int [(a + b*x^2)^2/(x^5*sqrt(c + d*x^2)), x, 5, -((a^2*sqrt(c + d*x^2))/(4*c*x^4)) - (a*(8*b*c - 3*a*d)*sqrt(c + d*x^2))/(8*c^2*x^2) - ((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(8*c^(5/2))]
    @test_int [(a + b*x^2)^2/(x^6*sqrt(c + d*x^2)), x, 3, -((a^2*sqrt(c + d*x^2))/(5*c*x^5)) - (2*a*(5*b*c - 2*a*d)*sqrt(c + d*x^2))/(15*c^2*x^3) - ((15*b^2*c^2 - 4*a*d*(5*b*c - 2*a*d))*sqrt(c + d*x^2))/(15*c^3*x)]
    @test_int [(a + b*x^2)^2/(x^7*sqrt(c + d*x^2)), x, 6, -((a^2*sqrt(c + d*x^2))/(6*c*x^6)) - (a*(12*b*c - 5*a*d)*sqrt(c + d*x^2))/(24*c^2*x^4) - ((8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*sqrt(c + d*x^2))/(16*c^3*x^2) + (d*(8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(16*c^(7/2))]

    @test_int [(x^4*(a + b*x^2)^2)/(c + d*x^2)^(3/2), x, 6, ((b*c - a*d)^2*x^5)/(c*d^2*sqrt(c + d*x^2)) + ((35*b^2*c^2 - 60*a*b*c*d + 24*a^2*d^2)*x*sqrt(c + d*x^2))/(16*d^4) - ((35*b^2*c^2 - 60*a*b*c*d + 24*a^2*d^2)*x^3*sqrt(c + d*x^2))/(24*c*d^3) + (b^2*x^5*sqrt(c + d*x^2))/(6*d^2) - (c*(35*b^2*c^2 - 60*a*b*c*d + 24*a^2*d^2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(16*d^(9/2))]
    @test_int [(x^3*(a + b*x^2)^2)/(c + d*x^2)^(3/2), x, 3, (c*(b*c - a*d)^2)/(d^4*sqrt(c + d*x^2)) + ((b*c - a*d)*(3*b*c - a*d)*sqrt(c + d*x^2))/d^4 - (b*(3*b*c - 2*a*d)*(c + d*x^2)^(3/2))/(3*d^4) + (b^2*(c + d*x^2)^(5/2))/(5*d^4)]
    @test_int [(x^2*(a + b*x^2)^2)/(c + d*x^2)^(3/2), x, 5, ((b*c - a*d)^2*x^3)/(c*d^2*sqrt(c + d*x^2)) - ((15*b^2*c^2 - 24*a*b*c*d + 8*a^2*d^2)*x*sqrt(c + d*x^2))/(8*c*d^3) + (b^2*x^3*sqrt(c + d*x^2))/(4*d^2) + ((15*b^2*c^2 - 24*a*b*c*d + 8*a^2*d^2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(8*d^(7/2))]
    @test_int [(x*(a + b*x^2)^2)/(c + d*x^2)^(3/2), x, 3, -((b*c - a*d)^2/(d^3*sqrt(c + d*x^2))) - (2*b*(b*c - a*d)*sqrt(c + d*x^2))/d^3 + (b^2*(c + d*x^2)^(3/2))/(3*d^3)]
    @test_int [(a + b*x^2)^2/(c + d*x^2)^(3/2), x, 4, -(((b*c - a*d)*x*(a + b*x^2))/(c*d*sqrt(c + d*x^2))) + (b*(3*b*c - 2*a*d)*x*sqrt(c + d*x^2))/(2*c*d^2) - (b*(3*b*c - 4*a*d)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(2*d^(5/2))]
    @test_int [(a + b*x^2)^2/(x*(c + d*x^2)^(3/2)), x, 5, (b*c - a*d)^2/(c*d^2*sqrt(c + d*x^2)) + (b^2*sqrt(c + d*x^2))/d^2 - (a^2*atanh(sqrt(c + d*x^2)/sqrt(c)))/c^(3/2)]
    @test_int [(a + b*x^2)^2/(x^2*(c + d*x^2)^(3/2)), x, 4, -(a^2/(c*x*sqrt(c + d*x^2))) - ((b^2*c^2 - 2*a*d*(b*c - a*d))*x)/(c^2*d*sqrt(c + d*x^2)) + (b^2*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/d^(3/2)]
    @test_int [(a + b*x^2)^2/(x^3*(c + d*x^2)^(3/2)), x, 5, (4*a*b - (2*b^2*c)/d - (3*a^2*d)/c)/(2*c*sqrt(c + d*x^2)) - a^2/(2*c*x^2*sqrt(c + d*x^2)) - (a*(4*b*c - 3*a*d)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(2*c^(5/2))]
    @test_int [(a + b*x^2)^2/(x^4*(c + d*x^2)^(3/2)), x, 3, -(a^2/(3*c*x^3*sqrt(c + d*x^2))) - (2*a*(3*b*c - 2*a*d))/(3*c^2*x*sqrt(c + d*x^2)) + ((3*b^2*c^2 - 4*a*d*(3*b*c - 2*a*d))*x)/(3*c^3*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^2/(x^5*(c + d*x^2)^(3/2)), x, 6, (8*b^2*c^2 - 3*a*d*(8*b*c - 5*a*d))/(8*c^3*sqrt(c + d*x^2)) - a^2/(4*c*x^4*sqrt(c + d*x^2)) - (a*(8*b*c - 5*a*d))/(8*c^2*x^2*sqrt(c + d*x^2)) - ((8*b^2*c^2 - 3*a*d*(8*b*c - 5*a*d))*atanh(sqrt(c + d*x^2)/sqrt(c)))/(8*c^(7/2))]
    @test_int [(a + b*x^2)^2/(x^6*(c + d*x^2)^(3/2)), x, 4, -(a^2/(5*c*x^5*sqrt(c + d*x^2))) - (2*a*(5*b*c - 3*a*d))/(15*c^2*x^3*sqrt(c + d*x^2)) - (15*b^2*c^2 - 8*a*d*(5*b*c - 3*a*d))/(15*c^3*x*sqrt(c + d*x^2)) - (2*d*(15*b^2*c^2 - 8*a*d*(5*b*c - 3*a*d))*x)/(15*c^4*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^2/(x^7*(c + d*x^2)^(3/2)), x, 7, -((d*(24*b^2*c^2 - 5*a*d*(12*b*c - 7*a*d)))/(16*c^4*sqrt(c + d*x^2))) - a^2/(6*c*x^6*sqrt(c + d*x^2)) - (a*(12*b*c - 7*a*d))/(24*c^2*x^4*sqrt(c + d*x^2)) - (24*b^2*c^2 - 5*a*d*(12*b*c - 7*a*d))/(48*c^3*x^2*sqrt(c + d*x^2)) + (d*(24*b^2*c^2 - 5*a*d*(12*b*c - 7*a*d))*atanh(sqrt(c + d*x^2)/sqrt(c)))/(16*c^(9/2))]

    @test_int [(x^4*(a + b*x^2)^2)/(c + d*x^2)^(5/2), x, 6, ((b*c - a*d)^2*x^5)/(3*c*d^2*(c + d*x^2)^(3/2)) + ((35*b^2*c^2 - 40*a*b*c*d + 8*a^2*d^2)*x^3)/(12*c*d^3*sqrt(c + d*x^2)) + (b^2*x^5)/(4*d^2*sqrt(c + d*x^2)) - ((35*b^2*c^2 - 40*a*b*c*d + 8*a^2*d^2)*x*sqrt(c + d*x^2))/(8*c*d^4) + ((35*b^2*c^2 - 40*a*b*c*d + 8*a^2*d^2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(8*d^(9/2))]
    @test_int [(x^3*(a + b*x^2)^2)/(c + d*x^2)^(5/2), x, 3, (c*(b*c - a*d)^2)/(3*d^4*(c + d*x^2)^(3/2)) - ((b*c - a*d)*(3*b*c - a*d))/(d^4*sqrt(c + d*x^2)) - (b*(3*b*c - 2*a*d)*sqrt(c + d*x^2))/d^4 + (b^2*(c + d*x^2)^(3/2))/(3*d^4)]
    @test_int [(x^2*(a + b*x^2)^2)/(c + d*x^2)^(5/2), x, 5, ((b*c - a*d)^2*x^3)/(3*c*d^2*(c + d*x^2)^(3/2)) + (2*b*(b*c - a*d)*x)/(d^3*sqrt(c + d*x^2)) + (b^2*x*sqrt(c + d*x^2))/(2*d^3) - (b*(5*b*c - 4*a*d)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(2*d^(7/2))]
    @test_int [(x*(a + b*x^2)^2)/(c + d*x^2)^(5/2), x, 3, -((b*c - a*d)^2/(3*d^3*(c + d*x^2)^(3/2))) + (2*b*(b*c - a*d))/(d^3*sqrt(c + d*x^2)) + (b^2*sqrt(c + d*x^2))/d^3]
    @test_int [(a + b*x^2)^2/(c + d*x^2)^(5/2), x, 4, -(((b*c - a*d)*x*(a + b*x^2))/(3*c*d*(c + d*x^2)^(3/2))) - ((b*c - a*d)*(3*b*c + 2*a*d)*x)/(3*c^2*d^2*sqrt(c + d*x^2)) + (b^2*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/d^(5/2)]
    @test_int [(a + b*x^2)^2/(x*(c + d*x^2)^(5/2)), x, 5, (b*c - a*d)^2/(3*c*d^2*(c + d*x^2)^(3/2)) + (a^2/c^2 - b^2/d^2)/sqrt(c + d*x^2) - (a^2*atanh(sqrt(c + d*x^2)/sqrt(c)))/c^(5/2)]
    @test_int [(a + b*x^2)^2/(x^2*(c + d*x^2)^(5/2)), x, 3, -(a^2/(c*x*(c + d*x^2)^(3/2))) + (x*(2*a*(b*c - 2*a*d) + b^2*c*x^2))/(3*c^2*(c + d*x^2)^(3/2)) + (4*a*(b*c - 2*a*d)*x)/(3*c^3*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^2/(x^3*(c + d*x^2)^(5/2)), x, 6, (4*a*b - (2*b^2*c)/d - (5*a^2*d)/c)/(6*c*(c + d*x^2)^(3/2)) - a^2/(2*c*x^2*(c + d*x^2)^(3/2)) + (a*(4*b*c - 5*a*d))/(2*c^3*sqrt(c + d*x^2)) - (a*(4*b*c - 5*a*d)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(2*c^(7/2))]
    @test_int [(a + b*x^2)^2/(x^4*(c + d*x^2)^(5/2)), x, 4, -(a^2/(3*c*x^3*(c + d*x^2)^(3/2))) - (2*a*(b*c - a*d))/(c^2*x*(c + d*x^2)^(3/2)) + ((b^2*c^2 - 8*a*d*(b*c - a*d))*x)/(3*c^3*(c + d*x^2)^(3/2)) + (2*(b^2*c^2 - 8*a*d*(b*c - a*d))*x)/(3*c^4*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^2/(x^5*(c + d*x^2)^(5/2)), x, 7, (8*b^2*c^2 - 5*a*d*(8*b*c - 7*a*d))/(24*c^3*(c + d*x^2)^(3/2)) - a^2/(4*c*x^4*(c + d*x^2)^(3/2)) - (a*(8*b*c - 7*a*d))/(8*c^2*x^2*(c + d*x^2)^(3/2)) + (8*b^2*c^2 - 5*a*d*(8*b*c - 7*a*d))/(8*c^4*sqrt(c + d*x^2)) - ((8*b^2*c^2 - 5*a*d*(8*b*c - 7*a*d))*atanh(sqrt(c + d*x^2)/sqrt(c)))/(8*c^(9/2))]
    @test_int [(a + b*x^2)^2/(x^6*(c + d*x^2)^(5/2)), x, 5, -(a^2/(5*c*x^5*(c + d*x^2)^(3/2))) - (2*a*(5*b*c - 4*a*d))/(15*c^2*x^3*(c + d*x^2)^(3/2)) - (5*b^2*c^2 - 4*a*d*(5*b*c - 4*a*d))/(5*c^3*x*(c + d*x^2)^(3/2)) - (4*d*(5*b^2*c^2 - 4*a*d*(5*b*c - 4*a*d))*x)/(15*c^4*(c + d*x^2)^(3/2)) - (8*d*(5*b^2*c^2 - 4*a*d*(5*b*c - 4*a*d))*x)/(15*c^5*sqrt(c + d*x^2))]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^2)^(p/2)*(c+d*x^2)^3=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(b*x^2)^(p/2) / (c+d*x^2)^1=#

    @test_int [x^5/((a + b*x^2)*sqrt(d*x^2)), x, 4, -((a*x^2)/(b^2*sqrt(d*x^2))) + x^4/(3*b*sqrt(d*x^2)) + (a^(3/2)*x*atan((sqrt(b)*x)/sqrt(a)))/(b^(5/2)*sqrt(d*x^2))]
    @test_int [x^3/((a + b*x^2)*sqrt(d*x^2)), x, 3, x^2/(b*sqrt(d*x^2)) - (sqrt(a)*x*atan((sqrt(b)*x)/sqrt(a)))/(b^(3/2)*sqrt(d*x^2))]
    @test_int [x^1/((a + b*x^2)*sqrt(d*x^2)), x, 2, (x*atan((sqrt(b)*x)/sqrt(a)))/(sqrt(a)*sqrt(b)*sqrt(d*x^2))]
    @test_int [1/(x^1*(a + b*x^2)*sqrt(d*x^2)), x, 3, -(1/(a*sqrt(d*x^2))) - (sqrt(b)*x*atan((sqrt(b)*x)/sqrt(a)))/(a^(3/2)*sqrt(d*x^2))]
    @test_int [1/(x^3*(a + b*x^2)*sqrt(d*x^2)), x, 4, b/(a^2*sqrt(d*x^2)) - 1/(3*a*x^2*sqrt(d*x^2)) + (b^(3/2)*x*atan((sqrt(b)*x)/sqrt(a)))/(a^(5/2)*sqrt(d*x^2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^2)^(p/2) / (c+d*x^2)^1=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(x^4*sqrt(c + d*x^2))/(a + b*x^2), x, 7, ((b*c - 4*a*d)*x*sqrt(c + d*x^2))/(8*b^2*d) + (x^3*sqrt(c + d*x^2))/(4*b) + (a^(3/2)*sqrt(b*c - a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/b^3 - ((b^2*c^2 + 4*a*b*c*d - 8*a^2*d^2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(8*b^3*d^(3/2))]
    @test_int [(x^3*sqrt(c + d*x^2))/(a + b*x^2), x, 5, -((a*sqrt(c + d*x^2))/b^2) + (c + d*x^2)^(3/2)/(3*b*d) + (a*sqrt(b*c - a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/b^(5/2)]
    @test_int [(x^2*sqrt(c + d*x^2))/(a + b*x^2), x, 6, (x*sqrt(c + d*x^2))/(2*b) - (sqrt(a)*sqrt(b*c - a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/b^2 + ((b*c - 2*a*d)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(2*b^2*sqrt(d))]
    @test_int [(x*sqrt(c + d*x^2))/(a + b*x^2), x, 4, sqrt(c + d*x^2)/b - (sqrt(b*c - a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/b^(3/2)]
    @test_int [sqrt(c + d*x^2)/(a + b*x^2), x, 5, (sqrt(b*c - a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(sqrt(a)*b) + (sqrt(d)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/b]
    @test_int [sqrt(c + d*x^2)/(x*(a + b*x^2)), x, 6, -((sqrt(c)*atanh(sqrt(c + d*x^2)/sqrt(c)))/a) + (sqrt(b*c - a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(a*sqrt(b))]
    @test_int [sqrt(c + d*x^2)/(x^2*(a + b*x^2)), x, 4, -(sqrt(c + d*x^2)/(a*x)) - (sqrt(b*c - a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/a^(3/2)]
    @test_int [sqrt(c + d*x^2)/(x^3*(a + b*x^2)), x, 7, -sqrt(c + d*x^2)/(2*a*x^2) + ((2*b*c - a*d)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(2*a^2*sqrt(c)) - (sqrt(b)*sqrt(b*c - a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/a^2]
    @test_int [sqrt(c + d*x^2)/(x^4*(a + b*x^2)), x, 5, -sqrt(c + d*x^2)/(3*a*x^3) + ((3*b*c - a*d)*sqrt(c + d*x^2))/(3*a^2*c*x) + (b*sqrt(b*c - a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/a^(5/2)]

    @test_int [(x^4*(c + d*x^2)^(3/2))/(a + b*x^2), x, 8, ((b^2*c^2 - 10*a*b*c*d + 8*a^2*d^2)*x*sqrt(c + d*x^2))/(16*b^3*d) + ((7*b*c - 6*a*d)*x^3*sqrt(c + d*x^2))/(24*b^2) + (d*x^5*sqrt(c + d*x^2))/(6*b) + (a^(3/2)*(b*c - a*d)^(3/2)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/b^4 - ((b*c - 2*a*d)*(b^2*c^2 + 8*a*b*c*d - 8*a^2*d^2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(16*b^4*d^(3/2))]
    @test_int [(x^3*(c + d*x^2)^(3/2))/(a + b*x^2), x, 6, -((a*(b*c - a*d)*sqrt(c + d*x^2))/b^3) - (a*(c + d*x^2)^(3/2))/(3*b^2) + (c + d*x^2)^(5/2)/(5*b*d) + (a*(b*c - a*d)^(3/2)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/b^(7/2)]
    @test_int [(x^2*(c + d*x^2)^(3/2))/(a + b*x^2), x, 7, ((5*b*c - 4*a*d)*x*sqrt(c + d*x^2))/(8*b^2) + (d*x^3*sqrt(c + d*x^2))/(4*b) - (sqrt(a)*(b*c - a*d)^(3/2)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/b^3 + ((3*b^2*c^2 - 12*a*b*c*d + 8*a^2*d^2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(8*b^3*sqrt(d))]
    @test_int [(x*(c + d*x^2)^(3/2))/(a + b*x^2), x, 5, ((b*c - a*d)*sqrt(c + d*x^2))/b^2 + (c + d*x^2)^(3/2)/(3*b) - ((b*c - a*d)^(3/2)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/b^(5/2)]
    @test_int [(c + d*x^2)^(3/2)/(a + b*x^2), x, 6, (d*x*sqrt(c + d*x^2))/(2*b) + ((b*c - a*d)^(3/2)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(sqrt(a)*b^2) + (sqrt(d)*(3*b*c - 2*a*d)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(2*b^2)]
    @test_int [(c + d*x^2)^(3/2)/(x*(a + b*x^2)), x, 7, (d*sqrt(c + d*x^2))/b - (c^(3/2)*atanh(sqrt(c + d*x^2)/sqrt(c)))/a + ((b*c - a*d)^(3/2)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(a*b^(3/2))]
    @test_int [(c + d*x^2)^(3/2)/(x^2*(a + b*x^2)), x, 6, -((c*sqrt(c + d*x^2))/(a*x)) - ((b*c - a*d)^(3/2)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(a^(3/2)*b) + (d^(3/2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/b]
    @test_int [(c + d*x^2)^(3/2)/(x^3*(a + b*x^2)), x, 7, -((c*sqrt(c + d*x^2))/(2*a*x^2)) + (sqrt(c)*(2*b*c - 3*a*d)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(2*a^2) - ((b*c - a*d)^(3/2)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(a^2*sqrt(b))]
    @test_int [(c + d*x^2)^(3/2)/(x^4*(a + b*x^2)), x, 5, -(c*sqrt(c + d*x^2))/(3*a*x^3) + ((3*b*c - 4*a*d)*sqrt(c + d*x^2))/(3*a^2*x) + ((b*c - a*d)^(3/2)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/a^(5/2)]

    @test_int [(x^4*(c + d*x^2)^(5/2))/(a + b*x^2), x, 9, ((5*b^3*c^3 - 88*a*b^2*c^2*d + 144*a^2*b*c*d^2 - 64*a^3*d^3)*x*sqrt(c + d*x^2))/(128*b^4*d) + ((59*b^2*c^2 - 104*a*b*c*d + 48*a^2*d^2)*x^3*sqrt(c + d*x^2))/(192*b^3) + (d*(11*b*c - 8*a*d)*x^5*sqrt(c + d*x^2))/(48*b^2) + (d*x^5*(c + d*x^2)^(3/2))/(8*b) + (a^(3/2)*(b*c - a*d)^(5/2)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/b^5 - ((5*b^4*c^4 + 40*a*b^3*c^3*d - 240*a^2*b^2*c^2*d^2 + 320*a^3*b*c*d^3 - 128*a^4*d^4)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(128*b^5*d^(3/2))]
    @test_int [(x^3*(c + d*x^2)^(5/2))/(a + b*x^2), x, 7, -((a*(b*c - a*d)^2*sqrt(c + d*x^2))/b^4) - (a*(b*c - a*d)*(c + d*x^2)^(3/2))/(3*b^3) - (a*(c + d*x^2)^(5/2))/(5*b^2) + (c + d*x^2)^(7/2)/(7*b*d) + (a*(b*c - a*d)^(5/2)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/b^(9/2)]
    @test_int [(x^2*(c + d*x^2)^(5/2))/(a + b*x^2), x, 8, ((11*b^2*c^2 - 18*a*b*c*d + 8*a^2*d^2)*x*sqrt(c + d*x^2))/(16*b^3) + (d*(3*b*c - 2*a*d)*x^3*sqrt(c + d*x^2))/(8*b^2) + (d*x^3*(c + d*x^2)^(3/2))/(6*b) - (sqrt(a)*(b*c - a*d)^(5/2)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/b^4 + ((5*b^3*c^3 - 30*a*b^2*c^2*d + 40*a^2*b*c*d^2 - 16*a^3*d^3)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(16*b^4*sqrt(d))]
    @test_int [(x*(c + d*x^2)^(5/2))/(a + b*x^2), x, 6, ((b*c - a*d)^2*sqrt(c + d*x^2))/b^3 + ((b*c - a*d)*(c + d*x^2)^(3/2))/(3*b^2) + (c + d*x^2)^(5/2)/(5*b) - ((b*c - a*d)^(5/2)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/b^(7/2)]
    @test_int [(c + d*x^2)^(5/2)/(a + b*x^2), x, 7, (d*(7*b*c - 4*a*d)*x*sqrt(c + d*x^2))/(8*b^2) + (d*x*(c + d*x^2)^(3/2))/(4*b) + ((b*c - a*d)^(5/2)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(sqrt(a)*b^3) + (sqrt(d)*(15*b^2*c^2 - 20*a*b*c*d + 8*a^2*d^2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(8*b^3)]
    @test_int [(c + d*x^2)^(5/2)/(x*(a + b*x^2)), x, 8, (d*(2*b*c - a*d)*sqrt(c + d*x^2))/b^2 + (d*(c + d*x^2)^(3/2))/(3*b) - (c^(5/2)*atanh(sqrt(c + d*x^2)/sqrt(c)))/a + ((b*c - a*d)^(5/2)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(a*b^(5/2))]
    @test_int [(c + d*x^2)^(5/2)/(x^2*(a + b*x^2)), x, 7, (d*(2*b*c + a*d)*x*sqrt(c + d*x^2))/(2*a*b) - (c*(c + d*x^2)^(3/2))/(a*x) - ((b*c - a*d)^(5/2)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(a^(3/2)*b^2) + (d^(3/2)*(5*b*c - 2*a*d)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(2*b^2)]
    @test_int [(c + d*x^2)^(5/2)/(x^3*(a + b*x^2)), x, 8, (d*(b*c + 2*a*d)*sqrt(c + d*x^2))/(2*a*b) - (c*(c + d*x^2)^(3/2))/(2*a*x^2) + (c^(3/2)*(2*b*c - 5*a*d)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(2*a^2) - ((b*c - a*d)^(5/2)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(a^2*b^(3/2))]
    @test_int [(c + d*x^2)^(5/2)/(x^4*(a + b*x^2)), x, 7, (c*(b*c - 2*a*d)*sqrt(c + d*x^2))/(a^2*x) - (c*(c + d*x^2)^(3/2))/(3*a*x^3) + ((b*c - a*d)^(5/2)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(a^(5/2)*b) + (d^(5/2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/b]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^5/((a + b*x^2)*sqrt(c + d*x^2)), x, 5, -(((b*c + a*d)*sqrt(c + d*x^2))/(b^2*d^2)) + (c + d*x^2)^(3/2)/(3*b*d^2) - (a^2*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(b^(5/2)*sqrt(b*c - a*d))]
    @test_int [x^3/((a + b*x^2)*sqrt(c + d*x^2)), x, 4, sqrt(c + d*x^2)/(b*d) + (a*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(b^(3/2)*sqrt(b*c - a*d))]
    @test_int [x^1/((a + b*x^2)*sqrt(c + d*x^2)), x, 3, -(atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d))/(sqrt(b)*sqrt(b*c - a*d)))]
    @test_int [1/(x^1*(a + b*x^2)*sqrt(c + d*x^2)), x, 6, -(atanh(sqrt(c + d*x^2)/sqrt(c))/(a*sqrt(c))) + (sqrt(b)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(a*sqrt(b*c - a*d))]
    @test_int [1/(x^3*(a + b*x^2)*sqrt(c + d*x^2)), x, 7, -sqrt(c + d*x^2)/(2*a*c*x^2) + ((2*b*c + a*d)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(2*a^2*c^(3/2)) - (b^(3/2)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(a^2*sqrt(b*c - a*d))]

    @test_int [x^4/((a + b*x^2)*sqrt(c + d*x^2)), x, 6, (x*sqrt(c + d*x^2))/(2*b*d) + (a^(3/2)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(b^2*sqrt(b*c - a*d)) - ((b*c + 2*a*d)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(2*b^2*d^(3/2))]
    @test_int [x^2/((a + b*x^2)*sqrt(c + d*x^2)), x, 5, -((sqrt(a)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(b*sqrt(b*c - a*d))) + atanh((sqrt(d)*x)/sqrt(c + d*x^2))/(b*sqrt(d))]
    @test_int [x^0/((a + b*x^2)*sqrt(c + d*x^2)), x, 2, atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2)))/(sqrt(a)*sqrt(b*c - a*d))]
    @test_int [1/(x^2*(a + b*x^2)*sqrt(c + d*x^2)), x, 4, -(sqrt(c + d*x^2)/(a*c*x)) - (b*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(a^(3/2)*sqrt(b*c - a*d))]
    @test_int [1/(x^4*(a + b*x^2)*sqrt(c + d*x^2)), x, 5, -sqrt(c + d*x^2)/(3*a*c*x^3) + ((3*b*c + 2*a*d)*sqrt(c + d*x^2))/(3*a^2*c^2*x) + (b^2*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(a^(5/2)*sqrt(b*c - a*d))]

    @test_int [x^4/((a + b*x^2)*(c + d*x^2)^(3/2)), x, 6, -((c*x)/(d*(b*c - a*d)*sqrt(c + d*x^2))) + (a^(3/2)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(b*(b*c - a*d)^(3/2)) + atanh((sqrt(d)*x)/sqrt(c + d*x^2))/(b*d^(3/2))]
    @test_int [x^3/((a + b*x^2)*(c + d*x^2)^(3/2)), x, 4, -(c/(d*(b*c - a*d)*sqrt(c + d*x^2))) + (a*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(sqrt(b)*(b*c - a*d)^(3/2))]
    @test_int [x^2/((a + b*x^2)*(c + d*x^2)^(3/2)), x, 4, x/((b*c - a*d)*sqrt(c + d*x^2)) - (sqrt(a)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(b*c - a*d)^(3/2)]
    @test_int [x^1/((a + b*x^2)*(c + d*x^2)^(3/2)), x, 4, 1/((b*c - a*d)*sqrt(c + d*x^2)) - (sqrt(b)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(b*c - a*d)^(3/2)]
    @test_int [x^0/((a + b*x^2)*(c + d*x^2)^(3/2)), x, 3, -((d*x)/(c*(b*c - a*d)*sqrt(c + d*x^2))) + (b*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(sqrt(a)*(b*c - a*d)^(3/2))]
    @test_int [1/(x^1*(a + b*x^2)*(c + d*x^2)^(3/2)), x, 7, -(d/(c*(b*c - a*d)*sqrt(c + d*x^2))) - atanh(sqrt(c + d*x^2)/sqrt(c))/(a*c^(3/2)) + (b^(3/2)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(a*(b*c - a*d)^(3/2))]
    @test_int [1/(x^2*(a + b*x^2)*(c + d*x^2)^(3/2)), x, 5, -(d/(c*(b*c - a*d)*x*sqrt(c + d*x^2))) - ((b*c - 2*a*d)*sqrt(c + d*x^2))/(a*c^2*(b*c - a*d)*x) - (b^2*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(a^(3/2)*(b*c - a*d)^(3/2))]
    @test_int [1/(x^3*(a + b*x^2)*(c + d*x^2)^(3/2)), x, 8, -(d*(b*c - 3*a*d))/(2*a*c^2*(b*c - a*d)*sqrt(c + d*x^2)) - 1/(2*a*c*x^2*sqrt(c + d*x^2)) + ((2*b*c + 3*a*d)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(2*a^2*c^(5/2)) - (b^(5/2)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(a^2*(b*c - a*d)^(3/2))]
    @test_int [1/(x^4*(a + b*x^2)*(c + d*x^2)^(3/2)), x, 6, -(d/(c*(b*c - a*d)*x^3*sqrt(c + d*x^2))) - ((b*c - 4*a*d)*sqrt(c + d*x^2))/(3*a*c^2*(b*c - a*d)*x^3) + ((3*b*c - 4*a*d)*(b*c + 2*a*d)*sqrt(c + d*x^2))/(3*a^2*c^3*(b*c - a*d)*x) + (b^3*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(a^(5/2)*(b*c - a*d)^(3/2))]

    @test_int [x^4/((a + b*x^2)*(c + d*x^2)^(5/2)), x, 5, -(c*x)/(3*d*(b*c - a*d)*(c + d*x^2)^(3/2)) + ((b*c - 4*a*d)*x)/(3*d*(b*c - a*d)^2*sqrt(c + d*x^2)) + (a^(3/2)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(b*c - a*d)^(5/2)]
    @test_int [x^3/((a + b*x^2)*(c + d*x^2)^(5/2)), x, 5, -c/(3*d*(b*c - a*d)*(c + d*x^2)^(3/2)) - a/((b*c - a*d)^2*sqrt(c + d*x^2)) + (a*sqrt(b)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(b*c - a*d)^(5/2)]
    @test_int [x^2/((a + b*x^2)*(c + d*x^2)^(5/2)), x, 5, x/(3*(b*c - a*d)*(c + d*x^2)^(3/2)) + ((2*b*c + a*d)*x)/(3*c*(b*c - a*d)^2*sqrt(c + d*x^2)) - (sqrt(a)*b*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(b*c - a*d)^(5/2)]
    @test_int [x^1/((a + b*x^2)*(c + d*x^2)^(5/2)), x, 5, 1/(3*(b*c - a*d)*(c + d*x^2)^(3/2)) + b/((b*c - a*d)^2*sqrt(c + d*x^2)) - (b^(3/2)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(b*c - a*d)^(5/2)]
    @test_int [x^0/((a + b*x^2)*(c + d*x^2)^(5/2)), x, 5, -(d*x)/(3*c*(b*c - a*d)*(c + d*x^2)^(3/2)) - (d*(5*b*c - 2*a*d)*x)/(3*c^2*(b*c - a*d)^2*sqrt(c + d*x^2)) + (b^2*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(sqrt(a)*(b*c - a*d)^(5/2))]
    @test_int [1/(x^1*(a + b*x^2)*(c + d*x^2)^(5/2)), x, 8, -d/(3*c*(b*c - a*d)*(c + d*x^2)^(3/2)) - (d*(2*b*c - a*d))/(c^2*(b*c - a*d)^2*sqrt(c + d*x^2)) - atanh(sqrt(c + d*x^2)/sqrt(c))/(a*c^(5/2)) + (b^(5/2)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(a*(b*c - a*d)^(5/2))]
    @test_int [1/(x^2*(a + b*x^2)*(c + d*x^2)^(5/2)), x, 6, -(d/(3*c*(b*c - a*d)*x*(c + d*x^2)^(3/2))) - (d*(7*b*c - 4*a*d))/(3*c^2*(b*c - a*d)^2*x*sqrt(c + d*x^2)) - ((b*c - 4*a*d)*(3*b*c - 2*a*d)*sqrt(c + d*x^2))/(3*a*c^3*(b*c - a*d)^2*x) - (b^3*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(a^(3/2)*(b*c - a*d)^(5/2))]
    @test_int [1/(x^3*(a + b*x^2)*(c + d*x^2)^(5/2)), x, 9, -(d*(3*b*c - 5*a*d))/(6*a*c^2*(b*c - a*d)*(c + d*x^2)^(3/2)) - 1/(2*a*c*x^2*(c + d*x^2)^(3/2)) - (d*(b^2*c^2 - 8*a*b*c*d + 5*a^2*d^2))/(2*a*c^3*(b*c - a*d)^2*sqrt(c + d*x^2)) + ((2*b*c + 5*a*d)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(2*a^2*c^(7/2)) - (b^(7/2)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(a^2*(b*c - a*d)^(5/2))]
    @test_int [1/(x^4*(a + b*x^2)*(c + d*x^2)^(5/2)), x, 7, -(d/(3*c*(b*c - a*d)*x^3*(c + d*x^2)^(3/2))) - (d*(3*b*c - 2*a*d))/(c^2*(b*c - a*d)^2*x^3*sqrt(c + d*x^2)) - ((b^2*c^2 - 12*a*b*c*d + 8*a^2*d^2)*sqrt(c + d*x^2))/(3*a*c^3*(b*c - a*d)^2*x^3) + ((b*c - 2*a*d)*(3*b^2*c^2 + 8*a*b*c*d - 8*a^2*d^2)*sqrt(c + d*x^2))/(3*a^2*c^4*(b*c - a*d)^2*x) + (b^4*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(a^(5/2)*(b*c - a*d)^(5/2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^2)^(p/2) / (c+d*x^2)^2=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(x^4*sqrt(c + d*x^2))/(a + b*x^2)^2, x, 7, (x*sqrt(c + d*x^2))/b^2 - (x^3*sqrt(c + d*x^2))/(2*b*(a + b*x^2)) - (sqrt(a)*(3*b*c - 4*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*b^3*sqrt(b*c - a*d)) + ((b*c - 4*a*d)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(2*b^3*sqrt(d))]
    @test_int [(x^3*sqrt(c + d*x^2))/(a + b*x^2)^2, x, 5, ((2*b*c - 3*a*d)*sqrt(c + d*x^2))/(2*b^2*(b*c - a*d)) + (a*(c + d*x^2)^(3/2))/(2*b*(b*c - a*d)*(a + b*x^2)) - ((2*b*c - 3*a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*b^(5/2)*sqrt(b*c - a*d))]
    @test_int [(x^2*sqrt(c + d*x^2))/(a + b*x^2)^2, x, 6, -(x*sqrt(c + d*x^2))/(2*b*(a + b*x^2)) + ((b*c - 2*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*sqrt(a)*b^2*sqrt(b*c - a*d)) + (sqrt(d)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/b^2]
    @test_int [(x*sqrt(c + d*x^2))/(a + b*x^2)^2, x, 4, -sqrt(c + d*x^2)/(2*b*(a + b*x^2)) - (d*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*b^(3/2)*sqrt(b*c - a*d))]
    @test_int [sqrt(c + d*x^2)/(a + b*x^2)^2, x, 3, (x*sqrt(c + d*x^2))/(2*a*(a + b*x^2)) + (c*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*a^(3/2)*sqrt(b*c - a*d))]
    @test_int [sqrt(c + d*x^2)/(x*(a + b*x^2)^2), x, 7, sqrt(c + d*x^2)/(2*a*(a + b*x^2)) - (sqrt(c)*atanh(sqrt(c + d*x^2)/sqrt(c)))/a^2 + ((2*b*c - a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*a^2*sqrt(b)*sqrt(b*c - a*d))]
    @test_int [sqrt(c + d*x^2)/(x^2*(a + b*x^2)^2), x, 5, (-3*sqrt(c + d*x^2))/(2*a^2*x) + sqrt(c + d*x^2)/(2*a*x*(a + b*x^2)) - ((3*b*c - 2*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*a^(5/2)*sqrt(b*c - a*d))]
    @test_int [sqrt(c + d*x^2)/(x^3*(a + b*x^2)^2), x, 8, -((b*sqrt(c + d*x^2))/(a^2*(a + b*x^2))) - sqrt(c + d*x^2)/(2*a*x^2*(a + b*x^2)) + ((4*b*c - a*d)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(2*a^3*sqrt(c)) - (sqrt(b)*(4*b*c - 3*a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*a^3*sqrt(b*c - a*d))]
    @test_int [sqrt(c + d*x^2)/(x^4*(a + b*x^2)^2), x, 6, (-5*sqrt(c + d*x^2))/(6*a^2*x^3) + ((15*b*c - 2*a*d)*sqrt(c + d*x^2))/(6*a^3*c*x) + sqrt(c + d*x^2)/(2*a*x^3*(a + b*x^2)) + (b*(5*b*c - 4*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*a^(7/2)*sqrt(b*c - a*d))]

    @test_int [(x^4*(c + d*x^2)^(3/2))/(a + b*x^2)^2, x, 8, (3*(3*b*c - 4*a*d)*x*sqrt(c + d*x^2))/(8*b^3) + (3*d*x^3*sqrt(c + d*x^2))/(4*b^2) - (x^3*(c + d*x^2)^(3/2))/(2*b*(a + b*x^2)) - (3*sqrt(a)*(b*c - 2*a*d)*sqrt(b*c - a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*b^4) + (3*(b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(8*b^4*sqrt(d))]
    @test_int [(x^3*(c + d*x^2)^(3/2))/(a + b*x^2)^2, x, 6, ((2*b*c - 5*a*d)*sqrt(c + d*x^2))/(2*b^3) + ((2*b*c - 5*a*d)*(c + d*x^2)^(3/2))/(6*b^2*(b*c - a*d)) + (a*(c + d*x^2)^(5/2))/(2*b*(b*c - a*d)*(a + b*x^2)) - ((2*b*c - 5*a*d)*sqrt(b*c - a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*b^(7/2))]
    @test_int [(x^2*(c + d*x^2)^(3/2))/(a + b*x^2)^2, x, 7, (d*x*sqrt(c + d*x^2))/b^2 - (x*(c + d*x^2)^(3/2))/(2*b*(a + b*x^2)) + ((b*c - 4*a*d)*sqrt(b*c - a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*sqrt(a)*b^3) + (sqrt(d)*(3*b*c - 4*a*d)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(2*b^3)]
    @test_int [(x*(c + d*x^2)^(3/2))/(a + b*x^2)^2, x, 5, (3*d*sqrt(c + d*x^2))/(2*b^2) - (c + d*x^2)^(3/2)/(2*b*(a + b*x^2)) - (3*d*sqrt(b*c - a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*b^(5/2))]
    @test_int [(c + d*x^2)^(3/2)/(a + b*x^2)^2, x, 6, ((b*c - a*d)*x*sqrt(c + d*x^2))/(2*a*b*(a + b*x^2)) + (sqrt(b*c - a*d)*(b*c + 2*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*a^(3/2)*b^2) + (d^(3/2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/b^2]
    @test_int [(c + d*x^2)^(3/2)/(x*(a + b*x^2)^2), x, 7, ((b*c - a*d)*sqrt(c + d*x^2))/(2*a*b*(a + b*x^2)) - (c^(3/2)*atanh(sqrt(c + d*x^2)/sqrt(c)))/a^2 + (sqrt(b*c - a*d)*(2*b*c + a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*a^2*b^(3/2))]
    @test_int [(c + d*x^2)^(3/2)/(x^2*(a + b*x^2)^2), x, 5, -((3*b*c - a*d)*sqrt(c + d*x^2))/(2*a^2*b*x) + ((b*c - a*d)*sqrt(c + d*x^2))/(2*a*b*x*(a + b*x^2)) - (3*c*sqrt(b*c - a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*a^(5/2))]
    @test_int [(c + d*x^2)^(3/2)/(x^3*(a + b*x^2)^2), x, 8, -(((2*b*c - a*d)*sqrt(c + d*x^2))/(2*a^2*(a + b*x^2))) - (c*sqrt(c + d*x^2))/(2*a*x^2*(a + b*x^2)) + (sqrt(c)*(4*b*c - 3*a*d)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(2*a^3) - (sqrt(b*c - a*d)*(4*b*c - a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*a^3*sqrt(b))]
    @test_int [(c + d*x^2)^(3/2)/(x^4*(a + b*x^2)^2), x, 6, -(((5*b*c - 3*a*d)*sqrt(c + d*x^2))/(6*a^2*b*x^3)) + ((15*b*c - 11*a*d)*sqrt(c + d*x^2))/(6*a^3*x) + ((b*c - a*d)*sqrt(c + d*x^2))/(2*a*b*x^3*(a + b*x^2)) + ((5*b*c - 2*a*d)*sqrt(b*c - a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*a^(7/2))]

    @test_int [(x^4*(c + d*x^2)^(5/2))/(a + b*x^2)^2, x, 9, ((19*b^2*c^2 - 52*a*b*c*d + 32*a^2*d^2)*x*sqrt(c + d*x^2))/(16*b^4) + (d*(7*b*c - 8*a*d)*x^3*sqrt(c + d*x^2))/(8*b^3) + (2*d*x^3*(c + d*x^2)^(3/2))/(3*b^2) - (x^3*(c + d*x^2)^(5/2))/(2*b*(a + b*x^2)) - (sqrt(a)*(3*b*c - 8*a*d)*(b*c - a*d)^(3/2)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*b^5) + ((5*b^3*c^3 - 60*a*b^2*c^2*d + 120*a^2*b*c*d^2 - 64*a^3*d^3)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(16*b^5*sqrt(d))]
    @test_int [(x^3*(c + d*x^2)^(5/2))/(a + b*x^2)^2, x, 7, ((2*b*c - 7*a*d)*(b*c - a*d)*sqrt(c + d*x^2))/(2*b^4) + ((2*b*c - 7*a*d)*(c + d*x^2)^(3/2))/(6*b^3) + ((2*b*c - 7*a*d)*(c + d*x^2)^(5/2))/(10*b^2*(b*c - a*d)) + (a*(c + d*x^2)^(7/2))/(2*b*(b*c - a*d)*(a + b*x^2)) - ((2*b*c - 7*a*d)*(b*c - a*d)^(3/2)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*b^(9/2))]
    @test_int [(x^2*(c + d*x^2)^(5/2))/(a + b*x^2)^2, x, 8, (d*(11*b*c - 12*a*d)*x*sqrt(c + d*x^2))/(8*b^3) + (3*d*x*(c + d*x^2)^(3/2))/(4*b^2) - (x*(c + d*x^2)^(5/2))/(2*b*(a + b*x^2)) + ((b*c - 6*a*d)*(b*c - a*d)^(3/2)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*sqrt(a)*b^4) + (sqrt(d)*(15*b^2*c^2 - 40*a*b*c*d + 24*a^2*d^2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(8*b^4)]
    @test_int [(x*(c + d*x^2)^(5/2))/(a + b*x^2)^2, x, 6, (5*d*(b*c - a*d)*sqrt(c + d*x^2))/(2*b^3) + (5*d*(c + d*x^2)^(3/2))/(6*b^2) - (c + d*x^2)^(5/2)/(2*b*(a + b*x^2)) - (5*d*(b*c - a*d)^(3/2)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*b^(7/2))]
    @test_int [(c + d*x^2)^(5/2)/(a + b*x^2)^2, x, 7, -(d*(b*c - 2*a*d)*x*sqrt(c + d*x^2))/(2*a*b^2) + ((b*c - a*d)*x*(c + d*x^2)^(3/2))/(2*a*b*(a + b*x^2)) + ((b*c - a*d)^(3/2)*(b*c + 4*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*a^(3/2)*b^3) + (d^(3/2)*(5*b*c - 4*a*d)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(2*b^3)]
    @test_int [(c + d*x^2)^(5/2)/(x*(a + b*x^2)^2), x, 8, -((d*(b*c - 3*a*d)*sqrt(c + d*x^2))/(2*a*b^2)) + ((b*c - a*d)*(c + d*x^2)^(3/2))/(2*a*b*(a + b*x^2)) - (c^(5/2)*atanh(sqrt(c + d*x^2)/sqrt(c)))/a^2 + ((b*c - a*d)^(3/2)*(2*b*c + 3*a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*a^2*b^(5/2))]
    @test_int [(c + d*x^2)^(5/2)/(x^2*(a + b*x^2)^2), x, 7, -(c*(3*b*c - a*d)*sqrt(c + d*x^2))/(2*a^2*b*x) + ((b*c - a*d)*(c + d*x^2)^(3/2))/(2*a*b*x*(a + b*x^2)) - ((b*c - a*d)^(3/2)*(3*b*c + 2*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*a^(5/2)*b^2) + (d^(5/2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/b^2]
    @test_int [(c + d*x^2)^(5/2)/(x^3*(a + b*x^2)^2), x, 8, -(((b*c - a*d)*(2*b*c - a*d)*sqrt(c + d*x^2))/(2*a^2*b*(a + b*x^2))) - (c*(c + d*x^2)^(3/2))/(2*a*x^2*(a + b*x^2)) + (c^(3/2)*(4*b*c - 5*a*d)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(2*a^3) - ((b*c - a*d)^(3/2)*(4*b*c + a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*a^3*b^(3/2))]
    @test_int [(c + d*x^2)^(5/2)/(x^4*(a + b*x^2)^2), x, 6, -(c*(5*b*c - 3*a*d)*sqrt(c + d*x^2))/(6*a^2*b*x^3) + ((15*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*sqrt(c + d*x^2))/(6*a^3*b*x) + ((b*c - a*d)*(c + d*x^2)^(3/2))/(2*a*b*x^3*(a + b*x^2)) + (5*c*(b*c - a*d)^(3/2)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*a^(7/2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^4/((a + b*x^2)^2*sqrt(c + d*x^2)), x, 6, (a*x*sqrt(c + d*x^2))/(2*b*(b*c - a*d)*(a + b*x^2)) - (sqrt(a)*(3*b*c - 2*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*b^2*(b*c - a*d)^(3/2)) + atanh((sqrt(d)*x)/sqrt(c + d*x^2))/(b^2*sqrt(d))]
    @test_int [x^3/((a + b*x^2)^2*sqrt(c + d*x^2)), x, 4, (a*sqrt(c + d*x^2))/(2*b*(b*c - a*d)*(a + b*x^2)) - ((2*b*c - a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*b^(3/2)*(b*c - a*d)^(3/2))]
    @test_int [x^2/((a + b*x^2)^2*sqrt(c + d*x^2)), x, 4, -(x*sqrt(c + d*x^2))/(2*(b*c - a*d)*(a + b*x^2)) + (c*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*sqrt(a)*(b*c - a*d)^(3/2))]
    @test_int [x^1/((a + b*x^2)^2*sqrt(c + d*x^2)), x, 4, -sqrt(c + d*x^2)/(2*(b*c - a*d)*(a + b*x^2)) + (d*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*sqrt(b)*(b*c - a*d)^(3/2))]
    @test_int [x^0/((a + b*x^2)^2*sqrt(c + d*x^2)), x, 3, (b*x*sqrt(c + d*x^2))/(2*a*(b*c - a*d)*(a + b*x^2)) + ((b*c - 2*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*a^(3/2)*(b*c - a*d)^(3/2))]
    @test_int [1/(x^1*(a + b*x^2)^2*sqrt(c + d*x^2)), x, 7, (b*sqrt(c + d*x^2))/(2*a*(b*c - a*d)*(a + b*x^2)) - atanh(sqrt(c + d*x^2)/sqrt(c))/(a^2*sqrt(c)) + (sqrt(b)*(2*b*c - 3*a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*a^2*(b*c - a*d)^(3/2))]
    @test_int [1/(x^2*(a + b*x^2)^2*sqrt(c + d*x^2)), x, 5, -((3*b*c - 2*a*d)*sqrt(c + d*x^2))/(2*a^2*c*(b*c - a*d)*x) + (b*sqrt(c + d*x^2))/(2*a*(b*c - a*d)*x*(a + b*x^2)) - (b*(3*b*c - 4*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*a^(5/2)*(b*c - a*d)^(3/2))]
    @test_int [1/(x^3*(a + b*x^2)^2*sqrt(c + d*x^2)), x, 8, -(b*(2*b*c - a*d)*sqrt(c + d*x^2))/(2*a^2*c*(b*c - a*d)*(a + b*x^2)) - sqrt(c + d*x^2)/(2*a*c*x^2*(a + b*x^2)) + ((4*b*c + a*d)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(2*a^3*c^(3/2)) - (b^(3/2)*(4*b*c - 5*a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*a^3*(b*c - a*d)^(3/2))]
    @test_int [1/(x^4*(a + b*x^2)^2*sqrt(c + d*x^2)), x, 6, -((5*b*c - 2*a*d)*sqrt(c + d*x^2))/(6*a^2*c*(b*c - a*d)*x^3) + ((15*b^2*c^2 - 8*a*b*c*d - 4*a^2*d^2)*sqrt(c + d*x^2))/(6*a^3*c^2*(b*c - a*d)*x) + (b*sqrt(c + d*x^2))/(2*a*(b*c - a*d)*x^3*(a + b*x^2)) + (b^2*(5*b*c - 6*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*a^(7/2)*(b*c - a*d)^(3/2))]

    @test_int [x^4/((a + b*x^2)^2*(c + d*x^2)^(3/2)), x, 5, ((2*b*c + a*d)*x)/(2*b*(b*c - a*d)^2*sqrt(c + d*x^2)) + (a*x)/(2*b*(b*c - a*d)*(a + b*x^2)*sqrt(c + d*x^2)) - (3*sqrt(a)*c*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*(b*c - a*d)^(5/2))]
    @test_int [x^3/((a + b*x^2)^2*(c + d*x^2)^(3/2)), x, 5, (2*b*c + a*d)/(2*b*(b*c - a*d)^2*sqrt(c + d*x^2)) + a/(2*b*(b*c - a*d)*(a + b*x^2)*sqrt(c + d*x^2)) - ((2*b*c + a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*sqrt(b)*(b*c - a*d)^(5/2))]
    @test_int [x^2/((a + b*x^2)^2*(c + d*x^2)^(3/2)), x, 5, (-3*d*x)/(2*(b*c - a*d)^2*sqrt(c + d*x^2)) - x/(2*(b*c - a*d)*(a + b*x^2)*sqrt(c + d*x^2)) + ((b*c + 2*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*sqrt(a)*(b*c - a*d)^(5/2))]
    @test_int [x^1/((a + b*x^2)^2*(c + d*x^2)^(3/2)), x, 5, (-3*d)/(2*(b*c - a*d)^2*sqrt(c + d*x^2)) - 1/(2*(b*c - a*d)*(a + b*x^2)*sqrt(c + d*x^2)) + (3*sqrt(b)*d*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*(b*c - a*d)^(5/2))]
    @test_int [x^0/((a + b*x^2)^2*(c + d*x^2)^(3/2)), x, 5, (d*(b*c + 2*a*d)*x)/(2*a*c*(b*c - a*d)^2*sqrt(c + d*x^2)) + (b*x)/(2*a*(b*c - a*d)*(a + b*x^2)*sqrt(c + d*x^2)) + (b*(b*c - 4*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*a^(3/2)*(b*c - a*d)^(5/2))]
    @test_int [1/(x^1*(a + b*x^2)^2*(c + d*x^2)^(3/2)), x, 8, (d*(b*c + 2*a*d))/(2*a*c*(b*c - a*d)^2*sqrt(c + d*x^2)) + b/(2*a*(b*c - a*d)*(a + b*x^2)*sqrt(c + d*x^2)) - atanh(sqrt(c + d*x^2)/sqrt(c))/(a^2*c^(3/2)) + (b^(3/2)*(2*b*c - 5*a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*a^2*(b*c - a*d)^(5/2))]
    @test_int [1/(x^2*(a + b*x^2)^2*(c + d*x^2)^(3/2)), x, 6, (d*(b*c + 2*a*d))/(2*a*c*(b*c - a*d)^2*x*sqrt(c + d*x^2)) + b/(2*a*(b*c - a*d)*x*(a + b*x^2)*sqrt(c + d*x^2)) - ((3*b^2*c^2 - 4*a*b*c*d + 4*a^2*d^2)*sqrt(c + d*x^2))/(2*a^2*c^2*(b*c - a*d)^2*x) - (3*b^2*(b*c - 2*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*a^(5/2)*(b*c - a*d)^(5/2))]
    @test_int [1/(x^3*(a + b*x^2)^2*(c + d*x^2)^(3/2)), x, 9, -(d*(2*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2))/(2*a^2*c^2*(b*c - a*d)^2*sqrt(c + d*x^2)) - (b*(2*b*c - a*d))/(2*a^2*c*(b*c - a*d)*(a + b*x^2)*sqrt(c + d*x^2)) - 1/(2*a*c*x^2*(a + b*x^2)*sqrt(c + d*x^2)) + ((4*b*c + 3*a*d)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(2*a^3*c^(5/2)) - (b^(5/2)*(4*b*c - 7*a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*a^3*(b*c - a*d)^(5/2))]
    @test_int [1/(x^4*(a + b*x^2)^2*(c + d*x^2)^(3/2)), x, 7, (d*(b*c + 2*a*d))/(2*a*c*(b*c - a*d)^2*x^3*sqrt(c + d*x^2)) + b/(2*a*(b*c - a*d)*x^3*(a + b*x^2)*sqrt(c + d*x^2)) - ((5*b^2*c^2 - 4*a*b*c*d + 8*a^2*d^2)*sqrt(c + d*x^2))/(6*a^2*c^2*(b*c - a*d)^2*x^3) + ((15*b^3*c^3 - 14*a*b^2*c^2*d - 8*a^2*b*c*d^2 + 16*a^3*d^3)*sqrt(c + d*x^2))/(6*a^3*c^3*(b*c - a*d)^2*x) + (b^3*(5*b*c - 8*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*a^(7/2)*(b*c - a*d)^(5/2))]

    @test_int [x^4/((a + b*x^2)^2*(c + d*x^2)^(5/2)), x, 6, ((2*b*c + 3*a*d)*x)/(6*b*(b*c - a*d)^2*(c + d*x^2)^(3/2)) + (a*x)/(2*b*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^(3/2)) + ((4*b*c + 11*a*d)*x)/(6*(b*c - a*d)^3*sqrt(c + d*x^2)) - (sqrt(a)*(3*b*c + 2*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*(b*c - a*d)^(7/2))]
    @test_int [x^3/((a + b*x^2)^2*(c + d*x^2)^(5/2)), x, 6, (2*b*c + 3*a*d)/(6*b*(b*c - a*d)^2*(c + d*x^2)^(3/2)) + a/(2*b*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^(3/2)) + (2*b*c + 3*a*d)/(2*(b*c - a*d)^3*sqrt(c + d*x^2)) - (sqrt(b)*(2*b*c + 3*a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*(b*c - a*d)^(7/2))]
    @test_int [x^2/((a + b*x^2)^2*(c + d*x^2)^(5/2)), x, 6, (-5*d*x)/(6*(b*c - a*d)^2*(c + d*x^2)^(3/2)) - x/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^(3/2)) - (d*(13*b*c + 2*a*d)*x)/(6*c*(b*c - a*d)^3*sqrt(c + d*x^2)) + (b*(b*c + 4*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*sqrt(a)*(b*c - a*d)^(7/2))]
    @test_int [x^1/((a + b*x^2)^2*(c + d*x^2)^(5/2)), x, 6, (-5*d)/(6*(b*c - a*d)^2*(c + d*x^2)^(3/2)) - 1/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^(3/2)) - (5*b*d)/(2*(b*c - a*d)^3*sqrt(c + d*x^2)) + (5*b^(3/2)*d*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*(b*c - a*d)^(7/2))]
    @test_int [x^0/((a + b*x^2)^2*(c + d*x^2)^(5/2)), x, 6, (d*(3*b*c + 2*a*d)*x)/(6*a*c*(b*c - a*d)^2*(c + d*x^2)^(3/2)) + (b*x)/(2*a*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^(3/2)) + (d*(3*b^2*c^2 + 16*a*b*c*d - 4*a^2*d^2)*x)/(6*a*c^2*(b*c - a*d)^3*sqrt(c + d*x^2)) + (b^2*(b*c - 6*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*a^(3/2)*(b*c - a*d)^(7/2))]
    @test_int [1/(x^1*(a + b*x^2)^2*(c + d*x^2)^(5/2)), x, 9, (d*(3*b*c + 2*a*d))/(6*a*c*(b*c - a*d)^2*(c + d*x^2)^(3/2)) + b/(2*a*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^(3/2)) + (d*(b^2*c^2 + 6*a*b*c*d - 2*a^2*d^2))/(2*a*c^2*(b*c - a*d)^3*sqrt(c + d*x^2)) - atanh(sqrt(c + d*x^2)/sqrt(c))/(a^2*c^(5/2)) + (b^(5/2)*(2*b*c - 7*a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*a^2*(b*c - a*d)^(7/2))]
    @test_int [1/(x^2*(a + b*x^2)^2*(c + d*x^2)^(5/2)), x, 7, (d*(3*b*c + 2*a*d))/(6*a*c*(b*c - a*d)^2*x*(c + d*x^2)^(3/2)) + b/(2*a*(b*c - a*d)*x*(a + b*x^2)*(c + d*x^2)^(3/2)) + (d*(3*b^2*c^2 + 20*a*b*c*d - 8*a^2*d^2))/(6*a*c^2*(b*c - a*d)^3*x*sqrt(c + d*x^2)) - ((9*b^3*c^3 - 18*a*b^2*c^2*d + 40*a^2*b*c*d^2 - 16*a^3*d^3)*sqrt(c + d*x^2))/(6*a^2*c^3*(b*c - a*d)^3*x) - (b^3*(3*b*c - 8*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*a^(5/2)*(b*c - a*d)^(7/2))]
    @test_int [1/(x^3*(a + b*x^2)^2*(c + d*x^2)^(5/2)), x, 10, -((d*(6*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2))/(6*a^2*c^2*(b*c - a*d)^2*(c + d*x^2)^(3/2))) - (b*(2*b*c - a*d))/(2*a^2*c*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^(3/2)) - 1/(2*a*c*x^2*(a + b*x^2)*(c + d*x^2)^(3/2)) - (d*(2*b*c - a*d)*(b^2*c^2 - a*b*c*d + 5*a^2*d^2))/(2*a^2*c^3*(b*c - a*d)^3*sqrt(c + d*x^2)) + ((4*b*c + 5*a*d)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(2*a^3*c^(7/2)) - (b^(7/2)*(4*b*c - 9*a*d)*atanh((sqrt(b)*sqrt(c + d*x^2))/sqrt(b*c - a*d)))/(2*a^3*(b*c - a*d)^(7/2))]
    @test_int [1/(x^4*(a + b*x^2)^2*(c + d*x^2)^(5/2)), x, 8, (d*(3*b*c + 2*a*d))/(6*a*c*(b*c - a*d)^2*x^3*(c + d*x^2)^(3/2)) + b/(2*a*(b*c - a*d)*x^3*(a + b*x^2)*(c + d*x^2)^(3/2)) + (d*(b^2*c^2 + 8*a*b*c*d - 4*a^2*d^2))/(2*a*c^2*(b*c - a*d)^3*x^3*sqrt(c + d*x^2)) - ((5*b^3*c^3 - 6*a*b^2*c^2*d + 32*a^2*b*c*d^2 - 16*a^3*d^3)*sqrt(c + d*x^2))/(6*a^2*c^3*(b*c - a*d)^3*x^3) + ((15*b^4*c^4 - 20*a*b^3*c^3*d - 12*a^2*b^2*c^2*d^2 + 64*a^3*b*c*d^3 - 32*a^4*d^4)*sqrt(c + d*x^2))/(6*a^3*c^4*(b*c - a*d)^3*x) + (5*b^4*(b*c - 2*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*a^(7/2)*(b*c - a*d)^(7/2))]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^2)^(p/2) / (c+d*x^2)^3=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(e*x)^(m/2)*(a+b*x^2)^(p/2)*(c+d*x^2)^q=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^2)^(p/2)*(c+d*x^2)^1=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(e*x)^(3/2)*sqrt(a + b*x^2)*(A + B*x^2), x, 5, (4*a*(11*A*b - 5*a*B)*e*sqrt(e*x)*sqrt(a + b*x^2))/(231*b^2) + (2*(11*A*b - 5*a*B)*(e*x)^(5/2)*sqrt(a + b*x^2))/(77*b*e) + (2*B*(e*x)^(5/2)*(a + b*x^2)^(3/2))/(11*b*e) - (2*a^(7/4)*(11*A*b - 5*a*B)*e^(3/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(231*b^(9/4)*sqrt(a + b*x^2))]
    @test_int [sqrt(e*x)*sqrt(a + b*x^2)*(A + B*x^2), x, 6, (2*(3*A*b - a*B)*(e*x)^(3/2)*sqrt(a + b*x^2))/(15*b*e) + (4*a*(3*A*b - a*B)*sqrt(e*x)*sqrt(a + b*x^2))/(15*b^(3/2)*(sqrt(a) + sqrt(b)*x)) + (2*B*(e*x)^(3/2)*(a + b*x^2)^(3/2))/(9*b*e) - (4*a^(5/4)*(3*A*b - a*B)*sqrt(e)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(15*b^(7/4)*sqrt(a + b*x^2)) + (2*a^(5/4)*(3*A*b - a*B)*sqrt(e)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(15*b^(7/4)*sqrt(a + b*x^2))]
    @test_int [(sqrt(a + b*x^2)*(A + B*x^2))/sqrt(e*x), x, 4, (2*(7*A*b - a*B)*sqrt(e*x)*sqrt(a + b*x^2))/(21*b*e) + (2*B*sqrt(e*x)*(a + b*x^2)^(3/2))/(7*b*e) + (2*a^(3/4)*(7*A*b - a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(21*b^(5/4)*sqrt(e)*sqrt(a + b*x^2))]
    @test_int [(sqrt(a + b*x^2)*(A + B*x^2))/(e*x)^(3/2), x, 6, (2*(5*A*b + a*B)*(e*x)^(3/2)*sqrt(a + b*x^2))/(5*a*e^3) + (4*(5*A*b + a*B)*sqrt(e*x)*sqrt(a + b*x^2))/(5*sqrt(b)*e^2*(sqrt(a) + sqrt(b)*x)) - (2*A*(a + b*x^2)^(3/2))/(a*e*sqrt(e*x)) - (4*a^(1/4)*(5*A*b + a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(5*b^(3/4)*e^(3/2)*sqrt(a + b*x^2)) + (2*a^(1/4)*(5*A*b + a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(5*b^(3/4)*e^(3/2)*sqrt(a + b*x^2))]
    @test_int [(sqrt(a + b*x^2)*(A + B*x^2))/(e*x)^(5/2), x, 4, (2*(A*b + a*B)*sqrt(e*x)*sqrt(a + b*x^2))/(3*a*e^3) - (2*A*(a + b*x^2)^(3/2))/(3*a*e*(e*x)^(3/2)) + (2*(A*b + a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(3*a^(1/4)*b^(1/4)*e^(5/2)*sqrt(a + b*x^2))]
    @test_int [(sqrt(a + b*x^2)*(A + B*x^2))/(e*x)^(7/2), x, 6, -((2*(A*b + 5*a*B)*sqrt(a + b*x^2))/(5*a*e^3*sqrt(e*x))) + (4*sqrt(b)*(A*b + 5*a*B)*sqrt(e*x)*sqrt(a + b*x^2))/(5*a*e^4*(sqrt(a) + sqrt(b)*x)) - (2*A*(a + b*x^2)^(3/2))/(5*a*e*(e*x)^(5/2)) - (4*b^(1/4)*(A*b + 5*a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(5*a^(3/4)*e^(7/2)*sqrt(a + b*x^2)) + (2*b^(1/4)*(A*b + 5*a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(5*a^(3/4)*e^(7/2)*sqrt(a + b*x^2))]
    @test_int [(sqrt(a + b*x^2)*(A + B*x^2))/x^(9/2), x, 4, (2*(A*b - 7*a*B)*sqrt(a + b*x^2))/(21*a*x^(3/2)) - (2*A*(a + b*x^2)^(3/2))/(7*a*x^(7/2)) - (2*b^(3/4)*(A*b - 7*a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(x))/a^(1/4)), 1/2))/(21*a^(5/4)*sqrt(a + b*x^2))]
    @test_int [(sqrt(a + b*x^2)*(A + B*x^2))/x^(11/2), x, 7, (2*(A*b - 3*a*B)*sqrt(a + b*x^2))/(15*a*x^(5/2)) + (4*b*(A*b - 3*a*B)*sqrt(a + b*x^2))/(15*a^2*sqrt(x)) - (4*b^(3/2)*(A*b - 3*a*B)*sqrt(x)*sqrt(a + b*x^2))/(15*a^2*(sqrt(a) + sqrt(b)*x)) - (2*A*(a + b*x^2)^(3/2))/(9*a*x^(9/2)) + (4*b^(5/4)*(A*b - 3*a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(x))/a^(1/4)), 1/2))/(15*a^(7/4)*sqrt(a + b*x^2)) - (2*b^(5/4)*(A*b - 3*a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(x))/a^(1/4)), 1/2))/(15*a^(7/4)*sqrt(a + b*x^2))]
    @test_int [(sqrt(a + b*x^2)*(A + B*x^2))/x^(13/2), x, 5, (2*(5*A*b - 11*a*B)*sqrt(a + b*x^2))/(77*a*x^(7/2)) + (4*b*(5*A*b - 11*a*B)*sqrt(a + b*x^2))/(231*a^2*x^(3/2)) - (2*A*(a + b*x^2)^(3/2))/(11*a*x^(11/2)) + (2*b^(7/4)*(5*A*b - 11*a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(x))/a^(1/4)), 1/2))/(231*a^(9/4)*sqrt(a + b*x^2))]

    @test_int [(e*x)^(3/2)*(a + b*x^2)^(3/2)*(A + B*x^2), x, 6, (8*a^2*(3*A*b - a*B)*e*sqrt(e*x)*sqrt(a + b*x^2))/(231*b^2) + (4*a*(3*A*b - a*B)*(e*x)^(5/2)*sqrt(a + b*x^2))/(77*b*e) + (2*(3*A*b - a*B)*(e*x)^(5/2)*(a + b*x^2)^(3/2))/(33*b*e) + (2*B*(e*x)^(5/2)*(a + b*x^2)^(5/2))/(15*b*e) - (4*a^(11/4)*(3*A*b - a*B)*e^(3/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(231*b^(9/4)*sqrt(a + b*x^2))]
    @test_int [sqrt(e*x)*(a + b*x^2)^(3/2)*(A + B*x^2), x, 7, (4*a*(13*A*b - 3*a*B)*(e*x)^(3/2)*sqrt(a + b*x^2))/(195*b*e) + (8*a^2*(13*A*b - 3*a*B)*sqrt(e*x)*sqrt(a + b*x^2))/(195*b^(3/2)*(sqrt(a) + sqrt(b)*x)) + (2*(13*A*b - 3*a*B)*(e*x)^(3/2)*(a + b*x^2)^(3/2))/(117*b*e) + (2*B*(e*x)^(3/2)*(a + b*x^2)^(5/2))/(13*b*e) - (8*a^(9/4)*(13*A*b - 3*a*B)*sqrt(e)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(195*b^(7/4)*sqrt(a + b*x^2)) + (4*a^(9/4)*(13*A*b - 3*a*B)*sqrt(e)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(195*b^(7/4)*sqrt(a + b*x^2))]
    @test_int [((a + b*x^2)^(3/2)*(A + B*x^2))/sqrt(e*x), x, 5, (4*a*(11*A*b - a*B)*sqrt(e*x)*sqrt(a + b*x^2))/(77*b*e) + (2*(11*A*b - a*B)*sqrt(e*x)*(a + b*x^2)^(3/2))/(77*b*e) + (2*B*sqrt(e*x)*(a + b*x^2)^(5/2))/(11*b*e) + (4*a^(7/4)*(11*A*b - a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(77*b^(5/4)*sqrt(e)*sqrt(a + b*x^2))]
    @test_int [((a + b*x^2)^(3/2)*(A + B*x^2))/(e*x)^(3/2), x, 7, (4*(9*A*b + a*B)*(e*x)^(3/2)*sqrt(a + b*x^2))/(15*e^3) + (8*a*(9*A*b + a*B)*sqrt(e*x)*sqrt(a + b*x^2))/(15*sqrt(b)*e^2*(sqrt(a) + sqrt(b)*x)) + (2*(9*A*b + a*B)*(e*x)^(3/2)*(a + b*x^2)^(3/2))/(9*a*e^3) - (2*A*(a + b*x^2)^(5/2))/(a*e*sqrt(e*x)) - (8*a^(5/4)*(9*A*b + a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(15*b^(3/4)*e^(3/2)*sqrt(a + b*x^2)) + (4*a^(5/4)*(9*A*b + a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(15*b^(3/4)*e^(3/2)*sqrt(a + b*x^2))]
    @test_int [((a + b*x^2)^(3/2)*(A + B*x^2))/(e*x)^(5/2), x, 5, (4*(7*A*b + 3*a*B)*sqrt(e*x)*sqrt(a + b*x^2))/(21*e^3) + (2*(7*A*b + 3*a*B)*sqrt(e*x)*(a + b*x^2)^(3/2))/(21*a*e^3) - (2*A*(a + b*x^2)^(5/2))/(3*a*e*(e*x)^(3/2)) + (4*a^(3/4)*(7*A*b + 3*a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(21*b^(1/4)*e^(5/2)*sqrt(a + b*x^2))]
    @test_int [((a + b*x^2)^(3/2)*(A + B*x^2))/(e*x)^(7/2), x, 7, (12*b*(A*b + a*B)*(e*x)^(3/2)*sqrt(a + b*x^2))/(5*a*e^5) + (24*sqrt(b)*(A*b + a*B)*sqrt(e*x)*sqrt(a + b*x^2))/(5*e^4*(sqrt(a) + sqrt(b)*x)) - (2*(A*b + a*B)*(a + b*x^2)^(3/2))/(a*e^3*sqrt(e*x)) - (2*A*(a + b*x^2)^(5/2))/(5*a*e*(e*x)^(5/2)) - (24*a^(1/4)*b^(1/4)*(A*b + a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(5*e^(7/2)*sqrt(a + b*x^2)) + (12*a^(1/4)*b^(1/4)*(A*b + a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(5*e^(7/2)*sqrt(a + b*x^2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [((e*x)^(5/2)*(A + B*x^2))/sqrt(a + b*x^2), x, 6, (2*(9*A*b - 7*a*B)*e*(e*x)^(3/2)*sqrt(a + b*x^2))/(45*b^2) + (2*B*(e*x)^(7/2)*sqrt(a + b*x^2))/(9*b*e) - (2*a*(9*A*b - 7*a*B)*e^2*sqrt(e*x)*sqrt(a + b*x^2))/(15*b^(5/2)*(sqrt(a) + sqrt(b)*x)) + (2*a^(5/4)*(9*A*b - 7*a*B)*e^(5/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(15*b^(11/4)*sqrt(a + b*x^2)) - (a^(5/4)*(9*A*b - 7*a*B)*e^(5/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(15*b^(11/4)*sqrt(a + b*x^2))]
    @test_int [((e*x)^(3/2)*(A + B*x^2))/sqrt(a + b*x^2), x, 4, (2*(7*A*b - 5*a*B)*e*sqrt(e*x)*sqrt(a + b*x^2))/(21*b^2) + (2*B*(e*x)^(5/2)*sqrt(a + b*x^2))/(7*b*e) - (a^(3/4)*(7*A*b - 5*a*B)*e^(3/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(21*b^(9/4)*sqrt(a + b*x^2))]
    @test_int [(sqrt(e*x)*(A + B*x^2))/sqrt(a + b*x^2), x, 5, (2*B*(e*x)^(3/2)*sqrt(a + b*x^2))/(5*b*e) + (2*(5*A*b - 3*a*B)*sqrt(e*x)*sqrt(a + b*x^2))/(5*b^(3/2)*(sqrt(a) + sqrt(b)*x)) - (2*a^(1/4)*(5*A*b - 3*a*B)*sqrt(e)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(5*b^(7/4)*sqrt(a + b*x^2)) + (a^(1/4)*(5*A*b - 3*a*B)*sqrt(e)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(5*b^(7/4)*sqrt(a + b*x^2))]
    @test_int [(A + B*x^2)/(sqrt(e*x)*sqrt(a + b*x^2)), x, 3, (2*B*sqrt(e*x)*sqrt(a + b*x^2))/(3*b*e) + ((3*A*b - a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(3*a^(1/4)*b^(5/4)*sqrt(e)*sqrt(a + b*x^2))]
    @test_int [(A + B*x^2)/((e*x)^(3/2)*sqrt(a + b*x^2)), x, 5, -((2*A*sqrt(a + b*x^2))/(a*e*sqrt(e*x))) + (2*(A*b + a*B)*sqrt(e*x)*sqrt(a + b*x^2))/(a*sqrt(b)*e^2*(sqrt(a) + sqrt(b)*x)) - (2*(A*b + a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(a^(3/4)*b^(3/4)*e^(3/2)*sqrt(a + b*x^2)) + ((A*b + a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(a^(3/4)*b^(3/4)*e^(3/2)*sqrt(a + b*x^2))]
    @test_int [(A + B*x^2)/((e*x)^(5/2)*sqrt(a + b*x^2)), x, 3, -((2*A*sqrt(a + b*x^2))/(3*a*e*(e*x)^(3/2))) - ((A*b - 3*a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(3*a^(5/4)*b^(1/4)*e^(5/2)*sqrt(a + b*x^2))]
    @test_int [(A + B*x^2)/((e*x)^(7/2)*sqrt(a + b*x^2)), x, 6, -((2*A*sqrt(a + b*x^2))/(5*a*e*(e*x)^(5/2))) + (2*(3*A*b - 5*a*B)*sqrt(a + b*x^2))/(5*a^2*e^3*sqrt(e*x)) - (2*sqrt(b)*(3*A*b - 5*a*B)*sqrt(e*x)*sqrt(a + b*x^2))/(5*a^2*e^4*(sqrt(a) + sqrt(b)*x)) + (2*b^(1/4)*(3*A*b - 5*a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(5*a^(7/4)*e^(7/2)*sqrt(a + b*x^2)) - (b^(1/4)*(3*A*b - 5*a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(5*a^(7/4)*e^(7/2)*sqrt(a + b*x^2))]

    @test_int [((e*x)^(7/2)*(A + B*x^2))/(a + b*x^2)^(3/2), x, 5, -(((7*A*b - 9*a*B)*e*(e*x)^(5/2))/(7*b^2*sqrt(a + b*x^2))) + (2*B*(e*x)^(9/2))/(7*b*e*sqrt(a + b*x^2)) + (5*(7*A*b - 9*a*B)*e^3*sqrt(e*x)*sqrt(a + b*x^2))/(21*b^3) - (5*a^(3/4)*(7*A*b - 9*a*B)*e^(7/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(42*b^(13/4)*sqrt(a + b*x^2))]
    @test_int [((e*x)^(5/2)*(A + B*x^2))/(a + b*x^2)^(3/2), x, 6, -(((5*A*b - 7*a*B)*e*(e*x)^(3/2))/(5*b^2*sqrt(a + b*x^2))) + (2*B*(e*x)^(7/2))/(5*b*e*sqrt(a + b*x^2)) + (3*(5*A*b - 7*a*B)*e^2*sqrt(e*x)*sqrt(a + b*x^2))/(5*b^(5/2)*(sqrt(a) + sqrt(b)*x)) - (3*a^(1/4)*(5*A*b - 7*a*B)*e^(5/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(5*b^(11/4)*sqrt(a + b*x^2)) + (3*a^(1/4)*(5*A*b - 7*a*B)*e^(5/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(10*b^(11/4)*sqrt(a + b*x^2))]
    @test_int [((e*x)^(3/2)*(A + B*x^2))/(a + b*x^2)^(3/2), x, 4, -(((3*A*b - 5*a*B)*e*sqrt(e*x))/(3*b^2*sqrt(a + b*x^2))) + (2*B*(e*x)^(5/2))/(3*b*e*sqrt(a + b*x^2)) + ((3*A*b - 5*a*B)*e^(3/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(6*a^(1/4)*b^(9/4)*sqrt(a + b*x^2))]
    @test_int [(sqrt(e*x)*(A + B*x^2))/(a + b*x^2)^(3/2), x, 5, ((A*b - a*B)*(e*x)^(3/2))/(a*b*e*sqrt(a + b*x^2)) - ((A*b - 3*a*B)*sqrt(e*x)*sqrt(a + b*x^2))/(a*b^(3/2)*(sqrt(a) + sqrt(b)*x)) + ((A*b - 3*a*B)*sqrt(e)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(a^(3/4)*b^(7/4)*sqrt(a + b*x^2)) - ((A*b - 3*a*B)*sqrt(e)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(2*a^(3/4)*b^(7/4)*sqrt(a + b*x^2))]
    @test_int [(A + B*x^2)/(sqrt(e*x)*(a + b*x^2)^(3/2)), x, 3, ((A*b - a*B)*sqrt(e*x))/(a*b*e*sqrt(a + b*x^2)) + ((A*b + a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(2*a^(5/4)*b^(5/4)*sqrt(e)*sqrt(a + b*x^2))]
    @test_int [(A + B*x^2)/((e*x)^(3/2)*(a + b*x^2)^(3/2)), x, 6, -((2*A)/(a*e*sqrt(e*x)*sqrt(a + b*x^2))) - ((3*A*b - a*B)*(e*x)^(3/2))/(a^2*e^3*sqrt(a + b*x^2)) + ((3*A*b - a*B)*sqrt(e*x)*sqrt(a + b*x^2))/(a^2*sqrt(b)*e^2*(sqrt(a) + sqrt(b)*x)) - ((3*A*b - a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(a^(7/4)*b^(3/4)*e^(3/2)*sqrt(a + b*x^2)) + ((3*A*b - a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(2*a^(7/4)*b^(3/4)*e^(3/2)*sqrt(a + b*x^2))]
    @test_int [(A + B*x^2)/((e*x)^(5/2)*(a + b*x^2)^(3/2)), x, 4, -((2*A)/(3*a*e*(e*x)^(3/2)*sqrt(a + b*x^2))) - ((5*A*b - 3*a*B)*sqrt(e*x))/(3*a^2*e^3*sqrt(a + b*x^2)) - ((5*A*b - 3*a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(6*a^(9/4)*b^(1/4)*e^(5/2)*sqrt(a + b*x^2))]
    @test_int [(A + B*x^2)/((e*x)^(7/2)*(a + b*x^2)^(3/2)), x, 7, -((2*A)/(5*a*e*(e*x)^(5/2)*sqrt(a + b*x^2))) - (7*A*b - 5*a*B)/(5*a^2*e^3*sqrt(e*x)*sqrt(a + b*x^2)) + (3*(7*A*b - 5*a*B)*sqrt(a + b*x^2))/(5*a^3*e^3*sqrt(e*x)) - (3*sqrt(b)*(7*A*b - 5*a*B)*sqrt(e*x)*sqrt(a + b*x^2))/(5*a^3*e^4*(sqrt(a) + sqrt(b)*x)) + (3*b^(1/4)*(7*A*b - 5*a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(5*a^(11/4)*e^(7/2)*sqrt(a + b*x^2)) - (3*b^(1/4)*(7*A*b - 5*a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(10*a^(11/4)*e^(7/2)*sqrt(a + b*x^2))]

    @test_int [((e*x)^(7/2)*(A + B*x^2))/(a + b*x^2)^(5/2), x, 5, -(((A*b - 3*a*B)*e*(e*x)^(5/2))/(3*b^2*(a + b*x^2)^(3/2))) + (2*B*(e*x)^(9/2))/(3*b*e*(a + b*x^2)^(3/2)) - (5*(A*b - 3*a*B)*e^3*sqrt(e*x))/(6*b^3*sqrt(a + b*x^2)) + (5*(A*b - 3*a*B)*e^(7/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(12*a^(1/4)*b^(13/4)*sqrt(a + b*x^2))]
    @test_int [((e*x)^(5/2)*(A + B*x^2))/(a + b*x^2)^(5/2), x, 6, ((A*b - a*B)*(e*x)^(7/2))/(3*a*b*e*(a + b*x^2)^(3/2)) + ((A*b - 7*a*B)*e*(e*x)^(3/2))/(6*a*b^2*sqrt(a + b*x^2)) - ((A*b - 7*a*B)*e^2*sqrt(e*x)*sqrt(a + b*x^2))/(2*a*b^(5/2)*(sqrt(a) + sqrt(b)*x)) + ((A*b - 7*a*B)*e^(5/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(2*a^(3/4)*b^(11/4)*sqrt(a + b*x^2)) - ((A*b - 7*a*B)*e^(5/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(4*a^(3/4)*b^(11/4)*sqrt(a + b*x^2))]
    @test_int [((e*x)^(3/2)*(A + B*x^2))/(a + b*x^2)^(5/2), x, 4, ((A*b - a*B)*(e*x)^(5/2))/(3*a*b*e*(a + b*x^2)^(3/2)) - ((A*b + 5*a*B)*e*sqrt(e*x))/(6*a*b^2*sqrt(a + b*x^2)) + ((A*b + 5*a*B)*e^(3/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(12*a^(5/4)*b^(9/4)*sqrt(a + b*x^2))]
    @test_int [(sqrt(e*x)*(A + B*x^2))/(a + b*x^2)^(5/2), x, 6, ((A*b - a*B)*(e*x)^(3/2))/(3*a*b*e*(a + b*x^2)^(3/2)) + ((A*b + a*B)*(e*x)^(3/2))/(2*a^2*b*e*sqrt(a + b*x^2)) - ((A*b + a*B)*sqrt(e*x)*sqrt(a + b*x^2))/(2*a^2*b^(3/2)*(sqrt(a) + sqrt(b)*x)) + ((A*b + a*B)*sqrt(e)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(2*a^(7/4)*b^(7/4)*sqrt(a + b*x^2)) - ((A*b + a*B)*sqrt(e)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(4*a^(7/4)*b^(7/4)*sqrt(a + b*x^2))]
    @test_int [(A + B*x^2)/(sqrt(e*x)*(a + b*x^2)^(5/2)), x, 4, ((A*b - a*B)*sqrt(e*x))/(3*a*b*e*(a + b*x^2)^(3/2)) + ((5*A*b + a*B)*sqrt(e*x))/(6*a^2*b*e*sqrt(a + b*x^2)) + ((5*A*b + a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(12*a^(9/4)*b^(5/4)*sqrt(e)*sqrt(a + b*x^2))]
    @test_int [(A + B*x^2)/((e*x)^(3/2)*(a + b*x^2)^(5/2)), x, 7, -((2*A)/(a*e*sqrt(e*x)*(a + b*x^2)^(3/2))) - ((7*A*b - a*B)*(e*x)^(3/2))/(3*a^2*e^3*(a + b*x^2)^(3/2)) - ((7*A*b - a*B)*(e*x)^(3/2))/(2*a^3*e^3*sqrt(a + b*x^2)) + ((7*A*b - a*B)*sqrt(e*x)*sqrt(a + b*x^2))/(2*a^3*sqrt(b)*e^2*(sqrt(a) + sqrt(b)*x)) - ((7*A*b - a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(2*a^(11/4)*b^(3/4)*e^(3/2)*sqrt(a + b*x^2)) + ((7*A*b - a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(4*a^(11/4)*b^(3/4)*e^(3/2)*sqrt(a + b*x^2))]
    @test_int [(A + B*x^2)/((e*x)^(5/2)*(a + b*x^2)^(5/2)), x, 5, -((2*A)/(3*a*e*(e*x)^(3/2)*(a + b*x^2)^(3/2))) - ((3*A*b - a*B)*sqrt(e*x))/(3*a^2*e^3*(a + b*x^2)^(3/2)) - (5*(3*A*b - a*B)*sqrt(e*x))/(6*a^3*e^3*sqrt(a + b*x^2)) - (5*(3*A*b - a*B)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(e*x))/(a^(1/4)*sqrt(e))), 1/2))/(12*a^(13/4)*b^(1/4)*e^(5/2)*sqrt(a + b*x^2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^2)^(p/2)*(c+d*x^2)^2=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(e*x)^(3/2)*(a + b*x^2)^2*sqrt(c + d*x^2), x, 6, (4*c*(11*a^2*d^2 + b*c*(3*b*c - 10*a*d))*e*sqrt(e*x)*sqrt(c + d*x^2))/(231*d^3) + (2*(11*a^2*d^2 + b*c*(3*b*c - 10*a*d))*(e*x)^(5/2)*sqrt(c + d*x^2))/(77*d^2*e) - (2*b*(3*b*c - 10*a*d)*(e*x)^(5/2)*(c + d*x^2)^(3/2))/(55*d^2*e) + (2*b^2*(e*x)^(9/2)*(c + d*x^2)^(3/2))/(15*d*e^3) - (2*c^(7/4)*(11*a^2*d^2 + b*c*(3*b*c - 10*a*d))*e^(3/2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(231*d^(13/4)*sqrt(c + d*x^2))]
    @test_int [sqrt(e*x)*(a + b*x^2)^2*sqrt(c + d*x^2), x, 7, (2*(39*a^2*d^2 + b*c*(7*b*c - 26*a*d))*(e*x)^(3/2)*sqrt(c + d*x^2))/(195*d^2*e) + (4*c*(39*a^2*d^2 + b*c*(7*b*c - 26*a*d))*sqrt(e*x)*sqrt(c + d*x^2))/(195*d^(5/2)*(sqrt(c) + sqrt(d)*x)) - (2*b*(7*b*c - 26*a*d)*(e*x)^(3/2)*(c + d*x^2)^(3/2))/(117*d^2*e) + (2*b^2*(e*x)^(7/2)*(c + d*x^2)^(3/2))/(13*d*e^3) - (4*c^(5/4)*(39*a^2*d^2 + b*c*(7*b*c - 26*a*d))*sqrt(e)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(195*d^(11/4)*sqrt(c + d*x^2)) + (2*c^(5/4)*(39*a^2*d^2 + b*c*(7*b*c - 26*a*d))*sqrt(e)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(195*d^(11/4)*sqrt(c + d*x^2))]
    @test_int [((a + b*x^2)^2*sqrt(c + d*x^2))/sqrt(e*x), x, 5, (2*(5*b^2*c^2 - 22*a*b*c*d + 77*a^2*d^2)*sqrt(e*x)*sqrt(c + d*x^2))/(231*d^2*e) - (2*b*(5*b*c - 22*a*d)*sqrt(e*x)*(c + d*x^2)^(3/2))/(77*d^2*e) + (2*b^2*(e*x)^(5/2)*(c + d*x^2)^(3/2))/(11*d*e^3) + (2*c^(3/4)*(5*b^2*c^2 - 22*a*b*c*d + 77*a^2*d^2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(231*d^(9/4)*sqrt(e)*sqrt(c + d*x^2))]
    @test_int [((a + b*x^2)^2*sqrt(c + d*x^2))/(e*x)^(3/2), x, 7, -((2*(b^2*c^2 - 3*a*d*(2*b*c + 5*a*d))*(e*x)^(3/2)*sqrt(c + d*x^2))/(15*c*d*e^3)) - (4*(b^2*c^2 - 3*a*d*(2*b*c + 5*a*d))*sqrt(e*x)*sqrt(c + d*x^2))/(15*d^(3/2)*e^2*(sqrt(c) + sqrt(d)*x)) - (2*a^2*(c + d*x^2)^(3/2))/(c*e*sqrt(e*x)) + (2*b^2*(e*x)^(3/2)*(c + d*x^2)^(3/2))/(9*d*e^3) + (4*c^(1/4)*(b^2*c^2 - 3*a*d*(2*b*c + 5*a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(15*d^(7/4)*e^(3/2)*sqrt(c + d*x^2)) - (2*c^(1/4)*(b^2*c^2 - 3*a*d*(2*b*c + 5*a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(15*d^(7/4)*e^(3/2)*sqrt(c + d*x^2))]
    @test_int [((a + b*x^2)^2*sqrt(c + d*x^2))/(e*x)^(5/2), x, 5, -((2*(b^2*c^2 - 7*a*d*(2*b*c + a*d))*sqrt(e*x)*sqrt(c + d*x^2))/(21*c*d*e^3)) - (2*a^2*(c + d*x^2)^(3/2))/(3*c*e*(e*x)^(3/2)) + (2*b^2*sqrt(e*x)*(c + d*x^2)^(3/2))/(7*d*e^3) - (2*(b^2*c^2 - 7*a*d*(2*b*c + a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(21*c^(1/4)*d^(5/4)*e^(5/2)*sqrt(c + d*x^2))]
    @test_int [((a + b*x^2)^2*sqrt(c + d*x^2))/(e*x)^(7/2), x, 7, (2*(b^2*c^2 + a*d*(10*b*c + a*d))*(e*x)^(3/2)*sqrt(c + d*x^2))/(5*c^2*e^5) + (4*(b^2*c^2 + a*d*(10*b*c + a*d))*sqrt(e*x)*sqrt(c + d*x^2))/(5*c*sqrt(d)*e^4*(sqrt(c) + sqrt(d)*x)) - (2*a^2*(c + d*x^2)^(3/2))/(5*c*e*(e*x)^(5/2)) - (2*a*(10*b*c + a*d)*(c + d*x^2)^(3/2))/(5*c^2*e^3*sqrt(e*x)) - (4*(b^2*c^2 + a*d*(10*b*c + a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(5*c^(3/4)*d^(3/4)*e^(7/2)*sqrt(c + d*x^2)) + (2*(b^2*c^2 + a*d*(10*b*c + a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(5*c^(3/4)*d^(3/4)*e^(7/2)*sqrt(c + d*x^2))]
    @test_int [((a + b*x^2)^2*sqrt(c + d*x^2))/x^(9/2), x, 5, (2*(7*b^2*c^2 + a*d*(14*b*c - a*d))*sqrt(x)*sqrt(c + d*x^2))/(21*c^2) - (2*a^2*(c + d*x^2)^(3/2))/(7*c*x^(7/2)) - (2*a*(14*b*c - a*d)*(c + d*x^2)^(3/2))/(21*c^2*x^(3/2)) + (2*(7*b^2*c^2 + a*d*(14*b*c - a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(x))/c^(1/4)), 1/2))/(21*c^(5/4)*d^(1/4)*sqrt(c + d*x^2))]
    @test_int [((a + b*x^2)^2*sqrt(c + d*x^2))/x^(11/2), x, 7, -((2*(15*b^2*c^2 + a*d*(6*b*c - a*d))*sqrt(c + d*x^2))/(15*c^2*sqrt(x))) + (4*sqrt(d)*(15*b^2*c^2 + a*d*(6*b*c - a*d))*sqrt(x)*sqrt(c + d*x^2))/(15*c^2*(sqrt(c) + sqrt(d)*x)) - (2*a^2*(c + d*x^2)^(3/2))/(9*c*x^(9/2)) - (2*a*(6*b*c - a*d)*(c + d*x^2)^(3/2))/(15*c^2*x^(5/2)) - (4*d^(1/4)*(15*b^2*c^2 + a*d*(6*b*c - a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(x))/c^(1/4)), 1/2))/(15*c^(7/4)*sqrt(c + d*x^2)) + (2*d^(1/4)*(15*b^2*c^2 + a*d*(6*b*c - a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(x))/c^(1/4)), 1/2))/(15*c^(7/4)*sqrt(c + d*x^2))]
    @test_int [((a + b*x^2)^2*sqrt(c + d*x^2))/x^(13/2), x, 5, -((2*(77*b^2*c^2 - 22*a*b*c*d + 5*a^2*d^2)*sqrt(c + d*x^2))/(231*c^2*x^(3/2))) - (2*a^2*(c + d*x^2)^(3/2))/(11*c*x^(11/2)) - (2*a*(22*b*c - 5*a*d)*(c + d*x^2)^(3/2))/(77*c^2*x^(7/2)) + (2*d^(3/4)*(77*b^2*c^2 - 22*a*b*c*d + 5*a^2*d^2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(x))/c^(1/4)), 1/2))/(231*c^(9/4)*sqrt(c + d*x^2))]
    @test_int [((a + b*x^2)^2*sqrt(c + d*x^2))/x^(15/2), x, 8, -((2*(39*b^2*c^2 - 26*a*b*c*d + 7*a^2*d^2)*sqrt(c + d*x^2))/(195*c^2*x^(5/2))) - (4*d*(39*b^2*c^2 - 26*a*b*c*d + 7*a^2*d^2)*sqrt(c + d*x^2))/(195*c^3*sqrt(x)) + (4*d^(3/2)*(39*b^2*c^2 - 26*a*b*c*d + 7*a^2*d^2)*sqrt(x)*sqrt(c + d*x^2))/(195*c^3*(sqrt(c) + sqrt(d)*x)) - (2*a^2*(c + d*x^2)^(3/2))/(13*c*x^(13/2)) - (2*a*(26*b*c - 7*a*d)*(c + d*x^2)^(3/2))/(117*c^2*x^(9/2)) - (4*d^(5/4)*(39*b^2*c^2 - 26*a*b*c*d + 7*a^2*d^2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(x))/c^(1/4)), 1/2))/(195*c^(11/4)*sqrt(c + d*x^2)) + (2*d^(5/4)*(39*b^2*c^2 - 26*a*b*c*d + 7*a^2*d^2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(x))/c^(1/4)), 1/2))/(195*c^(11/4)*sqrt(c + d*x^2))]

    @test_int [(e*x)^(5/2)*(a + b*x^2)^2*(c + d*x^2)^(3/2), x, 9, (8*c^2*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*e*(e*x)^(3/2)*sqrt(c + d*x^2))/(9945*d^3) + (4*c*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*(e*x)^(7/2)*sqrt(c + d*x^2))/(1989*d^2*e) - (8*c^3*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*e^2*sqrt(e*x)*sqrt(c + d*x^2))/(3315*d^(7/2)*(sqrt(c) + sqrt(d)*x)) + (2*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*(e*x)^(7/2)*(c + d*x^2)^(3/2))/(663*d^2*e) - (2*b*(11*b*c - 42*a*d)*(e*x)^(7/2)*(c + d*x^2)^(5/2))/(357*d^2*e) + (2*b^2*(e*x)^(11/2)*(c + d*x^2)^(5/2))/(21*d*e^3) + (8*c^(13/4)*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*e^(5/2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(3315*d^(15/4)*sqrt(c + d*x^2)) - (4*c^(13/4)*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*e^(5/2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(3315*d^(15/4)*sqrt(c + d*x^2))]
    @test_int [(e*x)^(3/2)*(a + b*x^2)^2*(c + d*x^2)^(3/2), x, 7, (8*c^2*(57*a^2*d^2 + b*c*(9*b*c - 38*a*d))*e*sqrt(e*x)*sqrt(c + d*x^2))/(4389*d^3) + (4*c*(57*a^2*d^2 + b*c*(9*b*c - 38*a*d))*(e*x)^(5/2)*sqrt(c + d*x^2))/(1463*d^2*e) + (2*(57*a^2*d^2 + b*c*(9*b*c - 38*a*d))*(e*x)^(5/2)*(c + d*x^2)^(3/2))/(627*d^2*e) - (2*b*(9*b*c - 38*a*d)*(e*x)^(5/2)*(c + d*x^2)^(5/2))/(285*d^2*e) + (2*b^2*(e*x)^(9/2)*(c + d*x^2)^(5/2))/(19*d*e^3) - (4*c^(11/4)*(57*a^2*d^2 + b*c*(9*b*c - 38*a*d))*e^(3/2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(4389*d^(13/4)*sqrt(c + d*x^2))]
    @test_int [sqrt(e*x)*(a + b*x^2)^2*(c + d*x^2)^(3/2), x, 8, (4*c*(221*a^2*d^2 + 3*b*c*(7*b*c - 34*a*d))*(e*x)^(3/2)*sqrt(c + d*x^2))/(3315*d^2*e) + (8*c^2*(221*a^2*d^2 + 3*b*c*(7*b*c - 34*a*d))*sqrt(e*x)*sqrt(c + d*x^2))/(3315*d^(5/2)*(sqrt(c) + sqrt(d)*x)) + (2*(221*a^2*d^2 + 3*b*c*(7*b*c - 34*a*d))*(e*x)^(3/2)*(c + d*x^2)^(3/2))/(1989*d^2*e) - (2*b*(7*b*c - 34*a*d)*(e*x)^(3/2)*(c + d*x^2)^(5/2))/(221*d^2*e) + (2*b^2*(e*x)^(7/2)*(c + d*x^2)^(5/2))/(17*d*e^3) - (8*c^(9/4)*(221*a^2*d^2 + 3*b*c*(7*b*c - 34*a*d))*sqrt(e)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(3315*d^(11/4)*sqrt(c + d*x^2)) + (4*c^(9/4)*(221*a^2*d^2 + 3*b*c*(7*b*c - 34*a*d))*sqrt(e)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(3315*d^(11/4)*sqrt(c + d*x^2))]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^(3/2))/sqrt(e*x), x, 6, (4*c*(33*a^2*d^2 + b*c*(b*c - 6*a*d))*sqrt(e*x)*sqrt(c + d*x^2))/(231*d^2*e) + (2*(33*a^2*d^2 + b*c*(b*c - 6*a*d))*sqrt(e*x)*(c + d*x^2)^(3/2))/(231*d^2*e) - (2*b*(b*c - 6*a*d)*sqrt(e*x)*(c + d*x^2)^(5/2))/(33*d^2*e) + (2*b^2*(e*x)^(5/2)*(c + d*x^2)^(5/2))/(15*d*e^3) + (4*c^(7/4)*(33*a^2*d^2 + b*c*(b*c - 6*a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(231*d^(9/4)*sqrt(e)*sqrt(c + d*x^2))]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^(3/2))/(e*x)^(3/2), x, 8, -((4*(3*b^2*c^2 - 13*a*d*(2*b*c + 9*a*d))*(e*x)^(3/2)*sqrt(c + d*x^2))/(195*d*e^3)) - (8*c*(3*b^2*c^2 - 13*a*d*(2*b*c + 9*a*d))*sqrt(e*x)*sqrt(c + d*x^2))/(195*d^(3/2)*e^2*(sqrt(c) + sqrt(d)*x)) - (2*(3*b^2*c^2 - 13*a*d*(2*b*c + 9*a*d))*(e*x)^(3/2)*(c + d*x^2)^(3/2))/(117*c*d*e^3) - (2*a^2*(c + d*x^2)^(5/2))/(c*e*sqrt(e*x)) + (2*b^2*(e*x)^(3/2)*(c + d*x^2)^(5/2))/(13*d*e^3) + (8*c^(5/4)*(3*b^2*c^2 - 13*a*d*(2*b*c + 9*a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(195*d^(7/4)*e^(3/2)*sqrt(c + d*x^2)) - (4*c^(5/4)*(3*b^2*c^2 - 13*a*d*(2*b*c + 9*a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(195*d^(7/4)*e^(3/2)*sqrt(c + d*x^2))]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^(3/2))/(e*x)^(5/2), x, 6, -((4*(3*b^2*c^2 - 11*a*d*(6*b*c + 7*a*d))*sqrt(e*x)*sqrt(c + d*x^2))/(231*d*e^3)) - (2*(3*b^2*c^2 - 11*a*d*(6*b*c + 7*a*d))*sqrt(e*x)*(c + d*x^2)^(3/2))/(231*c*d*e^3) - (2*a^2*(c + d*x^2)^(5/2))/(3*c*e*(e*x)^(3/2)) + (2*b^2*sqrt(e*x)*(c + d*x^2)^(5/2))/(11*d*e^3) - (4*c^(3/4)*(3*b^2*c^2 - 11*a*d*(6*b*c + 7*a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(231*d^(5/4)*e^(5/2)*sqrt(c + d*x^2))]
    @test_int [((a + b*x^2)^2*(c + d*x^2)^(3/2))/(e*x)^(7/2), x, 8, (4*(b^2*c^2 + 9*a*d*(2*b*c + a*d))*(e*x)^(3/2)*sqrt(c + d*x^2))/(15*c*e^5) + (8*(b^2*c^2 + 9*a*d*(2*b*c + a*d))*sqrt(e*x)*sqrt(c + d*x^2))/(15*sqrt(d)*e^4*(sqrt(c) + sqrt(d)*x)) + (2*(b^2*c^2 + 9*a*d*(2*b*c + a*d))*(e*x)^(3/2)*(c + d*x^2)^(3/2))/(9*c^2*e^5) - (2*a^2*(c + d*x^2)^(5/2))/(5*c*e*(e*x)^(5/2)) - (2*a*(2*b*c + a*d)*(c + d*x^2)^(5/2))/(c^2*e^3*sqrt(e*x)) - (8*c^(1/4)*(b^2*c^2 + 9*a*d*(2*b*c + a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(15*d^(3/4)*e^(7/2)*sqrt(c + d*x^2)) + (4*c^(1/4)*(b^2*c^2 + 9*a*d*(2*b*c + a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(15*d^(3/4)*e^(7/2)*sqrt(c + d*x^2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [((e*x)^(5/2)*(a + b*x^2)^2)/sqrt(c + d*x^2), x, 7, (2*(117*a^2*d^2 + 7*b*c*(11*b*c - 26*a*d))*e*(e*x)^(3/2)*sqrt(c + d*x^2))/(585*d^3) - (2*b*(11*b*c - 26*a*d)*(e*x)^(7/2)*sqrt(c + d*x^2))/(117*d^2*e) + (2*b^2*(e*x)^(11/2)*sqrt(c + d*x^2))/(13*d*e^3) - (2*c*(117*a^2*d^2 + 7*b*c*(11*b*c - 26*a*d))*e^2*sqrt(e*x)*sqrt(c + d*x^2))/(195*d^(7/2)*(sqrt(c) + sqrt(d)*x)) + (2*c^(5/4)*(117*a^2*d^2 + 7*b*c*(11*b*c - 26*a*d))*e^(5/2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(195*d^(15/4)*sqrt(c + d*x^2)) - (c^(5/4)*(117*a^2*d^2 + 7*b*c*(11*b*c - 26*a*d))*e^(5/2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(195*d^(15/4)*sqrt(c + d*x^2))]
    @test_int [((e*x)^(3/2)*(a + b*x^2)^2)/sqrt(c + d*x^2), x, 5, (2*(77*a^2*d^2 + 5*b*c*(9*b*c - 22*a*d))*e*sqrt(e*x)*sqrt(c + d*x^2))/(231*d^3) - (2*b*(9*b*c - 22*a*d)*(e*x)^(5/2)*sqrt(c + d*x^2))/(77*d^2*e) + (2*b^2*(e*x)^(9/2)*sqrt(c + d*x^2))/(11*d*e^3) - (c^(3/4)*(77*a^2*d^2 + 5*b*c*(9*b*c - 22*a*d))*e^(3/2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(231*d^(13/4)*sqrt(c + d*x^2))]
    @test_int [(sqrt(e*x)*(a + b*x^2)^2)/sqrt(c + d*x^2), x, 6, -((2*b*(7*b*c - 18*a*d)*(e*x)^(3/2)*sqrt(c + d*x^2))/(45*d^2*e)) + (2*b^2*(e*x)^(7/2)*sqrt(c + d*x^2))/(9*d*e^3) + (2*(15*a^2*d^2 + b*c*(7*b*c - 18*a*d))*sqrt(e*x)*sqrt(c + d*x^2))/(15*d^(5/2)*(sqrt(c) + sqrt(d)*x)) - (2*c^(1/4)*(15*a^2*d^2 + b*c*(7*b*c - 18*a*d))*sqrt(e)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(15*d^(11/4)*sqrt(c + d*x^2)) + (c^(1/4)*(15*a^2*d^2 + b*c*(7*b*c - 18*a*d))*sqrt(e)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(15*d^(11/4)*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^2/(sqrt(e*x)*sqrt(c + d*x^2)), x, 4, -((2*b*(5*b*c - 14*a*d)*sqrt(e*x)*sqrt(c + d*x^2))/(21*d^2*e)) + (2*b^2*(e*x)^(5/2)*sqrt(c + d*x^2))/(7*d*e^3) + ((5*b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(21*c^(1/4)*d^(9/4)*sqrt(e)*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^2/((e*x)^(3/2)*sqrt(c + d*x^2)), x, 6, -((2*a^2*sqrt(c + d*x^2))/(c*e*sqrt(e*x))) + (2*b^2*(e*x)^(3/2)*sqrt(c + d*x^2))/(5*d*e^3) - (2*(3*b^2*c^2 - 5*a*d*(2*b*c + a*d))*sqrt(e*x)*sqrt(c + d*x^2))/(5*c*d^(3/2)*e^2*(sqrt(c) + sqrt(d)*x)) + (2*(3*b^2*c^2 - 5*a*d*(2*b*c + a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(5*c^(3/4)*d^(7/4)*e^(3/2)*sqrt(c + d*x^2)) - ((3*b^2*c^2 - 5*a*d*(2*b*c + a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(5*c^(3/4)*d^(7/4)*e^(3/2)*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^2/((e*x)^(5/2)*sqrt(c + d*x^2)), x, 4, -((2*a^2*sqrt(c + d*x^2))/(3*c*e*(e*x)^(3/2))) + (2*b^2*sqrt(e*x)*sqrt(c + d*x^2))/(3*d*e^3) - ((b^2*c^2 - 6*a*b*c*d + a^2*d^2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(3*c^(5/4)*d^(5/4)*e^(5/2)*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^2/((e*x)^(7/2)*sqrt(c + d*x^2)), x, 6, -((2*a^2*sqrt(c + d*x^2))/(5*c*e*(e*x)^(5/2))) - (2*a*(10*b*c - 3*a*d)*sqrt(c + d*x^2))/(5*c^2*e^3*sqrt(e*x)) + (2*(5*b^2*c^2 + 10*a*b*c*d - 3*a^2*d^2)*sqrt(e*x)*sqrt(c + d*x^2))/(5*c^2*sqrt(d)*e^4*(sqrt(c) + sqrt(d)*x)) - (2*(5*b^2*c^2 + 10*a*b*c*d - 3*a^2*d^2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(5*c^(7/4)*d^(3/4)*e^(7/2)*sqrt(c + d*x^2)) + ((5*b^2*c^2 + 10*a*b*c*d - 3*a^2*d^2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(5*c^(7/4)*d^(3/4)*e^(7/2)*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^2/((e*x)^(9/2)*sqrt(c + d*x^2)), x, 4, -((2*a^2*sqrt(c + d*x^2))/(7*c*e*(e*x)^(7/2))) - (2*a*(14*b*c - 5*a*d)*sqrt(c + d*x^2))/(21*c^2*e^3*(e*x)^(3/2)) + ((21*b^2*c^2 - 14*a*b*c*d + 5*a^2*d^2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(21*c^(9/4)*d^(1/4)*e^(9/2)*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^2/((e*x)^(11/2)*sqrt(c + d*x^2)), x, 7, -((2*a^2*sqrt(c + d*x^2))/(9*c*e*(e*x)^(9/2))) - (2*a*(18*b*c - 7*a*d)*sqrt(c + d*x^2))/(45*c^2*e^3*(e*x)^(5/2)) - (2*(15*b^2*c^2 - 18*a*b*c*d + 7*a^2*d^2)*sqrt(c + d*x^2))/(15*c^3*e^5*sqrt(e*x)) + (2*sqrt(d)*(15*b^2*c^2 - 18*a*b*c*d + 7*a^2*d^2)*sqrt(e*x)*sqrt(c + d*x^2))/(15*c^3*e^6*(sqrt(c) + sqrt(d)*x)) - (2*d^(1/4)*(15*b^2*c^2 - 18*a*b*c*d + 7*a^2*d^2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(15*c^(11/4)*e^(11/2)*sqrt(c + d*x^2)) + (d^(1/4)*(15*b^2*c^2 - 18*a*b*c*d + 7*a^2*d^2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(15*c^(11/4)*e^(11/2)*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^2/((e*x)^(13/2)*sqrt(c + d*x^2)), x, 5, -((2*a^2*sqrt(c + d*x^2))/(11*c*e*(e*x)^(11/2))) - (2*a*(22*b*c - 9*a*d)*sqrt(c + d*x^2))/(77*c^2*e^3*(e*x)^(7/2)) - (2*(77*b^2*c^2 - 5*a*d*(22*b*c - 9*a*d))*sqrt(c + d*x^2))/(231*c^3*e^5*(e*x)^(3/2)) - (d^(3/4)*(77*b^2*c^2 - 5*a*d*(22*b*c - 9*a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(231*c^(13/4)*e^(13/2)*sqrt(c + d*x^2))]

    @test_int [((e*x)^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2), x, 6, ((b*c - a*d)^2*(e*x)^(9/2))/(c*d^2*e*sqrt(c + d*x^2)) + (5*(117*b^2*c^2 - 198*a*b*c*d + 77*a^2*d^2)*e^3*sqrt(e*x)*sqrt(c + d*x^2))/(231*d^4) - ((117*b^2*c^2 - 198*a*b*c*d + 77*a^2*d^2)*e*(e*x)^(5/2)*sqrt(c + d*x^2))/(77*c*d^3) + (2*b^2*(e*x)^(9/2)*sqrt(c + d*x^2))/(11*d^2*e) - (5*c^(3/4)*(117*b^2*c^2 - 198*a*b*c*d + 77*a^2*d^2)*e^(7/2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(462*d^(17/4)*sqrt(c + d*x^2))]
    @test_int [((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2), x, 7, ((b*c - a*d)^2*(e*x)^(7/2))/(c*d^2*e*sqrt(c + d*x^2)) - ((77*b^2*c^2 - 126*a*b*c*d + 45*a^2*d^2)*e*(e*x)^(3/2)*sqrt(c + d*x^2))/(45*c*d^3) + (2*b^2*(e*x)^(7/2)*sqrt(c + d*x^2))/(9*d^2*e) + ((77*b^2*c^2 - 126*a*b*c*d + 45*a^2*d^2)*e^2*sqrt(e*x)*sqrt(c + d*x^2))/(15*d^(7/2)*(sqrt(c) + sqrt(d)*x)) - (c^(1/4)*(77*b^2*c^2 - 126*a*b*c*d + 45*a^2*d^2)*e^(5/2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(15*d^(15/4)*sqrt(c + d*x^2)) + (c^(1/4)*(77*b^2*c^2 - 126*a*b*c*d + 45*a^2*d^2)*e^(5/2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(30*d^(15/4)*sqrt(c + d*x^2))]
    @test_int [((e*x)^(3/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2), x, 5, ((b*c - a*d)^2*(e*x)^(5/2))/(c*d^2*e*sqrt(c + d*x^2)) - ((45*b^2*c^2 - 70*a*b*c*d + 21*a^2*d^2)*e*sqrt(e*x)*sqrt(c + d*x^2))/(21*c*d^3) + (2*b^2*(e*x)^(5/2)*sqrt(c + d*x^2))/(7*d^2*e) + ((45*b^2*c^2 - 70*a*b*c*d + 21*a^2*d^2)*e^(3/2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(42*c^(1/4)*d^(13/4)*sqrt(c + d*x^2))]
    @test_int [(sqrt(e*x)*(a + b*x^2)^2)/(c + d*x^2)^(3/2), x, 6, ((b*c - a*d)^2*(e*x)^(3/2))/(c*d^2*e*sqrt(c + d*x^2)) + (2*b^2*(e*x)^(3/2)*sqrt(c + d*x^2))/(5*d^2*e) - ((21*b^2*c^2 - 30*a*b*c*d + 5*a^2*d^2)*sqrt(e*x)*sqrt(c + d*x^2))/(5*c*d^(5/2)*(sqrt(c) + sqrt(d)*x)) + ((21*b^2*c^2 - 30*a*b*c*d + 5*a^2*d^2)*sqrt(e)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(5*c^(3/4)*d^(11/4)*sqrt(c + d*x^2)) - ((21*b^2*c^2 - 30*a*b*c*d + 5*a^2*d^2)*sqrt(e)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(10*c^(3/4)*d^(11/4)*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^2/(sqrt(e*x)*(c + d*x^2)^(3/2)), x, 4, ((b*c - a*d)^2*sqrt(e*x))/(c*d^2*e*sqrt(c + d*x^2)) + (2*b^2*sqrt(e*x)*sqrt(c + d*x^2))/(3*d^2*e) - ((5*b^2*c^2 - 6*a*b*c*d - 3*a^2*d^2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(6*c^(5/4)*d^(9/4)*sqrt(e)*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^2/((e*x)^(3/2)*(c + d*x^2)^(3/2)), x, 6, -((2*a^2)/(c*e*sqrt(e*x)*sqrt(c + d*x^2))) - ((b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2)*(e*x)^(3/2))/(c^2*d*e^3*sqrt(c + d*x^2)) + ((3*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2)*sqrt(e*x)*sqrt(c + d*x^2))/(c^2*d^(3/2)*e^2*(sqrt(c) + sqrt(d)*x)) - ((3*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(c^(7/4)*d^(7/4)*e^(3/2)*sqrt(c + d*x^2)) + ((3*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(2*c^(7/4)*d^(7/4)*e^(3/2)*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^2/((e*x)^(5/2)*(c + d*x^2)^(3/2)), x, 4, -((2*a^2)/(3*c*e*(e*x)^(3/2)*sqrt(c + d*x^2))) - ((3*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*sqrt(e*x))/(3*c^2*d*e^3*sqrt(c + d*x^2)) + ((3*b^2*c^2 + a*d*(6*b*c - 5*a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(6*c^(9/4)*d^(5/4)*e^(5/2)*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^2/((e*x)^(7/2)*(c + d*x^2)^(3/2)), x, 7, -((2*a^2)/(5*c*e*(e*x)^(5/2)*sqrt(c + d*x^2))) - (2*a*(10*b*c - 7*a*d))/(5*c^2*e^3*sqrt(e*x)*sqrt(c + d*x^2)) + ((5*b^2*c^2 - 3*a*d*(10*b*c - 7*a*d))*(e*x)^(3/2))/(5*c^3*e^5*sqrt(c + d*x^2)) - ((5*b^2*c^2 - 3*a*d*(10*b*c - 7*a*d))*sqrt(e*x)*sqrt(c + d*x^2))/(5*c^3*sqrt(d)*e^4*(sqrt(c) + sqrt(d)*x)) + ((5*b^2*c^2 - 3*a*d*(10*b*c - 7*a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(5*c^(11/4)*d^(3/4)*e^(7/2)*sqrt(c + d*x^2)) - ((5*b^2*c^2 - 3*a*d*(10*b*c - 7*a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(10*c^(11/4)*d^(3/4)*e^(7/2)*sqrt(c + d*x^2))]

    @test_int [((e*x)^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2), x, 6, ((b*c - a*d)^2*(e*x)^(9/2))/(3*c*d^2*e*(c + d*x^2)^(3/2)) + ((39*b^2*c^2 - 42*a*b*c*d + 7*a^2*d^2)*e*(e*x)^(5/2))/(14*c*d^3*sqrt(c + d*x^2)) + (2*b^2*(e*x)^(9/2))/(7*d^2*e*sqrt(c + d*x^2)) - (5*(39*b^2*c^2 - 42*a*b*c*d + 7*a^2*d^2)*e^3*sqrt(e*x)*sqrt(c + d*x^2))/(42*c*d^4) + (5*(39*b^2*c^2 - 42*a*b*c*d + 7*a^2*d^2)*e^(7/2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(84*c^(1/4)*d^(17/4)*sqrt(c + d*x^2))]
    @test_int [((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2), x, 7, ((b*c - a*d)^2*(e*x)^(7/2))/(3*c*d^2*e*(c + d*x^2)^(3/2)) + ((77*b^2*c^2 - 70*a*b*c*d + 5*a^2*d^2)*e*(e*x)^(3/2))/(30*c*d^3*sqrt(c + d*x^2)) + (2*b^2*(e*x)^(7/2))/(5*d^2*e*sqrt(c + d*x^2)) - ((77*b^2*c^2 - 70*a*b*c*d + 5*a^2*d^2)*e^2*sqrt(e*x)*sqrt(c + d*x^2))/(10*c*d^(7/2)*(sqrt(c) + sqrt(d)*x)) + ((77*b^2*c^2 - 70*a*b*c*d + 5*a^2*d^2)*e^(5/2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(10*c^(3/4)*d^(15/4)*sqrt(c + d*x^2)) - ((77*b^2*c^2 - 70*a*b*c*d + 5*a^2*d^2)*e^(5/2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(20*c^(3/4)*d^(15/4)*sqrt(c + d*x^2))]
    @test_int [((e*x)^(3/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2), x, 5, ((b*c - a*d)^2*(e*x)^(5/2))/(3*c*d^2*e*(c + d*x^2)^(3/2)) + ((15*b^2*c^2 - 10*a*b*c*d - a^2*d^2)*e*sqrt(e*x))/(6*c*d^3*sqrt(c + d*x^2)) + (2*b^2*(e*x)^(5/2))/(3*d^2*e*sqrt(c + d*x^2)) - ((15*b^2*c^2 - 10*a*b*c*d - a^2*d^2)*e^(3/2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(12*c^(5/4)*d^(13/4)*sqrt(c + d*x^2))]
    @test_int [(sqrt(e*x)*(a + b*x^2)^2)/(c + d*x^2)^(5/2), x, 6, ((b*c - a*d)^2*(e*x)^(3/2))/(3*c*d^2*e*(c + d*x^2)^(3/2)) - ((b*c - a*d)*(3*b*c + a*d)*(e*x)^(3/2))/(2*c^2*d^2*e*sqrt(c + d*x^2)) + ((7*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*sqrt(e*x)*sqrt(c + d*x^2))/(2*c^2*d^(5/2)*(sqrt(c) + sqrt(d)*x)) - ((7*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*sqrt(e)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(2*c^(7/4)*d^(11/4)*sqrt(c + d*x^2)) + ((7*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*sqrt(e)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(4*c^(7/4)*d^(11/4)*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^2/(sqrt(e*x)*(c + d*x^2)^(5/2)), x, 4, ((b*c - a*d)^2*sqrt(e*x))/(3*c*d^2*e*(c + d*x^2)^(3/2)) - ((b*c - a*d)*(7*b*c + 5*a*d)*sqrt(e*x))/(6*c^2*d^2*e*sqrt(c + d*x^2)) + ((5*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(12*c^(9/4)*d^(9/4)*sqrt(e)*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^2/((e*x)^(3/2)*(c + d*x^2)^(5/2)), x, 7, -((2*a^2)/(c*e*sqrt(e*x)*(c + d*x^2)^(3/2))) - ((b^2*c^2 - 2*a*b*c*d + 7*a^2*d^2)*(e*x)^(3/2))/(3*c^2*d*e^3*(c + d*x^2)^(3/2)) + ((b^2*c^2 + a*d*(2*b*c - 7*a*d))*(e*x)^(3/2))/(2*c^3*d*e^3*sqrt(c + d*x^2)) - ((b^2*c^2 + a*d*(2*b*c - 7*a*d))*sqrt(e*x)*sqrt(c + d*x^2))/(2*c^3*d^(3/2)*e^2*(sqrt(c) + sqrt(d)*x)) + ((b^2*c^2 + a*d*(2*b*c - 7*a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(2*c^(11/4)*d^(7/4)*e^(3/2)*sqrt(c + d*x^2)) - ((b^2*c^2 + a*d*(2*b*c - 7*a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(4*c^(11/4)*d^(7/4)*e^(3/2)*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^2/((e*x)^(5/2)*(c + d*x^2)^(5/2)), x, 5, -((2*a^2)/(3*c*e*(e*x)^(3/2)*(c + d*x^2)^(3/2))) - ((b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2)*sqrt(e*x))/(3*c^2*d*e^3*(c + d*x^2)^(3/2)) + ((b^2*c^2 + 5*a*d*(2*b*c - 3*a*d))*sqrt(e*x))/(6*c^3*d*e^3*sqrt(c + d*x^2)) + ((b^2*c^2 + 5*a*d*(2*b*c - 3*a*d))*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(12*c^(13/4)*d^(5/4)*e^(5/2)*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^2/((e*x)^(7/2)*(c + d*x^2)^(5/2)), x, 8, -((2*a^2)/(5*c*e*(e*x)^(5/2)*(c + d*x^2)^(3/2))) - (2*a*(10*b*c - 11*a*d))/(5*c^2*e^3*sqrt(e*x)*(c + d*x^2)^(3/2)) + ((5*b^2*c^2 - 70*a*b*c*d + 77*a^2*d^2)*(e*x)^(3/2))/(15*c^3*e^5*(c + d*x^2)^(3/2)) + ((5*b^2*c^2 - 70*a*b*c*d + 77*a^2*d^2)*(e*x)^(3/2))/(10*c^4*e^5*sqrt(c + d*x^2)) - ((5*b^2*c^2 - 70*a*b*c*d + 77*a^2*d^2)*sqrt(e*x)*sqrt(c + d*x^2))/(10*c^4*sqrt(d)*e^4*(sqrt(c) + sqrt(d)*x)) + ((5*b^2*c^2 - 70*a*b*c*d + 77*a^2*d^2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.E(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(10*c^(15/4)*d^(3/4)*e^(7/2)*sqrt(c + d*x^2)) - ((5*b^2*c^2 - 70*a*b*c*d + 77*a^2*d^2)*(sqrt(c) + sqrt(d)*x)*sqrt((c + d*x^2)/(sqrt(c) + sqrt(d)*x)^2)*Elliptic.F(2*atan((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), 1/2))/(20*c^(15/4)*d^(3/4)*e^(7/2)*sqrt(c + d*x^2))]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^2)^(p/2)*(c+d*x^2)^3=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^2)^(p/2) / (c+d*x^2)^1=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [((e*x)^(7/2)*sqrt(c - d*x^2))/(a - b*x^2), x, 11, (2*(2*b*c - 7*a*d)*e^3*sqrt(e*x)*sqrt(c - d*x^2))/(21*b^2*d) - (2*e*(e*x)^(5/2)*sqrt(c - d*x^2))/(7*b) - (2*c^(1/4)*(2*b^2*c^2 + 14*a*b*c*d - 21*a^2*d^2)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(21*b^3*d^(5/4)*sqrt(c - d*x^2)) + (a*c^(1/4)*(b*c - a*d)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b^3*d^(1/4)*sqrt(c - d*x^2)) + (a*c^(1/4)*(b*c - a*d)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b^3*d^(1/4)*sqrt(c - d*x^2))]
    @test_int [((e*x)^(5/2)*sqrt(c - d*x^2))/(a - b*x^2), x, 15, (-2*e*(e*x)^(3/2)*sqrt(c - d*x^2))/(5*b) - (2*c^(3/4)*(2*b*c - 5*a*d)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(5*b^2*d^(3/4)*sqrt(c - d*x^2)) + (2*c^(3/4)*(2*b*c - 5*a*d)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(5*b^2*d^(3/4)*sqrt(c - d*x^2)) - (sqrt(a)*c^(1/4)*(b*c - a*d)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b^(5/2)*d^(1/4)*sqrt(c - d*x^2)) + (sqrt(a)*c^(1/4)*(b*c - a*d)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b^(5/2)*d^(1/4)*sqrt(c - d*x^2))]
    @test_int [((e*x)^(3/2)*sqrt(c - d*x^2))/(a - b*x^2), x, 10, (-2*e*sqrt(e*x)*sqrt(c - d*x^2))/(3*b) - (2*c^(1/4)*(2*b*c - 3*a*d)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(3*b^2*d^(1/4)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c - a*d)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b^2*d^(1/4)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c - a*d)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b^2*d^(1/4)*sqrt(c - d*x^2))]
    @test_int [(sqrt(e*x)*sqrt(c - d*x^2))/(a - b*x^2), x, 13, (2*c^(3/4)*d^(1/4)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b*sqrt(c - d*x^2)) - (2*c^(3/4)*d^(1/4)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b*sqrt(c - d*x^2)) - (c^(1/4)*(b*c - a*d)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(sqrt(a)*b^(3/2)*d^(1/4)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c - a*d)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(sqrt(a)*b^(3/2)*d^(1/4)*sqrt(c - d*x^2))]
    @test_int [sqrt(c - d*x^2)/(sqrt(e*x)*(a - b*x^2)), x, 9, (2*c^(1/4)*d^(3/4)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b*sqrt(e)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c - a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a*b*d^(1/4)*sqrt(e)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c - a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a*b*d^(1/4)*sqrt(e)*sqrt(c - d*x^2))]
    @test_int [sqrt(c - d*x^2)/((e*x)^(3/2)*(a - b*x^2)), x, 15, (-2*sqrt(c - d*x^2))/(a*e*sqrt(e*x)) - (2*c^(3/4)*d^(1/4)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a*e^(3/2)*sqrt(c - d*x^2)) + (2*c^(3/4)*d^(1/4)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a*e^(3/2)*sqrt(c - d*x^2)) - (c^(1/4)*(b*c - a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^(3/2)*sqrt(b)*d^(1/4)*e^(3/2)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c - a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^(3/2)*sqrt(b)*d^(1/4)*e^(3/2)*sqrt(c - d*x^2))]
    @test_int [sqrt(c - d*x^2)/((e*x)^(5/2)*(a - b*x^2)), x, 10, (-2*sqrt(c - d*x^2))/(3*a*e*(e*x)^(3/2)) + (2*c^(1/4)*d^(3/4)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(3*a*e^(5/2)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c - a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^2*d^(1/4)*e^(5/2)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c - a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^2*d^(1/4)*e^(5/2)*sqrt(c - d*x^2))]
    @test_int [sqrt(c - d*x^2)/((e*x)^(7/2)*(a - b*x^2)), x, 16, (-2*sqrt(c - d*x^2))/(5*a*e*(e*x)^(5/2)) - (2*(5*b*c - 2*a*d)*sqrt(c - d*x^2))/(5*a^2*c*e^3*sqrt(e*x)) - (2*d^(1/4)*(5*b*c - 2*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(5*a^2*c^(1/4)*e^(7/2)*sqrt(c - d*x^2)) + (2*d^(1/4)*(5*b*c - 2*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(5*a^2*c^(1/4)*e^(7/2)*sqrt(c - d*x^2)) - (sqrt(b)*c^(1/4)*(b*c - a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^(5/2)*d^(1/4)*e^(7/2)*sqrt(c - d*x^2)) + (sqrt(b)*c^(1/4)*(b*c - a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^(5/2)*d^(1/4)*e^(7/2)*sqrt(c - d*x^2))]

    @test_int [((e*x)^(5/2)*(c - d*x^2)^(3/2))/(a - b*x^2), x, 16, (-2*(11*b*c - 9*a*d)*e*(e*x)^(3/2)*sqrt(c - d*x^2))/(45*b^2) + (2*d*(e*x)^(7/2)*sqrt(c - d*x^2))/(9*b*e) - (2*c^(3/4)*(4*b^2*c^2 - 21*a*b*c*d + 15*a^2*d^2)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(15*b^3*d^(3/4)*sqrt(c - d*x^2)) + (2*c^(3/4)*(4*b^2*c^2 - 21*a*b*c*d + 15*a^2*d^2)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(15*b^3*d^(3/4)*sqrt(c - d*x^2)) - (sqrt(a)*c^(1/4)*(b*c - a*d)^2*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b^(7/2)*d^(1/4)*sqrt(c - d*x^2)) + (sqrt(a)*c^(1/4)*(b*c - a*d)^2*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b^(7/2)*d^(1/4)*sqrt(c - d*x^2))]
    @test_int [((e*x)^(3/2)*(c - d*x^2)^(3/2))/(a - b*x^2), x, 11, (-2*(9*b*c - 7*a*d)*e*sqrt(e*x)*sqrt(c - d*x^2))/(21*b^2) + (2*d*(e*x)^(5/2)*sqrt(c - d*x^2))/(7*b*e) - (2*c^(1/4)*(12*b^2*c^2 - 35*a*b*c*d + 21*a^2*d^2)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(21*b^3*d^(1/4)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c - a*d)^2*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b^3*d^(1/4)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c - a*d)^2*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b^3*d^(1/4)*sqrt(c - d*x^2))]
    @test_int [(sqrt(e*x)*(c - d*x^2)^(3/2))/(a - b*x^2), x, 15, (2*d*(e*x)^(3/2)*sqrt(c - d*x^2))/(5*b*e) + (2*c^(3/4)*d^(1/4)*(7*b*c - 5*a*d)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(5*b^2*sqrt(c - d*x^2)) - (2*c^(3/4)*d^(1/4)*(7*b*c - 5*a*d)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(5*b^2*sqrt(c - d*x^2)) - (c^(1/4)*(b*c - a*d)^2*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(sqrt(a)*b^(5/2)*d^(1/4)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c - a*d)^2*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(sqrt(a)*b^(5/2)*d^(1/4)*sqrt(c - d*x^2))]
    @test_int [(c - d*x^2)^(3/2)/(sqrt(e*x)*(a - b*x^2)), x, 10, (2*d*sqrt(e*x)*sqrt(c - d*x^2))/(3*b*e) + (2*c^(1/4)*d^(3/4)*(5*b*c - 3*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(3*b^2*sqrt(e)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c - a*d)^2*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a*b^2*d^(1/4)*sqrt(e)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c - a*d)^2*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a*b^2*d^(1/4)*sqrt(e)*sqrt(c - d*x^2))]
    @test_int [(c - d*x^2)^(3/2)/((e*x)^(3/2)*(a - b*x^2)), x, 15, (-2*c*sqrt(c - d*x^2))/(a*e*sqrt(e*x)) - (2*c^(3/4)*d^(1/4)*(b*c + a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a*b*e^(3/2)*sqrt(c - d*x^2)) + (2*c^(3/4)*d^(1/4)*(b*c + a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a*b*e^(3/2)*sqrt(c - d*x^2)) - (c^(1/4)*(b*c - a*d)^2*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^(3/2)*b^(3/2)*d^(1/4)*e^(3/2)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c - a*d)^2*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^(3/2)*b^(3/2)*d^(1/4)*e^(3/2)*sqrt(c - d*x^2))]
    @test_int [(c - d*x^2)^(3/2)/((e*x)^(5/2)*(a - b*x^2)), x, 10, (-2*c*sqrt(c - d*x^2))/(3*a*e*(e*x)^(3/2)) + (2*c^(1/4)*d^(3/4)*(b*c - 3*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(3*a*b*e^(5/2)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c - a*d)^2*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^2*b*d^(1/4)*e^(5/2)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c - a*d)^2*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^2*b*d^(1/4)*e^(5/2)*sqrt(c - d*x^2))]
    @test_int [(c - d*x^2)^(3/2)/((e*x)^(7/2)*(a - b*x^2)), x, 16, (-2*c*sqrt(c - d*x^2))/(5*a*e*(e*x)^(5/2)) - (2*(5*b*c - 7*a*d)*sqrt(c - d*x^2))/(5*a^2*e^3*sqrt(e*x)) - (2*c^(3/4)*d^(1/4)*(5*b*c - 7*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(5*a^2*e^(7/2)*sqrt(c - d*x^2)) + (2*c^(3/4)*d^(1/4)*(5*b*c - 7*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(5*a^2*e^(7/2)*sqrt(c - d*x^2)) - (c^(1/4)*(b*c - a*d)^2*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^(5/2)*sqrt(b)*d^(1/4)*e^(7/2)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c - a*d)^2*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^(5/2)*sqrt(b)*d^(1/4)*e^(7/2)*sqrt(c - d*x^2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(e*x)^(7/2)/((a - b*x^2)*sqrt(c - d*x^2)), x, 10, (2*e^3*sqrt(e*x)*sqrt(c - d*x^2))/(3*b*d) - (2*c^(1/4)*(b*c + 3*a*d)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(3*b^2*d^(5/4)*sqrt(c - d*x^2)) + (a*c^(1/4)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b^2*d^(1/4)*sqrt(c - d*x^2)) + (a*c^(1/4)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b^2*d^(1/4)*sqrt(c - d*x^2))]
    @test_int [(e*x)^(5/2)/((a - b*x^2)*sqrt(c - d*x^2)), x, 13, (-2*c^(3/4)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b*d^(3/4)*sqrt(c - d*x^2)) + (2*c^(3/4)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b*d^(3/4)*sqrt(c - d*x^2)) - (sqrt(a)*c^(1/4)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b^(3/2)*d^(1/4)*sqrt(c - d*x^2)) + (sqrt(a)*c^(1/4)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b^(3/2)*d^(1/4)*sqrt(c - d*x^2))]
    @test_int [(e*x)^(3/2)/((a - b*x^2)*sqrt(c - d*x^2)), x, 9, (-2*c^(1/4)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b*d^(1/4)*sqrt(c - d*x^2)) + (c^(1/4)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b*d^(1/4)*sqrt(c - d*x^2)) + (c^(1/4)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b*d^(1/4)*sqrt(c - d*x^2))]
    @test_int [sqrt(e*x)/((a - b*x^2)*sqrt(c - d*x^2)), x, 6, -((c^(1/4)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(sqrt(a)*sqrt(b)*d^(1/4)*sqrt(c - d*x^2))) + (c^(1/4)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(sqrt(a)*sqrt(b)*d^(1/4)*sqrt(c - d*x^2))]
    @test_int [1/(sqrt(e*x)*(a - b*x^2)*sqrt(c - d*x^2)), x, 6, (c^(1/4)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a*d^(1/4)*sqrt(e)*sqrt(c - d*x^2)) + (c^(1/4)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a*d^(1/4)*sqrt(e)*sqrt(c - d*x^2))]
    @test_int [1/((e*x)^(3/2)*(a - b*x^2)*sqrt(c - d*x^2)), x, 15, (-2*sqrt(c - d*x^2))/(a*c*e*sqrt(e*x)) - (2*d^(1/4)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a*c^(1/4)*e^(3/2)*sqrt(c - d*x^2)) + (2*d^(1/4)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a*c^(1/4)*e^(3/2)*sqrt(c - d*x^2)) - (sqrt(b)*c^(1/4)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^(3/2)*d^(1/4)*e^(3/2)*sqrt(c - d*x^2)) + (sqrt(b)*c^(1/4)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^(3/2)*d^(1/4)*e^(3/2)*sqrt(c - d*x^2))]
    @test_int [1/((e*x)^(5/2)*(a - b*x^2)*sqrt(c - d*x^2)), x, 10, (-2*sqrt(c - d*x^2))/(3*a*c*e*(e*x)^(3/2)) + (2*d^(3/4)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(3*a*c^(3/4)*e^(5/2)*sqrt(c - d*x^2)) + (b*c^(1/4)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^2*d^(1/4)*e^(5/2)*sqrt(c - d*x^2)) + (b*c^(1/4)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^2*d^(1/4)*e^(5/2)*sqrt(c - d*x^2))]
    @test_int [1/((e*x)^(7/2)*(a - b*x^2)*sqrt(c - d*x^2)), x, 16, (-2*sqrt(c - d*x^2))/(5*a*c*e*(e*x)^(5/2)) - (2*(5*b*c + 3*a*d)*sqrt(c - d*x^2))/(5*a^2*c^2*e^3*sqrt(e*x)) - (2*d^(1/4)*(5*b*c + 3*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(5*a^2*c^(5/4)*e^(7/2)*sqrt(c - d*x^2)) + (2*d^(1/4)*(5*b*c + 3*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(5*a^2*c^(5/4)*e^(7/2)*sqrt(c - d*x^2)) - (b^(3/2)*c^(1/4)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^(5/2)*d^(1/4)*e^(7/2)*sqrt(c - d*x^2)) + (b^(3/2)*c^(1/4)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^(5/2)*d^(1/4)*e^(7/2)*sqrt(c - d*x^2))]

    @test_int [(e*x)^(9/2)/((a - b*x^2)*(c - d*x^2)^(3/2)), x, 15, -((c*e^3*(e*x)^(3/2))/(d*(b*c - a*d)*sqrt(c - d*x^2))) + (c^(3/4)*(3*b*c - 2*a*d)*e^(9/2)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b*d^(7/4)*(b*c - a*d)*sqrt(c - d*x^2)) - (c^(3/4)*(3*b*c - 2*a*d)*e^(9/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b*d^(7/4)*(b*c - a*d)*sqrt(c - d*x^2)) - (a^(3/2)*c^(1/4)*e^(9/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b^(3/2)*d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2)) + (a^(3/2)*c^(1/4)*e^(9/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b^(3/2)*d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2))]
    @test_int [(e*x)^(7/2)/((a - b*x^2)*(c - d*x^2)^(3/2)), x, 10, -((c*e^3*sqrt(e*x))/(d*(b*c - a*d)*sqrt(c - d*x^2))) + (c^(1/4)*(b*c - 2*a*d)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b*d^(5/4)*(b*c - a*d)*sqrt(c - d*x^2)) + (a*c^(1/4)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b*d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2)) + (a*c^(1/4)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(b*d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2))]
    @test_int [(e*x)^(5/2)/((a - b*x^2)*(c - d*x^2)^(3/2)), x, 15, -((e*(e*x)^(3/2))/((b*c - a*d)*sqrt(c - d*x^2))) + (c^(3/4)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(d^(3/4)*(b*c - a*d)*sqrt(c - d*x^2)) - (c^(3/4)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(d^(3/4)*(b*c - a*d)*sqrt(c - d*x^2)) - (sqrt(a)*c^(1/4)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(sqrt(b)*d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2)) + (sqrt(a)*c^(1/4)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(sqrt(b)*d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2))]
    @test_int [(e*x)^(3/2)/((a - b*x^2)*(c - d*x^2)^(3/2)), x, 10, -((e*sqrt(e*x))/((b*c - a*d)*sqrt(c - d*x^2))) - (c^(1/4)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2)) + (c^(1/4)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2)) + (c^(1/4)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2))]
    @test_int [sqrt(e*x)/((a - b*x^2)*(c - d*x^2)^(3/2)), x, 15, -((d*(e*x)^(3/2))/(c*(b*c - a*d)*e*sqrt(c - d*x^2))) + (d^(1/4)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(c^(1/4)*(b*c - a*d)*sqrt(c - d*x^2)) - (d^(1/4)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(c^(1/4)*(b*c - a*d)*sqrt(c - d*x^2)) - (sqrt(b)*c^(1/4)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(sqrt(a)*d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2)) + (sqrt(b)*c^(1/4)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(sqrt(a)*d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2))]
    @test_int [1/(sqrt(e*x)*(a - b*x^2)*(c - d*x^2)^(3/2)), x, 10, -((d*sqrt(e*x))/(c*(b*c - a*d)*e*sqrt(c - d*x^2))) - (d^(3/4)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(c^(3/4)*(b*c - a*d)*sqrt(e)*sqrt(c - d*x^2)) + (b*c^(1/4)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a*d^(1/4)*(b*c - a*d)*sqrt(e)*sqrt(c - d*x^2)) + (b*c^(1/4)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a*d^(1/4)*(b*c - a*d)*sqrt(e)*sqrt(c - d*x^2))]
    @test_int [1/((e*x)^(3/2)*(a - b*x^2)*(c - d*x^2)^(3/2)), x, 16, -(d/(c*(b*c - a*d)*e*sqrt(e*x)*sqrt(c - d*x^2))) - ((2*b*c - 3*a*d)*sqrt(c - d*x^2))/(a*c^2*(b*c - a*d)*e*sqrt(e*x)) - (d^(1/4)*(2*b*c - 3*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a*c^(5/4)*(b*c - a*d)*e^(3/2)*sqrt(c - d*x^2)) + (d^(1/4)*(2*b*c - 3*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a*c^(5/4)*(b*c - a*d)*e^(3/2)*sqrt(c - d*x^2)) - (b^(3/2)*c^(1/4)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^(3/2)*d^(1/4)*(b*c - a*d)*e^(3/2)*sqrt(c - d*x^2)) + (b^(3/2)*c^(1/4)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^(3/2)*d^(1/4)*(b*c - a*d)*e^(3/2)*sqrt(c - d*x^2))]
    @test_int [1/((e*x)^(5/2)*(a - b*x^2)*(c - d*x^2)^(3/2)), x, 11, -(d/(c*(b*c - a*d)*e*(e*x)^(3/2)*sqrt(c - d*x^2))) - ((2*b*c - 5*a*d)*sqrt(c - d*x^2))/(3*a*c^2*(b*c - a*d)*e*(e*x)^(3/2)) + (d^(3/4)*(2*b*c - 5*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(3*a*c^(7/4)*(b*c - a*d)*e^(5/2)*sqrt(c - d*x^2)) + (b^2*c^(1/4)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^2*d^(1/4)*(b*c - a*d)*e^(5/2)*sqrt(c - d*x^2)) + (b^2*c^(1/4)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(a^2*d^(1/4)*(b*c - a*d)*e^(5/2)*sqrt(c - d*x^2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^2)^(p/2) / (c+d*x^2)^2=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [((e*x)^(7/2)*sqrt(c - d*x^2))/(a - b*x^2)^2, x, 11, (7*e^3*sqrt(e*x)*sqrt(c - d*x^2))/(6*b^2) + (e*(e*x)^(5/2)*sqrt(c - d*x^2))/(2*b*(a - b*x^2)) + (c^(1/4)*(8*b*c - 21*a*d)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(6*b^3*d^(1/4)*sqrt(c - d*x^2)) - (c^(1/4)*(5*b*c - 7*a*d)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*b^3*d^(1/4)*sqrt(c - d*x^2)) - (c^(1/4)*(5*b*c - 7*a*d)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*b^3*d^(1/4)*sqrt(c - d*x^2))]
    @test_int [((e*x)^(5/2)*sqrt(c - d*x^2))/(a - b*x^2)^2, x, 15, (e*(e*x)^(3/2)*sqrt(c - d*x^2))/(2*b*(a - b*x^2)) - (5*c^(3/4)*d^(1/4)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*b^2*sqrt(c - d*x^2)) + (5*c^(3/4)*d^(1/4)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*b^2*sqrt(c - d*x^2)) + (c^(1/4)*(3*b*c - 5*a*d)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*sqrt(a)*b^(5/2)*d^(1/4)*sqrt(c - d*x^2)) - (c^(1/4)*(3*b*c - 5*a*d)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*sqrt(a)*b^(5/2)*d^(1/4)*sqrt(c - d*x^2))]
    @test_int [((e*x)^(3/2)*sqrt(c - d*x^2))/(a - b*x^2)^2, x, 10, (e*sqrt(e*x)*sqrt(c - d*x^2))/(2*b*(a - b*x^2)) - (3*c^(1/4)*d^(3/4)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*b^2*sqrt(c - d*x^2)) - (c^(1/4)*(b*c - 3*a*d)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a*b^2*d^(1/4)*sqrt(c - d*x^2)) - (c^(1/4)*(b*c - 3*a*d)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a*b^2*d^(1/4)*sqrt(c - d*x^2))]
    @test_int [((e*x)^(1/2)*sqrt(c - d*x^2))/(a - b*x^2)^2, x, 15, ((e*x)^(3/2)*sqrt(c - d*x^2))/(2*a*e*(a - b*x^2)) - (c^(3/4)*d^(1/4)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a*b*sqrt(c - d*x^2)) + (c^(3/4)*d^(1/4)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a*b*sqrt(c - d*x^2)) - (c^(1/4)*(b*c + a*d)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^(3/2)*b^(3/2)*d^(1/4)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c + a*d)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^(3/2)*b^(3/2)*d^(1/4)*sqrt(c - d*x^2))]
    @test_int [sqrt(c - d*x^2)/((e*x)^(1/2)*(a - b*x^2)^2), x, 10, (sqrt(e*x)*sqrt(c - d*x^2))/(2*a*e*(a - b*x^2)) + (c^(1/4)*d^(3/4)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a*b*sqrt(e)*sqrt(c - d*x^2)) + (c^(1/4)*(3*b*c - a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^2*b*d^(1/4)*sqrt(e)*sqrt(c - d*x^2)) + (c^(1/4)*(3*b*c - a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^2*b*d^(1/4)*sqrt(e)*sqrt(c - d*x^2))]
    @test_int [sqrt(c - d*x^2)/((e*x)^(3/2)*(a - b*x^2)^2), x, 16, (-5*sqrt(c - d*x^2))/(2*a^2*e*sqrt(e*x)) + sqrt(c - d*x^2)/(2*a*e*sqrt(e*x)*(a - b*x^2)) - (5*c^(3/4)*d^(1/4)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a^2*e^(3/2)*sqrt(c - d*x^2)) + (5*c^(3/4)*d^(1/4)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a^2*e^(3/2)*sqrt(c - d*x^2)) - (c^(1/4)*(5*b*c - 3*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^(5/2)*sqrt(b)*d^(1/4)*e^(3/2)*sqrt(c - d*x^2)) + (c^(1/4)*(5*b*c - 3*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^(5/2)*sqrt(b)*d^(1/4)*e^(3/2)*sqrt(c - d*x^2))]
    @test_int [sqrt(c - d*x^2)/((e*x)^(5/2)*(a - b*x^2)^2), x, 11, (-7*sqrt(c - d*x^2))/(6*a^2*e*(e*x)^(3/2)) + sqrt(c - d*x^2)/(2*a*e*(e*x)^(3/2)*(a - b*x^2)) + (7*c^(1/4)*d^(3/4)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(6*a^2*e^(5/2)*sqrt(c - d*x^2)) + (c^(1/4)*(7*b*c - 5*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^3*d^(1/4)*e^(5/2)*sqrt(c - d*x^2)) + (c^(1/4)*(7*b*c - 5*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^3*d^(1/4)*e^(5/2)*sqrt(c - d*x^2))]

    @test_int [((e*x)^(7/2)*(c - d*x^2)^(3/2))/(a - b*x^2)^2, x, 12, ((57*b*c - 77*a*d)*e^3*sqrt(e*x)*sqrt(c - d*x^2))/(42*b^3) - (11*d*e*(e*x)^(5/2)*sqrt(c - d*x^2))/(14*b^2) + (e*(e*x)^(5/2)*(c - d*x^2)^(3/2))/(2*b*(a - b*x^2)) + (c^(1/4)*(48*b^2*c^2 - 259*a*b*c*d + 231*a^2*d^2)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(42*b^4*d^(1/4)*sqrt(c - d*x^2)) - (c^(1/4)*(5*b*c - 11*a*d)*(b*c - a*d)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*b^4*d^(1/4)*sqrt(c - d*x^2)) - (c^(1/4)*(5*b*c - 11*a*d)*(b*c - a*d)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*b^4*d^(1/4)*sqrt(c - d*x^2))]
    @test_int [((e*x)^(5/2)*(c - d*x^2)^(3/2))/(a - b*x^2)^2, x, 16, (-9*d*e*(e*x)^(3/2)*sqrt(c - d*x^2))/(10*b^2) + (e*(e*x)^(3/2)*(c - d*x^2)^(3/2))/(2*b*(a - b*x^2)) - (3*c^(3/4)*d^(1/4)*(11*b*c - 15*a*d)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(10*b^3*sqrt(c - d*x^2)) + (3*c^(3/4)*d^(1/4)*(11*b*c - 15*a*d)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(10*b^3*sqrt(c - d*x^2)) + (3*c^(1/4)*(b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*sqrt(a)*b^(7/2)*d^(1/4)*sqrt(c - d*x^2)) - (3*c^(1/4)*(b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*sqrt(a)*b^(7/2)*d^(1/4)*sqrt(c - d*x^2))]
    @test_int [((e*x)^(3/2)*(c - d*x^2)^(3/2))/(a - b*x^2)^2, x, 11, (-7*d*e*sqrt(e*x)*sqrt(c - d*x^2))/(6*b^2) + (e*sqrt(e*x)*(c - d*x^2)^(3/2))/(2*b*(a - b*x^2)) - (c^(1/4)*d^(3/4)*(17*b*c - 21*a*d)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(6*b^3*sqrt(c - d*x^2)) - (c^(1/4)*(b*c - 7*a*d)*(b*c - a*d)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a*b^3*d^(1/4)*sqrt(c - d*x^2)) - (c^(1/4)*(b*c - 7*a*d)*(b*c - a*d)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a*b^3*d^(1/4)*sqrt(c - d*x^2))]
    @test_int [((e*x)^(1/2)*(c - d*x^2)^(3/2))/(a - b*x^2)^2, x, 15, ((b*c - a*d)*(e*x)^(3/2)*sqrt(c - d*x^2))/(2*a*b*e*(a - b*x^2)) - (c^(3/4)*d^(1/4)*(b*c - 5*a*d)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a*b^2*sqrt(c - d*x^2)) + (c^(3/4)*d^(1/4)*(b*c - 5*a*d)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a*b^2*sqrt(c - d*x^2)) - (c^(1/4)*(b^2*c^2 + 4*a*b*c*d - 5*a^2*d^2)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^(3/2)*b^(5/2)*d^(1/4)*sqrt(c - d*x^2)) + (c^(1/4)*(b^2*c^2 + 4*a*b*c*d - 5*a^2*d^2)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^(3/2)*b^(5/2)*d^(1/4)*sqrt(c - d*x^2))]
    @test_int [(c - d*x^2)^(3/2)/((e*x)^(1/2)*(a - b*x^2)^2), x, 10, ((b*c - a*d)*sqrt(e*x)*sqrt(c - d*x^2))/(2*a*b*e*(a - b*x^2)) + (c^(1/4)*d^(3/4)*(b*c + 3*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a*b^2*sqrt(e)*sqrt(c - d*x^2)) + (3*c^(1/4)*(b*c - a*d)*(b*c + a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^2*b^2*d^(1/4)*sqrt(e)*sqrt(c - d*x^2)) + (3*c^(1/4)*(b*c - a*d)*(b*c + a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^2*b^2*d^(1/4)*sqrt(e)*sqrt(c - d*x^2))]
    @test_int [(c - d*x^2)^(3/2)/((e*x)^(3/2)*(a - b*x^2)^2), x, 16, -((5*b*c - a*d)*sqrt(c - d*x^2))/(2*a^2*b*e*sqrt(e*x)) + ((b*c - a*d)*sqrt(c - d*x^2))/(2*a*b*e*sqrt(e*x)*(a - b*x^2)) - (c^(3/4)*d^(1/4)*(5*b*c - a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a^2*b*e^(3/2)*sqrt(c - d*x^2)) + (c^(3/4)*d^(1/4)*(5*b*c - a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a^2*b*e^(3/2)*sqrt(c - d*x^2)) - (c^(1/4)*(5*b^2*c^2 - 4*a*b*c*d - a^2*d^2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^(5/2)*b^(3/2)*d^(1/4)*e^(3/2)*sqrt(c - d*x^2)) + (c^(1/4)*(5*b^2*c^2 - 4*a*b*c*d - a^2*d^2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^(5/2)*b^(3/2)*d^(1/4)*e^(3/2)*sqrt(c - d*x^2))]
    @test_int [(c - d*x^2)^(3/2)/((e*x)^(5/2)*(a - b*x^2)^2), x, 11, -((7*b*c - 3*a*d)*sqrt(c - d*x^2))/(6*a^2*b*e*(e*x)^(3/2)) + ((b*c - a*d)*sqrt(c - d*x^2))/(2*a*b*e*(e*x)^(3/2)*(a - b*x^2)) + (c^(1/4)*d^(3/4)*(7*b*c - 3*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(6*a^2*b*e^(5/2)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c - a*d)*(7*b*c - a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^3*b*d^(1/4)*e^(5/2)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c - a*d)*(7*b*c - a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^3*b*d^(1/4)*e^(5/2)*sqrt(c - d*x^2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(e*x)^(9/2)/((a - b*x^2)^2*sqrt(c - d*x^2)), x, 15, (a*e^3*(e*x)^(3/2)*sqrt(c - d*x^2))/(2*b*(b*c - a*d)*(a - b*x^2)) + (c^(3/4)*(4*b*c - 5*a*d)*e^(9/2)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*b^2*d^(3/4)*(b*c - a*d)*sqrt(c - d*x^2)) - (c^(3/4)*(4*b*c - 5*a*d)*e^(9/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*b^2*d^(3/4)*(b*c - a*d)*sqrt(c - d*x^2)) + (sqrt(a)*c^(1/4)*(7*b*c - 5*a*d)*e^(9/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*b^(5/2)*d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2)) - (sqrt(a)*c^(1/4)*(7*b*c - 5*a*d)*e^(9/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*b^(5/2)*d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2))]
    @test_int [(e*x)^(7/2)/((a - b*x^2)^2*sqrt(c - d*x^2)), x, 10, (a*e^3*sqrt(e*x)*sqrt(c - d*x^2))/(2*b*(b*c - a*d)*(a - b*x^2)) + (c^(1/4)*(4*b*c - 3*a*d)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*b^2*d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2)) - (c^(1/4)*(5*b*c - 3*a*d)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*b^2*d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2)) - (c^(1/4)*(5*b*c - 3*a*d)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*b^2*d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2))]
    @test_int [(e*x)^(5/2)/((a - b*x^2)^2*sqrt(c - d*x^2)), x, 15, (e*(e*x)^(3/2)*sqrt(c - d*x^2))/(2*(b*c - a*d)*(a - b*x^2)) - (c^(3/4)*d^(1/4)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*b*(b*c - a*d)*sqrt(c - d*x^2)) + (c^(3/4)*d^(1/4)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*b*(b*c - a*d)*sqrt(c - d*x^2)) + (c^(1/4)*(3*b*c - a*d)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*sqrt(a)*b^(3/2)*d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2)) - (c^(1/4)*(3*b*c - a*d)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*sqrt(a)*b^(3/2)*d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2))]
    @test_int [(e*x)^(3/2)/((a - b*x^2)^2*sqrt(c - d*x^2)), x, 10, (e*sqrt(e*x)*sqrt(c - d*x^2))/(2*(b*c - a*d)*(a - b*x^2)) + (c^(1/4)*d^(3/4)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*b*(b*c - a*d)*sqrt(c - d*x^2)) - (c^(1/4)*(b*c + a*d)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a*b*d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2)) - (c^(1/4)*(b*c + a*d)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a*b*d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2))]
    @test_int [(e*x)^(1/2)/((a - b*x^2)^2*sqrt(c - d*x^2)), x, 15, (b*(e*x)^(3/2)*sqrt(c - d*x^2))/(2*a*(b*c - a*d)*e*(a - b*x^2)) - (c^(3/4)*d^(1/4)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a*(b*c - a*d)*sqrt(c - d*x^2)) + (c^(3/4)*d^(1/4)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a*(b*c - a*d)*sqrt(c - d*x^2)) - (c^(1/4)*(b*c - 3*a*d)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^(3/2)*sqrt(b)*d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2)) + (c^(1/4)*(b*c - 3*a*d)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^(3/2)*sqrt(b)*d^(1/4)*(b*c - a*d)*sqrt(c - d*x^2))]
    @test_int [1/((e*x)^(1/2)*(a - b*x^2)^2*sqrt(c - d*x^2)), x, 10, (b*sqrt(e*x)*sqrt(c - d*x^2))/(2*a*(b*c - a*d)*e*(a - b*x^2)) + (c^(1/4)*d^(3/4)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a*(b*c - a*d)*sqrt(e)*sqrt(c - d*x^2)) + (c^(1/4)*(3*b*c - 5*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^2*d^(1/4)*(b*c - a*d)*sqrt(e)*sqrt(c - d*x^2)) + (c^(1/4)*(3*b*c - 5*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^2*d^(1/4)*(b*c - a*d)*sqrt(e)*sqrt(c - d*x^2))]
    @test_int [1/((e*x)^(3/2)*(a - b*x^2)^2*sqrt(c - d*x^2)), x, 16, -((5*b*c - 4*a*d)*sqrt(c - d*x^2))/(2*a^2*c*(b*c - a*d)*e*sqrt(e*x)) + (b*sqrt(c - d*x^2))/(2*a*(b*c - a*d)*e*sqrt(e*x)*(a - b*x^2)) - (d^(1/4)*(5*b*c - 4*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a^2*c^(1/4)*(b*c - a*d)*e^(3/2)*sqrt(c - d*x^2)) + (d^(1/4)*(5*b*c - 4*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a^2*c^(1/4)*(b*c - a*d)*e^(3/2)*sqrt(c - d*x^2)) - (sqrt(b)*c^(1/4)*(5*b*c - 7*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^(5/2)*d^(1/4)*(b*c - a*d)*e^(3/2)*sqrt(c - d*x^2)) + (sqrt(b)*c^(1/4)*(5*b*c - 7*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^(5/2)*d^(1/4)*(b*c - a*d)*e^(3/2)*sqrt(c - d*x^2))]
    @test_int [1/((e*x)^(5/2)*(a - b*x^2)^2*sqrt(c - d*x^2)), x, 11, -((7*b*c - 4*a*d)*sqrt(c - d*x^2))/(6*a^2*c*(b*c - a*d)*e*(e*x)^(3/2)) + (b*sqrt(c - d*x^2))/(2*a*(b*c - a*d)*e*(e*x)^(3/2)*(a - b*x^2)) + (d^(3/4)*(7*b*c - 4*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(6*a^2*c^(3/4)*(b*c - a*d)*e^(5/2)*sqrt(c - d*x^2)) + (b*c^(1/4)*(7*b*c - 9*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^3*d^(1/4)*(b*c - a*d)*e^(5/2)*sqrt(c - d*x^2)) + (b*c^(1/4)*(7*b*c - 9*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^3*d^(1/4)*(b*c - a*d)*e^(5/2)*sqrt(c - d*x^2))]

    @test_int [(e*x)^(9/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)), x, 16, ((2*b*c + a*d)*e^3*(e*x)^(3/2))/(2*b*(b*c - a*d)^2*sqrt(c - d*x^2)) + (a*e^3*(e*x)^(3/2))/(2*b*(b*c - a*d)*(a - b*x^2)*sqrt(c - d*x^2)) - (c^(3/4)*(2*b*c + a*d)*e^(9/2)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*b*d^(3/4)*(b*c - a*d)^2*sqrt(c - d*x^2)) + (c^(3/4)*(2*b*c + a*d)*e^(9/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*b*d^(3/4)*(b*c - a*d)^2*sqrt(c - d*x^2)) + (sqrt(a)*c^(1/4)*(7*b*c - a*d)*e^(9/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*b^(3/2)*d^(1/4)*(b*c - a*d)^2*sqrt(c - d*x^2)) - (sqrt(a)*c^(1/4)*(7*b*c - a*d)*e^(9/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*b^(3/2)*d^(1/4)*(b*c - a*d)^2*sqrt(c - d*x^2))]
    @test_int [(e*x)^(7/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)), x, 11, ((2*b*c + a*d)*e^3*sqrt(e*x))/(2*b*(b*c - a*d)^2*sqrt(c - d*x^2)) + (a*e^3*sqrt(e*x))/(2*b*(b*c - a*d)*(a - b*x^2)*sqrt(c - d*x^2)) + (c^(1/4)*(2*b*c + a*d)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*b*d^(1/4)*(b*c - a*d)^2*sqrt(c - d*x^2)) - (c^(1/4)*(5*b*c + a*d)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*b*d^(1/4)*(b*c - a*d)^2*sqrt(c - d*x^2)) - (c^(1/4)*(5*b*c + a*d)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*b*d^(1/4)*(b*c - a*d)^2*sqrt(c - d*x^2))]
    @test_int [(e*x)^(5/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)), x, 16, (3*d*e*(e*x)^(3/2))/(2*(b*c - a*d)^2*sqrt(c - d*x^2)) + (e*(e*x)^(3/2))/(2*(b*c - a*d)*(a - b*x^2)*sqrt(c - d*x^2)) - (3*c^(3/4)*d^(1/4)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*(b*c - a*d)^2*sqrt(c - d*x^2)) + (3*c^(3/4)*d^(1/4)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*(b*c - a*d)^2*sqrt(c - d*x^2)) + (3*c^(1/4)*(b*c + a*d)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*sqrt(a)*sqrt(b)*d^(1/4)*(b*c - a*d)^2*sqrt(c - d*x^2)) - (3*c^(1/4)*(b*c + a*d)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*sqrt(a)*sqrt(b)*d^(1/4)*(b*c - a*d)^2*sqrt(c - d*x^2))]
    @test_int [(e*x)^(3/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)), x, 11, (3*d*e*sqrt(e*x))/(2*(b*c - a*d)^2*sqrt(c - d*x^2)) + (e*sqrt(e*x))/(2*(b*c - a*d)*(a - b*x^2)*sqrt(c - d*x^2)) + (3*c^(1/4)*d^(3/4)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*(b*c - a*d)^2*sqrt(c - d*x^2)) - (c^(1/4)*(b*c + 5*a*d)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a*d^(1/4)*(b*c - a*d)^2*sqrt(c - d*x^2)) - (c^(1/4)*(b*c + 5*a*d)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a*d^(1/4)*(b*c - a*d)^2*sqrt(c - d*x^2))]
    @test_int [(e*x)^(1/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)), x, 16, (d*(b*c + 2*a*d)*(e*x)^(3/2))/(2*a*c*(b*c - a*d)^2*e*sqrt(c - d*x^2)) + (b*(e*x)^(3/2))/(2*a*(b*c - a*d)*e*(a - b*x^2)*sqrt(c - d*x^2)) - (d^(1/4)*(b*c + 2*a*d)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a*c^(1/4)*(b*c - a*d)^2*sqrt(c - d*x^2)) + (d^(1/4)*(b*c + 2*a*d)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a*c^(1/4)*(b*c - a*d)^2*sqrt(c - d*x^2)) - (sqrt(b)*c^(1/4)*(b*c - 7*a*d)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^(3/2)*d^(1/4)*(b*c - a*d)^2*sqrt(c - d*x^2)) + (sqrt(b)*c^(1/4)*(b*c - 7*a*d)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^(3/2)*d^(1/4)*(b*c - a*d)^2*sqrt(c - d*x^2))]
    @test_int [1/((e*x)^(1/2)*(a - b*x^2)^2*(c - d*x^2)^(3/2)), x, 11, (d*(b*c + 2*a*d)*sqrt(e*x))/(2*a*c*(b*c - a*d)^2*e*sqrt(c - d*x^2)) + (b*sqrt(e*x))/(2*a*(b*c - a*d)*e*(a - b*x^2)*sqrt(c - d*x^2)) + (d^(3/4)*(b*c + 2*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a*c^(3/4)*(b*c - a*d)^2*sqrt(e)*sqrt(c - d*x^2)) + (3*b*c^(1/4)*(b*c - 3*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^2*d^(1/4)*(b*c - a*d)^2*sqrt(e)*sqrt(c - d*x^2)) + (3*b*c^(1/4)*(b*c - 3*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^2*d^(1/4)*(b*c - a*d)^2*sqrt(e)*sqrt(c - d*x^2))]
    @test_int [1/((e*x)^(3/2)*(a - b*x^2)^2*(c - d*x^2)^(3/2)), x, 17, (d*(b*c + 2*a*d))/(2*a*c*(b*c - a*d)^2*e*sqrt(e*x)*sqrt(c - d*x^2)) + b/(2*a*(b*c - a*d)*e*sqrt(e*x)*(a - b*x^2)*sqrt(c - d*x^2)) - ((5*b^2*c^2 - 8*a*b*c*d + 6*a^2*d^2)*sqrt(c - d*x^2))/(2*a^2*c^2*(b*c - a*d)^2*e*sqrt(e*x)) - (d^(1/4)*(5*b^2*c^2 - 8*a*b*c*d + 6*a^2*d^2)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a^2*c^(5/4)*(b*c - a*d)^2*e^(3/2)*sqrt(c - d*x^2)) + (d^(1/4)*(5*b^2*c^2 - 8*a*b*c*d + 6*a^2*d^2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a^2*c^(5/4)*(b*c - a*d)^2*e^(3/2)*sqrt(c - d*x^2)) - (b^(3/2)*c^(1/4)*(5*b*c - 11*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^(5/2)*d^(1/4)*(b*c - a*d)^2*e^(3/2)*sqrt(c - d*x^2)) + (b^(3/2)*c^(1/4)*(5*b*c - 11*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^(5/2)*d^(1/4)*(b*c - a*d)^2*e^(3/2)*sqrt(c - d*x^2))]
    @test_int [1/((e*x)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(3/2)), x, 12, (d*(b*c + 2*a*d))/(2*a*c*(b*c - a*d)^2*e*(e*x)^(3/2)*sqrt(c - d*x^2)) + b/(2*a*(b*c - a*d)*e*(e*x)^(3/2)*(a - b*x^2)*sqrt(c - d*x^2)) - ((7*b^2*c^2 - 8*a*b*c*d + 10*a^2*d^2)*sqrt(c - d*x^2))/(6*a^2*c^2*(b*c - a*d)^2*e*(e*x)^(3/2)) + (d^(3/4)*(7*b^2*c^2 - 8*a*b*c*d + 10*a^2*d^2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(6*a^2*c^(7/4)*(b*c - a*d)^2*e^(5/2)*sqrt(c - d*x^2)) + (b^2*c^(1/4)*(7*b*c - 13*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^3*d^(1/4)*(b*c - a*d)^2*e^(5/2)*sqrt(c - d*x^2)) + (b^2*c^(1/4)*(7*b*c - 13*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^3*d^(1/4)*(b*c - a*d)^2*e^(5/2)*sqrt(c - d*x^2))]

    @test_int [(e*x)^(9/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)), x, 17, ((2*b*c + 3*a*d)*e^3*(e*x)^(3/2))/(6*b*(b*c - a*d)^2*(c - d*x^2)^(3/2)) + (a*e^3*(e*x)^(3/2))/(2*b*(b*c - a*d)*(a - b*x^2)*(c - d*x^2)^(3/2)) + ((b*c + 4*a*d)*e^3*(e*x)^(3/2))/(2*(b*c - a*d)^3*sqrt(c - d*x^2)) - (c^(3/4)*(b*c + 4*a*d)*e^(9/2)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*d^(3/4)*(b*c - a*d)^3*sqrt(c - d*x^2)) + (c^(3/4)*(b*c + 4*a*d)*e^(9/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*d^(3/4)*(b*c - a*d)^3*sqrt(c - d*x^2)) + (sqrt(a)*c^(1/4)*(7*b*c + 3*a*d)*e^(9/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*sqrt(b)*d^(1/4)*(b*c - a*d)^3*sqrt(c - d*x^2)) - (sqrt(a)*c^(1/4)*(7*b*c + 3*a*d)*e^(9/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*sqrt(b)*d^(1/4)*(b*c - a*d)^3*sqrt(c - d*x^2))]
    @test_int [(e*x)^(7/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)), x, 12, ((2*b*c + 3*a*d)*e^3*sqrt(e*x))/(6*b*(b*c - a*d)^2*(c - d*x^2)^(3/2)) + (a*e^3*sqrt(e*x))/(2*b*(b*c - a*d)*(a - b*x^2)*(c - d*x^2)^(3/2)) + (5*(b*c + 2*a*d)*e^3*sqrt(e*x))/(6*(b*c - a*d)^3*sqrt(c - d*x^2)) + (5*c^(1/4)*(b*c + 2*a*d)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(6*d^(1/4)*(b*c - a*d)^3*sqrt(c - d*x^2)) - (5*c^(1/4)*(b*c + a*d)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*d^(1/4)*(b*c - a*d)^3*sqrt(c - d*x^2)) - (5*c^(1/4)*(b*c + a*d)*e^(7/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*d^(1/4)*(b*c - a*d)^3*sqrt(c - d*x^2))]
    @test_int [(e*x)^(5/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)), x, 17, (5*d*e*(e*x)^(3/2))/(6*(b*c - a*d)^2*(c - d*x^2)^(3/2)) + (e*(e*x)^(3/2))/(2*(b*c - a*d)*(a - b*x^2)*(c - d*x^2)^(3/2)) + (d*(4*b*c + a*d)*e*(e*x)^(3/2))/(2*c*(b*c - a*d)^3*sqrt(c - d*x^2)) - (d^(1/4)*(4*b*c + a*d)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*c^(1/4)*(b*c - a*d)^3*sqrt(c - d*x^2)) + (d^(1/4)*(4*b*c + a*d)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*c^(1/4)*(b*c - a*d)^3*sqrt(c - d*x^2)) + (sqrt(b)*c^(1/4)*(3*b*c + 7*a*d)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*sqrt(a)*d^(1/4)*(b*c - a*d)^3*sqrt(c - d*x^2)) - (sqrt(b)*c^(1/4)*(3*b*c + 7*a*d)*e^(5/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*sqrt(a)*d^(1/4)*(b*c - a*d)^3*sqrt(c - d*x^2))]
    @test_int [(e*x)^(3/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)), x, 12, (5*d*e*sqrt(e*x))/(6*(b*c - a*d)^2*(c - d*x^2)^(3/2)) + (e*sqrt(e*x))/(2*(b*c - a*d)*(a - b*x^2)*(c - d*x^2)^(3/2)) + (d*(14*b*c + a*d)*e*sqrt(e*x))/(6*c*(b*c - a*d)^3*sqrt(c - d*x^2)) + (d^(3/4)*(14*b*c + a*d)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(6*c^(3/4)*(b*c - a*d)^3*sqrt(c - d*x^2)) - (b*c^(1/4)*(b*c + 9*a*d)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a*d^(1/4)*(b*c - a*d)^3*sqrt(c - d*x^2)) - (b*c^(1/4)*(b*c + 9*a*d)*e^(3/2)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a*d^(1/4)*(b*c - a*d)^3*sqrt(c - d*x^2))]
    @test_int [(e*x)^(1/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)), x, 17, (d*(3*b*c + 2*a*d)*(e*x)^(3/2))/(6*a*c*(b*c - a*d)^2*e*(c - d*x^2)^(3/2)) + (b*(e*x)^(3/2))/(2*a*(b*c - a*d)*e*(a - b*x^2)*(c - d*x^2)^(3/2)) + (d*(b^2*c^2 + 5*a*b*c*d - a^2*d^2)*(e*x)^(3/2))/(2*a*c^2*(b*c - a*d)^3*e*sqrt(c - d*x^2)) - (d^(1/4)*(b^2*c^2 + 5*a*b*c*d - a^2*d^2)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a*c^(5/4)*(b*c - a*d)^3*sqrt(c - d*x^2)) + (d^(1/4)*(b^2*c^2 + 5*a*b*c*d - a^2*d^2)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a*c^(5/4)*(b*c - a*d)^3*sqrt(c - d*x^2)) - (b^(3/2)*c^(1/4)*(b*c - 11*a*d)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^(3/2)*d^(1/4)*(b*c - a*d)^3*sqrt(c - d*x^2)) + (b^(3/2)*c^(1/4)*(b*c - 11*a*d)*sqrt(e)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^(3/2)*d^(1/4)*(b*c - a*d)^3*sqrt(c - d*x^2))]
    @test_int [1/((e*x)^(1/2)*(a - b*x^2)^2*(c - d*x^2)^(5/2)), x, 12, (d*(3*b*c + 2*a*d)*sqrt(e*x))/(6*a*c*(b*c - a*d)^2*e*(c - d*x^2)^(3/2)) + (b*sqrt(e*x))/(2*a*(b*c - a*d)*e*(a - b*x^2)*(c - d*x^2)^(3/2)) + (d*(3*b^2*c^2 + 17*a*b*c*d - 5*a^2*d^2)*sqrt(e*x))/(6*a*c^2*(b*c - a*d)^3*e*sqrt(c - d*x^2)) + (d^(3/4)*(3*b^2*c^2 + 17*a*b*c*d - 5*a^2*d^2)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(6*a*c^(7/4)*(b*c - a*d)^3*sqrt(e)*sqrt(c - d*x^2)) + (b^2*c^(1/4)*(3*b*c - 13*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^2*d^(1/4)*(b*c - a*d)^3*sqrt(e)*sqrt(c - d*x^2)) + (b^2*c^(1/4)*(3*b*c - 13*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^2*d^(1/4)*(b*c - a*d)^3*sqrt(e)*sqrt(c - d*x^2))]
    @test_int [1/((e*x)^(3/2)*(a - b*x^2)^2*(c - d*x^2)^(5/2)), x, 18, (d*(3*b*c + 2*a*d))/(6*a*c*(b*c - a*d)^2*e*sqrt(e*x)*(c - d*x^2)^(3/2)) + b/(2*a*(b*c - a*d)*e*sqrt(e*x)*(a - b*x^2)*(c - d*x^2)^(3/2)) + (d*(3*b^2*c^2 + 19*a*b*c*d - 7*a^2*d^2))/(6*a*c^2*(b*c - a*d)^3*e*sqrt(e*x)*sqrt(c - d*x^2)) - ((5*b^3*c^3 - 12*a*b^2*c^2*d + 19*a^2*b*c*d^2 - 7*a^3*d^3)*sqrt(c - d*x^2))/(2*a^2*c^3*(b*c - a*d)^3*e*sqrt(e*x)) - (d^(1/4)*(5*b^3*c^3 - 12*a*b^2*c^2*d + 19*a^2*b*c*d^2 - 7*a^3*d^3)*sqrt(1 - (d*x^2)/c)*Elliptic.E(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a^2*c^(9/4)*(b*c - a*d)^3*e^(3/2)*sqrt(c - d*x^2)) + (d^(1/4)*(5*b^3*c^3 - 12*a*b^2*c^2*d + 19*a^2*b*c*d^2 - 7*a^3*d^3)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(2*a^2*c^(9/4)*(b*c - a*d)^3*e^(3/2)*sqrt(c - d*x^2)) - (5*b^(5/2)*c^(1/4)*(b*c - 3*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^(5/2)*d^(1/4)*(b*c - a*d)^3*e^(3/2)*sqrt(c - d*x^2)) + (5*b^(5/2)*c^(1/4)*(b*c - 3*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^(5/2)*d^(1/4)*(b*c - a*d)^3*e^(3/2)*sqrt(c - d*x^2))]
    @test_int [1/((e*x)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(5/2)), x, 13, (d*(3*b*c + 2*a*d))/(6*a*c*(b*c - a*d)^2*e*(e*x)^(3/2)*(c - d*x^2)^(3/2)) + b/(2*a*(b*c - a*d)*e*(e*x)^(3/2)*(a - b*x^2)*(c - d*x^2)^(3/2)) + (d*(b^2*c^2 + 7*a*b*c*d - 3*a^2*d^2))/(2*a*c^2*(b*c - a*d)^3*e*(e*x)^(3/2)*sqrt(c - d*x^2)) - ((7*b^3*c^3 - 12*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 15*a^3*d^3)*sqrt(c - d*x^2))/(6*a^2*c^3*(b*c - a*d)^3*e*(e*x)^(3/2)) + (d^(3/4)*(7*b^3*c^3 - 12*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 15*a^3*d^3)*sqrt(1 - (d*x^2)/c)*Elliptic.F(asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(6*a^2*c^(11/4)*(b*c - a*d)^3*e^(5/2)*sqrt(c - d*x^2)) + (b^3*c^(1/4)*(7*b*c - 17*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi(-((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d))), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^3*d^(1/4)*(b*c - a*d)^3*e^(5/2)*sqrt(c - d*x^2)) + (b^3*c^(1/4)*(7*b*c - 17*a*d)*sqrt(1 - (d*x^2)/c)*Elliptic.Pi((sqrt(b)*sqrt(c))/(sqrt(a)*sqrt(d)), asin((d^(1/4)*sqrt(e*x))/(c^(1/4)*sqrt(e))), -1))/(4*a^3*d^(1/4)*(b*c - a*d)^3*e^(5/2)*sqrt(c - d*x^2))]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^2)^(p/2) / (c+d*x^2)^1=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^2)^(p/2) / (c+d*x^2)^3=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(e*x)^m*(a+b*x^2)^(p/2)*(c+d*x^2)^(q/2)=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^2)^(p/2)*(c+d*x^2)^(1/2)=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^2)^(p/2)*(c+d*x^2)^(3/2)=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^2)^(p/2)*(c+d*x^2)^(5/2)=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^2)^(p/2) / (c+d*x^2)^(1/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^5*(a + b*x^2)^(1/2)/sqrt(c + d*x^2), x, 7, ((5*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(16*b^2*d^3) - ((5*b*c + 3*a*d)*(a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(24*b^2*d^2) + (x^2*(a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(6*b*d) - ((b*c - a*d)*(5*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2))))/(16*b^(5/2)*d^(7/2))]
    @test_int [x^3*(a + b*x^2)^(1/2)/sqrt(c + d*x^2), x, 6, -((3*b*c + a*d)*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(8*b*d^2) + ((a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(4*b*d) + ((b*c - a*d)*(3*b*c + a*d)*atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2))))/(8*b^(3/2)*d^(5/2))]
    @test_int [x^1*(a + b*x^2)^(1/2)/sqrt(c + d*x^2), x, 5, (sqrt(a + b*x^2)*sqrt(c + d*x^2))/(2*d) - ((b*c - a*d)*atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2))))/(2*sqrt(b)*d^(3/2))]
    @test_int [1/x^1*(a + b*x^2)^(1/2)/sqrt(c + d*x^2), x, 8, -((sqrt(a)*atanh((sqrt(c)*sqrt(a + b*x^2))/(sqrt(a)*sqrt(c + d*x^2))))/sqrt(c)) + (sqrt(b)*atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2))))/sqrt(d)]
    @test_int [1/x^3*(a + b*x^2)^(1/2)/sqrt(c + d*x^2), x, 4, -((sqrt(a + b*x^2)*sqrt(c + d*x^2))/(2*c*x^2)) - ((b*c - a*d)*atanh((sqrt(c)*sqrt(a + b*x^2))/(sqrt(a)*sqrt(c + d*x^2))))/(2*sqrt(a)*c^(3/2))]
    @test_int [1/x^5*(a + b*x^2)^(1/2)/sqrt(c + d*x^2), x, 5, ((b*c + 3*a*d)*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(8*a*c^2*x^2) - ((a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(4*a*c*x^4) + ((b*c - a*d)*(b*c + 3*a*d)*atanh((sqrt(c)*sqrt(a + b*x^2))/(sqrt(a)*sqrt(c + d*x^2))))/(8*a^(3/2)*c^(5/2))]

    @test_int [x^4*(a + b*x^2)^(1/2)/sqrt(c + d*x^2), x, 6, ((8*b^2*c^2 - 3*a*b*c*d - 2*a^2*d^2)*x*sqrt(a + b*x^2))/(15*b^2*d^2*sqrt(c + d*x^2)) - ((4*b*c - a*d)*x*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(15*b*d^2) + (x^3*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(5*d) - (sqrt(c)*(8*b^2*c^2 - 3*a*b*c*d - 2*a^2*d^2)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*b^2*d^(5/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) + (c^(3/2)*(4*b*c - a*d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*b*d^(5/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [x^2*(a + b*x^2)^(1/2)/sqrt(c + d*x^2), x, 5, -(((2*b*c - a*d)*x*sqrt(a + b*x^2))/(3*b*d*sqrt(c + d*x^2))) + (x*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(3*d) + (sqrt(c)*(2*b*c - a*d)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*b*d^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) - (c^(3/2)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*d^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [1/x^2*(a + b*x^2)^(1/2)/sqrt(c + d*x^2), x, 6, (d*x*sqrt(a + b*x^2))/(c*sqrt(c + d*x^2)) - (sqrt(a + b*x^2)*sqrt(c + d*x^2))/(c*x) - (sqrt(d)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(sqrt(c)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) + (b*sqrt(c)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(a*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [1/x^4*(a + b*x^2)^(1/2)/sqrt(c + d*x^2), x, 6, (d*(b*c - 2*a*d)*x*sqrt(a + b*x^2))/(3*a*c^2*sqrt(c + d*x^2)) - (sqrt(a + b*x^2)*sqrt(c + d*x^2))/(3*c*x^3) - ((b*c - 2*a*d)*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(3*a*c^2*x) - (sqrt(d)*(b*c - 2*a*d)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*a*c^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) - (b*sqrt(d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*a*sqrt(c)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]

    @test_int [x^5*(a + b*x^2)^(3/2)/sqrt(c + d*x^2), x, 8, -(((b*c - a*d)*(35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(128*b^2*d^4)) + ((35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*(a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(192*b^2*d^3) - ((7*b*c + 3*a*d)*(a + b*x^2)^(5/2)*sqrt(c + d*x^2))/(48*b^2*d^2) + (x^2*(a + b*x^2)^(5/2)*sqrt(c + d*x^2))/(8*b*d) + ((b*c - a*d)^2*(35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2))))/(128*b^(5/2)*d^(9/2))]
    @test_int [x^3*(a + b*x^2)^(3/2)/sqrt(c + d*x^2), x, 7, ((b*c - a*d)*(5*b*c + a*d)*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(16*b*d^3) - ((5*b*c + a*d)*(a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(24*b*d^2) + ((a + b*x^2)^(5/2)*sqrt(c + d*x^2))/(6*b*d) - ((b*c - a*d)^2*(5*b*c + a*d)*atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2))))/(16*b^(3/2)*d^(7/2))]
    @test_int [x^1*(a + b*x^2)^(3/2)/sqrt(c + d*x^2), x, 6, -((3*(b*c - a*d)*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(8*d^2)) + ((a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(4*d) + (3*(b*c - a*d)^2*atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2))))/(8*sqrt(b)*d^(5/2))]
    @test_int [1/x^1*(a + b*x^2)^(3/2)/sqrt(c + d*x^2), x, 8, (b*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(2*d) - (a^(3/2)*atanh((sqrt(c)*sqrt(a + b*x^2))/(sqrt(a)*sqrt(c + d*x^2))))/sqrt(c) - (sqrt(b)*(b*c - 3*a*d)*atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2))))/(2*d^(3/2))]
    @test_int [1/x^3*(a + b*x^2)^(3/2)/sqrt(c + d*x^2), x, 8, -((a*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(2*c*x^2)) - (sqrt(a)*(3*b*c - a*d)*atanh((sqrt(c)*sqrt(a + b*x^2))/(sqrt(a)*sqrt(c + d*x^2))))/(2*c^(3/2)) + (b^(3/2)*atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2))))/sqrt(d)]
    @test_int [1/x^5*(a + b*x^2)^(3/2)/sqrt(c + d*x^2), x, 5, -((3*(b*c - a*d)*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(8*c^2*x^2)) - ((a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(4*c*x^4) - (3*(b*c - a*d)^2*atanh((sqrt(c)*sqrt(a + b*x^2))/(sqrt(a)*sqrt(c + d*x^2))))/(8*sqrt(a)*c^(5/2))]

    @test_int [x^4*(a + b*x^2)^(3/2)/sqrt(c + d*x^2), x, 7, -((2*(2*b*c - a*d)*(4*b^2*c^2 - 4*a*b*c*d - a^2*d^2)*x*sqrt(a + b*x^2))/(35*b^2*d^3*sqrt(c + d*x^2))) + ((8*b^2*c^2 - 11*a*b*c*d + a^2*d^2)*x*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(35*b*d^3) - (2*(3*b*c - 4*a*d)*x^3*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(35*d^2) + (b*x^5*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(7*d) + (2*sqrt(c)*(2*b*c - a*d)*(4*b^2*c^2 - 4*a*b*c*d - a^2*d^2)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(35*b^2*d^(7/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) - (c^(3/2)*(8*b^2*c^2 - 11*a*b*c*d + a^2*d^2)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(35*b*d^(7/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [x^2*(a + b*x^2)^(3/2)/sqrt(c + d*x^2), x, 6, -(((13*a*c - (8*b*c^2)/d - (3*a^2*d)/b)*x*sqrt(a + b*x^2))/(15*d*sqrt(c + d*x^2))) - (2*(2*b*c - 3*a*d)*x*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(15*d^2) + (b*x^3*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(5*d) - (sqrt(c)*(8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*b*d^(5/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) + (2*c^(3/2)*(2*b*c - 3*a*d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*d^(5/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [1/x^2*(a + b*x^2)^(3/2)/sqrt(c + d*x^2), x, 5, ((b*c + a*d)*x*sqrt(a + b*x^2))/(c*sqrt(c + d*x^2)) - (a*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(c*x) - ((b*c + a*d)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(sqrt(c)*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) + (2*b*sqrt(c)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [1/x^4*(a + b*x^2)^(3/2)/sqrt(c + d*x^2), x, 6, (2*d*(2*b*c - a*d)*x*sqrt(a + b*x^2))/(3*c^2*sqrt(c + d*x^2)) - (a*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(3*c*x^3) - (2*(2*b*c - a*d)*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(3*c^2*x) - (2*sqrt(d)*(2*b*c - a*d)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*c^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) + (b*(3*b*c - a*d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*a*sqrt(c)*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]

    @test_int [x^5*(a + b*x^2)^(5/2)/sqrt(c + d*x^2), x, 9, ((b*c - a*d)^2*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(256*b^2*d^5) - ((b*c - a*d)*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*(a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(384*b^2*d^4) + ((63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*(a + b*x^2)^(5/2)*sqrt(c + d*x^2))/(480*b^2*d^3) - (3*(3*b*c + a*d)*(a + b*x^2)^(7/2)*sqrt(c + d*x^2))/(80*b^2*d^2) + (x^2*(a + b*x^2)^(7/2)*sqrt(c + d*x^2))/(10*b*d) - ((b*c - a*d)^3*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2))))/(256*b^(5/2)*d^(11/2))]
    @test_int [x^3*(a + b*x^2)^(5/2)/sqrt(c + d*x^2), x, 8, (-5*(b*c - a*d)^2*(7*b*c + a*d)*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(128*b*d^4) + (5*(b*c - a*d)*(7*b*c + a*d)*(a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(192*b*d^3) - ((7*b*c + a*d)*(a + b*x^2)^(5/2)*sqrt(c + d*x^2))/(48*b*d^2) + ((a + b*x^2)^(7/2)*sqrt(c + d*x^2))/(8*b*d) + (5*(b*c - a*d)^3*(7*b*c + a*d)*atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2))))/(128*b^(3/2)*d^(9/2))]
    @test_int [x^1*(a + b*x^2)^(5/2)/sqrt(c + d*x^2), x, 7, (5*(b*c - a*d)^2*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(16*d^3) - (5*(b*c - a*d)*(a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(24*d^2) + ((a + b*x^2)^(5/2)*sqrt(c + d*x^2))/(6*d) - (5*(b*c - a*d)^3*atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2))))/(16*sqrt(b)*d^(7/2))]
    @test_int [1/x^1*(a + b*x^2)^(5/2)/sqrt(c + d*x^2), x, 9, -((b*(3*b*c - 7*a*d)*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(8*d^2)) + (b*(a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(4*d) - (a^(5/2)*atanh((sqrt(c)*sqrt(a + b*x^2))/(sqrt(a)*sqrt(c + d*x^2))))/sqrt(c) + (sqrt(b)*(3*b^2*c^2 - 10*a*b*c*d + 15*a^2*d^2)*atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2))))/(8*d^(5/2))]
    @test_int [1/x^3*(a + b*x^2)^(5/2)/sqrt(c + d*x^2), x, 9, (b*(b*c + a*d)*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(2*c*d) - (a*(a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(2*c*x^2) - (a^(3/2)*(5*b*c - a*d)*atanh((sqrt(c)*sqrt(a + b*x^2))/(sqrt(a)*sqrt(c + d*x^2))))/(2*c^(3/2)) - (b^(3/2)*(b*c - 5*a*d)*atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2))))/(2*d^(3/2))]
    @test_int [1/x^5*(a + b*x^2)^(5/2)/sqrt(c + d*x^2), x, 9, -((a*(7*b*c - 3*a*d)*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(8*c^2*x^2)) - (a*(a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(4*c*x^4) - (sqrt(a)*(15*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*atanh((sqrt(c)*sqrt(a + b*x^2))/(sqrt(a)*sqrt(c + d*x^2))))/(8*c^(5/2)) + (b^(5/2)*atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2))))/sqrt(d)]

    @test_int [x^4*(a + b*x^2)^(5/2)/sqrt(c + d*x^2), x, 8, ((128*b^4*c^4 - 328*a*b^3*c^3*d + 243*a^2*b^2*c^2*d^2 - 25*a^3*b*c*d^3 - 10*a^4*d^4)*x*sqrt(a + b*x^2))/(315*b^2*d^4*sqrt(c + d*x^2)) - ((64*b^3*c^3 - 156*a*b^2*c^2*d + 105*a^2*b*c*d^2 - 5*a^3*d^3)*x*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(315*b*d^4) + ((48*b^2*c^2 - 115*a*b*c*d + 75*a^2*d^2)*x^3*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(315*d^3) - (4*b*(2*b*c - 3*a*d)*x^5*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(63*d^2) + (b*x^5*(a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(9*d) - (sqrt(c)*(128*b^4*c^4 - 328*a*b^3*c^3*d + 243*a^2*b^2*c^2*d^2 - 25*a^3*b*c*d^3 - 10*a^4*d^4)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(315*b^2*d^(9/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) + (c^(3/2)*(64*b^3*c^3 - 156*a*b^2*c^2*d + 105*a^2*b*c*d^2 - 5*a^3*d^3)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(315*b*d^(9/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [x^2*(a + b*x^2)^(5/2)/sqrt(c + d*x^2), x, 7, -(((48*b^3*c^3 - 128*a*b^2*c^2*d + 103*a^2*b*c*d^2 - 15*a^3*d^3)*x*sqrt(a + b*x^2))/(105*b*d^3*sqrt(c + d*x^2))) + ((24*b^2*c^2 - 61*a*b*c*d + 45*a^2*d^2)*x*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(105*d^3) - (2*b*(3*b*c - 5*a*d)*x^3*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(35*d^2) + (b*x^3*(a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(7*d) + (sqrt(c)*(48*b^3*c^3 - 128*a*b^2*c^2*d + 103*a^2*b*c*d^2 - 15*a^3*d^3)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(105*b*d^(7/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) - (c^(3/2)*(24*b^2*c^2 - 61*a*b*c*d + 45*a^2*d^2)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(105*d^(7/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [1/x^2*(a + b*x^2)^(5/2)/sqrt(c + d*x^2), x, 6, ((7*a*b - (2*b^2*c)/d + (3*a^2*d)/c)*x*sqrt(a + b*x^2))/(3*sqrt(c + d*x^2)) + (b*(b*c + 3*a*d)*x*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(3*c*d) - (a*(a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(c*x) + ((2*b^2*c^2 - 7*a*b*c*d - 3*a^2*d^2)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*sqrt(c)*d^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) - (b*sqrt(c)*(b*c - 9*a*d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*d^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [1/x^4*(a + b*x^2)^(5/2)/sqrt(c + d*x^2), x, 6, ((3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*x*sqrt(a + b*x^2))/(3*c^2*sqrt(c + d*x^2)) - (2*a*(3*b*c - a*d)*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(3*c^2*x) - (a*(a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(3*c*x^3) - ((3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*c^(3/2)*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) + (b*(9*b*c - a*d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*sqrt(c)*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]

    @test_int [x^4*sqrt(-1 + 3*x^2)/sqrt(2 - 3*x^2), x, 5, (-(7/135))*x*sqrt(2 - 3*x^2)*sqrt(-1 + 3*x^2) - (1/15)*x^3*sqrt(2 - 3*x^2)*sqrt(-1 + 3*x^2) - (8*Elliptic.E(acos(sqrt(3/2)*x), 2))/(45*sqrt(3)) - (2*Elliptic.F(acos(sqrt(3/2)*x), 2))/(27*sqrt(3))]
    @test_int [x^3*sqrt(-1 + 3*x^2)/sqrt(2 - 3*x^2), x, 6, (-(7/72))*sqrt(2 - 3*x^2)*sqrt(-1 + 3*x^2) - (1/36)*sqrt(2 - 3*x^2)*(-1 + 3*x^2)^(3/2) - (7/144)*asin(3 - 6*x^2)]
    @test_int [x^2*sqrt(-1 + 3*x^2)/sqrt(2 - 3*x^2), x, 4, (-(1/9))*x*sqrt(2 - 3*x^2)*sqrt(-1 + 3*x^2) - Elliptic.E(acos(sqrt(3/2)*x), 2)/(3*sqrt(3)) - Elliptic.F(acos(sqrt(3/2)*x), 2)/(9*sqrt(3))]
    @test_int [x^1*sqrt(-1 + 3*x^2)/sqrt(2 - 3*x^2), x, 5, (-(1/6))*sqrt(2 - 3*x^2)*sqrt(-1 + 3*x^2) - (1/12)*asin(3 - 6*x^2)]

    @test_int [x^2*sqrt(2 + b*x^2)/sqrt(3 + d*x^2), x, 5, -((2*(3*b - d)*x*sqrt(2 + b*x^2))/(3*b*d*sqrt(3 + d*x^2))) + (x*sqrt(2 + b*x^2)*sqrt(3 + d*x^2))/(3*d) + (2*sqrt(2)*(3*b - d)*sqrt(2 + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(3)), 1 - (3*b)/(2*d)))/(3*b*d^(3/2)*sqrt((2 + b*x^2)/(3 + d*x^2))*sqrt(3 + d*x^2)) - (sqrt(2)*sqrt(2 + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(3)), 1 - (3*b)/(2*d)))/(d^(3/2)*sqrt((2 + b*x^2)/(3 + d*x^2))*sqrt(3 + d*x^2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^5/((a + b*x^2)^(1/2)*sqrt(c + d*x^2)), x, 6, -((3*(b*c + a*d)*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(8*b^2*d^2)) + (x^2*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(4*b*d) - ((4*a*b*c*d - 3*(b*c + a*d)^2)*atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2))))/(8*b^(5/2)*d^(5/2))]
    @test_int [x^3/((a + b*x^2)^(1/2)*sqrt(c + d*x^2)), x, 5, (sqrt(a + b*x^2)*sqrt(c + d*x^2))/(2*b*d) - ((b*c + a*d)*atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2))))/(2*b^(3/2)*d^(3/2))]
    @test_int [x^1/((a + b*x^2)^(1/2)*sqrt(c + d*x^2)), x, 4, atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2)))/(sqrt(b)*sqrt(d))]
    @test_int [1/(x^1*(a + b*x^2)^(1/2)*sqrt(c + d*x^2)), x, 3, -(atanh((sqrt(c)*sqrt(a + b*x^2))/(sqrt(a)*sqrt(c + d*x^2)))/(sqrt(a)*sqrt(c)))]
    @test_int [1/(x^3*(a + b*x^2)^(1/2)*sqrt(c + d*x^2)), x, 4, -((sqrt(a + b*x^2)*sqrt(c + d*x^2))/(2*a*c*x^2)) + ((b*c + a*d)*atanh((sqrt(c)*sqrt(a + b*x^2))/(sqrt(a)*sqrt(c + d*x^2))))/(2*a^(3/2)*c^(3/2))]
    @test_int [1/(x^5*(a + b*x^2)^(1/2)*sqrt(c + d*x^2)), x, 6, -((sqrt(a + b*x^2)*sqrt(c + d*x^2))/(4*a*c*x^4)) + (3*(b*c + a*d)*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(8*a^2*c^2*x^2) - ((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*atanh((sqrt(c)*sqrt(a + b*x^2))/(sqrt(a)*sqrt(c + d*x^2))))/(8*a^(5/2)*c^(5/2))]

    @test_int [x^6/((a + b*x^2)^(1/2)*sqrt(c + d*x^2)), x, 6, ((8*b^2*c^2 + 7*a*b*c*d + 8*a^2*d^2)*x*sqrt(a + b*x^2))/(15*b^3*d^2*sqrt(c + d*x^2)) - (4*(b*c + a*d)*x*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(15*b^2*d^2) + (x^3*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(5*b*d) - (sqrt(c)*(8*b^2*c^2 + 7*a*b*c*d + 8*a^2*d^2)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*b^3*d^(5/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) + (4*c^(3/2)*(b*c + a*d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*b^2*d^(5/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [x^4/((a + b*x^2)^(1/2)*sqrt(c + d*x^2)), x, 5, -((2*(b*c + a*d)*x*sqrt(a + b*x^2))/(3*b^2*d*sqrt(c + d*x^2))) + (x*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(3*b*d) + (2*sqrt(c)*(b*c + a*d)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*b^2*d^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) - (c^(3/2)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*b*d^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [x^2/((a + b*x^2)^(1/2)*sqrt(c + d*x^2)), x, 2, (x*sqrt(a + b*x^2))/(b*sqrt(c + d*x^2)) - (sqrt(c)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(b*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [1/(x^2*(a + b*x^2)^(1/2)*sqrt(c + d*x^2)), x, 4, (d*x*sqrt(a + b*x^2))/(a*c*sqrt(c + d*x^2)) - (sqrt(a + b*x^2)*sqrt(c + d*x^2))/(a*c*x) - (sqrt(d)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(a*sqrt(c)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [1/(x^4*(a + b*x^2)^(1/2)*sqrt(c + d*x^2)), x, 6, -((2*d*(b*c + a*d)*x*sqrt(a + b*x^2))/(3*a^2*c^2*sqrt(c + d*x^2))) - (sqrt(a + b*x^2)*sqrt(c + d*x^2))/(3*a*c*x^3) + (2*(b*c + a*d)*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(3*a^2*c^2*x) + (2*sqrt(d)*(b*c + a*d)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*a^2*c^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) - (b*sqrt(d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*a^2*sqrt(c)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]

    @test_int [x^5/((a + b*x^2)^(3/2)*sqrt(c + d*x^2)), x, 6, -((a^2*sqrt(c + d*x^2))/(b^2*(b*c - a*d)*sqrt(a + b*x^2))) + (sqrt(a + b*x^2)*sqrt(c + d*x^2))/(2*b^2*d) - ((b*c + 3*a*d)*atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2))))/(2*b^(5/2)*d^(3/2))]
    @test_int [x^3/((a + b*x^2)^(3/2)*sqrt(c + d*x^2)), x, 5, (a*sqrt(c + d*x^2))/(b*(b*c - a*d)*sqrt(a + b*x^2)) + atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2)))/(b^(3/2)*sqrt(d))]
    @test_int [x^1/((a + b*x^2)^(3/2)*sqrt(c + d*x^2)), x, 2, -(sqrt(c + d*x^2)/((b*c - a*d)*sqrt(a + b*x^2)))]

    @test_int [x^5/((a + b*x^2)^(5/2)*sqrt(c + d*x^2)), x, 6, -((a^2*sqrt(c + d*x^2))/(3*b^2*(b*c - a*d)*(a + b*x^2)^(3/2))) + (2*a*(3*b*c - 2*a*d)*sqrt(c + d*x^2))/(3*b^2*(b*c - a*d)^2*sqrt(a + b*x^2)) + atanh((sqrt(d)*sqrt(a + b*x^2))/(sqrt(b)*sqrt(c + d*x^2)))/(b^(5/2)*sqrt(d))]
    @test_int [x^3/((a + b*x^2)^(5/2)*sqrt(c + d*x^2)), x, 3, (a*sqrt(c + d*x^2))/(3*b*(b*c - a*d)*(a + b*x^2)^(3/2)) - ((3*b*c - a*d)*sqrt(c + d*x^2))/(3*b*(b*c - a*d)^2*sqrt(a + b*x^2))]
    @test_int [x^1/((a + b*x^2)^(5/2)*sqrt(c + d*x^2)), x, 3, -(sqrt(c + d*x^2)/(3*(b*c - a*d)*(a + b*x^2)^(3/2))) + (2*d*sqrt(c + d*x^2))/(3*(b*c - a*d)^2*sqrt(a + b*x^2))]

    @test_int [x^5/((a + b*x^2)^(7/2)*sqrt(c + d*x^2)), x, 4, -((a^2*sqrt(c + d*x^2))/(5*b^2*(b*c - a*d)*(a + b*x^2)^(5/2))) + (2*a*(5*b*c - 3*a*d)*sqrt(c + d*x^2))/(15*b^2*(b*c - a*d)^2*(a + b*x^2)^(3/2)) - ((15*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*sqrt(c + d*x^2))/(15*b^2*(b*c - a*d)^3*sqrt(a + b*x^2))]
    @test_int [x^3/((a + b*x^2)^(7/2)*sqrt(c + d*x^2)), x, 4, (a*sqrt(c + d*x^2))/(5*b*(b*c - a*d)*(a + b*x^2)^(5/2)) - ((5*b*c - a*d)*sqrt(c + d*x^2))/(15*b*(b*c - a*d)^2*(a + b*x^2)^(3/2)) + (2*d*(5*b*c - a*d)*sqrt(c + d*x^2))/(15*b*(b*c - a*d)^3*sqrt(a + b*x^2))]
    @test_int [x^1/((a + b*x^2)^(7/2)*sqrt(c + d*x^2)), x, 4, -(sqrt(c + d*x^2)/(5*(b*c - a*d)*(a + b*x^2)^(5/2))) + (4*d*sqrt(c + d*x^2))/(15*(b*c - a*d)^2*(a + b*x^2)^(3/2)) - (8*d^2*sqrt(c + d*x^2))/(15*(b*c - a*d)^3*sqrt(a + b*x^2))]

    @test_int [x^5/((a + b*x^2)^(9/2)*sqrt(c + d*x^2)), x, 5, -((a^2*sqrt(c + d*x^2))/(7*b^2*(b*c - a*d)*(a + b*x^2)^(7/2))) + (2*a*(7*b*c - 4*a*d)*sqrt(c + d*x^2))/(35*b^2*(b*c - a*d)^2*(a + b*x^2)^(5/2)) - ((35*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*sqrt(c + d*x^2))/(105*b^2*(b*c - a*d)^3*(a + b*x^2)^(3/2)) + (2*d*(35*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*sqrt(c + d*x^2))/(105*b^2*(b*c - a*d)^4*sqrt(a + b*x^2))]

    @test_int [x/(sqrt(a - b*x^2)*sqrt(c + d*x^2)), x, 4, -(atan((sqrt(d)*sqrt(a - b*x^2))/(sqrt(b)*sqrt(c + d*x^2)))/(sqrt(b)*sqrt(d)))]
    @test_int [x/(sqrt(a - b*x^2)*sqrt(c - d*x^2)), x, 4, -(atanh((sqrt(d)*sqrt(a - b*x^2))/(sqrt(b)*sqrt(c - d*x^2)))/(sqrt(b)*sqrt(d)))]

    @test_int [x^2/(sqrt(2 + b*x^2)*sqrt(3 + d*x^2)), x, 2, (x*sqrt(2 + b*x^2))/(b*sqrt(3 + d*x^2)) - (sqrt(2)*sqrt(2 + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(3)), 1 - (3*b)/(2*d)))/(b*sqrt(d)*sqrt((2 + b*x^2)/(3 + d*x^2))*sqrt(3 + d*x^2))]
    @test_int [x^2/(sqrt(4 - x^2)*sqrt(c + d*x^2)), x, 5, (sqrt(c + d*x^2)*Elliptic.E(asin(x/2), -((4*d)/c)))/(d*sqrt(1 + (d*x^2)/c)) - (c*sqrt(1 + (d*x^2)/c)*Elliptic.F(asin(x/2), -((4*d)/c)))/(d*sqrt(c + d*x^2))]
    @test_int [x^2/(sqrt(4 + x^2)*sqrt(c + d*x^2)), x, 2, (x*sqrt(c + d*x^2))/(d*sqrt(4 + x^2)) - (sqrt(c + d*x^2)*Elliptic.E(atan(x/2), 1 - (4*d)/c))/(d*sqrt(4 + x^2)*sqrt((c + d*x^2)/(c*(4 + x^2))))]

    @test_int [x^2/(sqrt(1 - x^2)*sqrt(2 + 3*x^2)), x, 3, (1/3)*sqrt(2)*Elliptic.E(asin(x), -(3/2)) - (1/3)*sqrt(2)*Elliptic.F(asin(x), -(3/2))]
    @test_int [x^2/(sqrt(1 - x^2)*sqrt(2 - 3*x^2)), x, 3, (-(1/3))*sqrt(2)*Elliptic.E(asin(x), 3/2) + (1/3)*sqrt(2)*Elliptic.F(asin(x), 3/2)]
    @test_int [x^2/(sqrt(4 - x^2)*sqrt(2 + 3*x^2)), x, 3, (1/3)*sqrt(2)*Elliptic.E(asin(x/2), -6) - (1/3)*sqrt(2)*Elliptic.F(asin(x/2), -6)]
    @test_int [x^2/(sqrt(4 - x^2)*sqrt(2 - 3*x^2)), x, 3, (-(1/3))*sqrt(2)*Elliptic.E(asin(x/2), 6) + (1/3)*sqrt(2)*Elliptic.F(asin(x/2), 6)]
    @test_int [x^2/(sqrt(1 - 4*x^2)*sqrt(2 + 3*x^2)), x, 3, Elliptic.E(asin(2*x), -(3/8))/(3*sqrt(2)) - Elliptic.F(asin(2*x), -(3/8))/(3*sqrt(2))]
    @test_int [x^2/(sqrt(1 - 4*x^2)*sqrt(2 - 3*x^2)), x, 3, -(Elliptic.E(asin(2*x), 3/8)/(3*sqrt(2))) + Elliptic.F(asin(2*x), 3/8)/(3*sqrt(2))]

    @test_int [x^2/(sqrt(1 + x^2)*sqrt(2 - 3*x^2)), x, 3, Elliptic.E(asin(sqrt(3/2)*x), -(2/3))/sqrt(3) - Elliptic.F(asin(sqrt(3/2)*x), -(2/3))/sqrt(3)]
    @test_int [x^2/(sqrt(4 + x^2)*sqrt(2 - 3*x^2)), x, 3, (2*Elliptic.E(asin(sqrt(3/2)*x), -(1/6)))/sqrt(3) - (2*Elliptic.F(asin(sqrt(3/2)*x), -(1/6)))/sqrt(3)]
    @test_int [x^2/(sqrt(1 + 4*x^2)*sqrt(2 - 3*x^2)), x, 3, Elliptic.E(asin(sqrt(3/2)*x), -(8/3))/(4*sqrt(3)) - Elliptic.F(asin(sqrt(3/2)*x), -(8/3))/(4*sqrt(3))]

    @test_int [x^2/(sqrt(1 + x^2)*sqrt(2 + 3*x^2)), x, 2, (x*sqrt(2 + 3*x^2))/(3*sqrt(1 + x^2)) - (sqrt(2)*sqrt(2 + 3*x^2)*Elliptic.E(atan(x), -(1/2)))/(3*sqrt(1 + x^2)*sqrt((2 + 3*x^2)/(1 + x^2)))]
    @test_int [x^2/(sqrt(4 + x^2)*sqrt(2 + 3*x^2)), x, 2, (x*sqrt(2 + 3*x^2))/(3*sqrt(4 + x^2)) - (sqrt(2)*sqrt(2 + 3*x^2)*Elliptic.E(atan(x/2), -5))/(3*sqrt(4 + x^2)*sqrt((2 + 3*x^2)/(4 + x^2)))]
    @test_int [x^2/(sqrt(1 + 4*x^2)*sqrt(2 + 3*x^2)), x, 2, (x*sqrt(2 + 3*x^2))/(3*sqrt(1 + 4*x^2)) - (sqrt(2 + 3*x^2)*Elliptic.E(atan(2*x), 5/8))/(3*sqrt(2)*sqrt((2 + 3*x^2)/(1 + 4*x^2))*sqrt(1 + 4*x^2))]

    @test_int [x^2/(sqrt(1 - x^2)*sqrt(-1 + 2*x^2)), x, 3, (-(1/2))*Elliptic.E(acos(x), 2) - (1/2)*Elliptic.F(acos(x), 2)]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^2)^(p/2) / (c+d*x^2)^(3/2)=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^2)^(p/2) / (c+d*x^2)^(5/2)=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(e*x)^(m/2)*(a+b*x^2)^(p/2)*(c+d*x^2)^(q/2)=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^2)^(p/2)*(c+d*x^2)^(1/2)=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^2)^(p/2)*(c+d*x^2)^(3/2)=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^2)^(p/2)*(c+d*x^2)^(5/2)=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^2)^(p/2) / (c+d*x^2)^(1/2)=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^2)^(p/2) / (c+d*x^2)^(3/2)=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^2)^(p/2) / (c+d*x^2)^(5/2)=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(e*x)^m*(a+b*x^2)^(p/3)*(c+d*x^2)^q=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(e*x)^m / (a+b*x^2)^(1/3)*(c+d*x^2)^q*when*b*c+3*a*d=0=#

    #= ::Subsubsection:: =#
    #=q>0=#

    #= ::Subsubsection::Closed:: =#
    #=q<0=#

    @test_int [x^5/((1 - x^2)^(1/3)*(3 + x^2)), x, 7, (3/2)*(1 - x^2)^(2/3) + (3/10)*(1 - x^2)^(5/3) + (9*sqrt(3)*atan((1 + (2 - 2*x^2)^(1/3))/sqrt(3)))/(2*2^(2/3)) - (9*log(3 + x^2))/(4*2^(2/3)) + (27*log(2^(2/3) - (1 - x^2)^(1/3)))/(4*2^(2/3))]
    @test_int [x^3/((1 - x^2)^(1/3)*(3 + x^2)), x, 6, (-(3/4))*(1 - x^2)^(2/3) - (3*sqrt(3)*atan((1 + (2 - 2*x^2)^(1/3))/sqrt(3)))/(2*2^(2/3)) + (3*log(3 + x^2))/(4*2^(2/3)) - (9*log(2^(2/3) - (1 - x^2)^(1/3)))/(4*2^(2/3))]
    @test_int [x^1/((1 - x^2)^(1/3)*(3 + x^2)), x, 5, (sqrt(3)*atan((1 + (2 - 2*x^2)^(1/3))/sqrt(3)))/(2*2^(2/3)) - log(3 + x^2)/(4*2^(2/3)) + (3*log(2^(2/3) - (1 - x^2)^(1/3)))/(4*2^(2/3))]
    @test_int [1/(x^1*(1 - x^2)^(1/3)*(3 + x^2)), x, 10, -(atan((1 + (2 - 2*x^2)^(1/3))/sqrt(3))/(2*2^(2/3)*sqrt(3))) + atan((1 + 2*(1 - x^2)^(1/3))/sqrt(3))/(2*sqrt(3)) - log(x)/6 + log(3 + x^2)/(12*2^(2/3)) + (1/4)*log(1 - (1 - x^2)^(1/3)) - log(2^(2/3) - (1 - x^2)^(1/3))/(4*2^(2/3))]
    @test_int [1/(x^3*(1 - x^2)^(1/3)*(3 + x^2)), x, 7, -((1 - x^2)^(2/3)/(6*x^2)) + atan((1 + (2 - 2*x^2)^(1/3))/sqrt(3))/(6*2^(2/3)*sqrt(3)) - log(3 + x^2)/(36*2^(2/3)) + log(2^(2/3) - (1 - x^2)^(1/3))/(12*2^(2/3))]
    @test_int [1/(x^5*(1 - x^2)^(1/3)*(3 + x^2)), x, 12, -((1 - x^2)^(2/3)/(12*x^4)) - (1 - x^2)^(2/3)/(18*x^2) - atan((1 + (2 - 2*x^2)^(1/3))/sqrt(3))/(18*2^(2/3)*sqrt(3)) + atan((1 + 2*(1 - x^2)^(1/3))/sqrt(3))/(9*sqrt(3)) - log(x)/27 + log(3 + x^2)/(108*2^(2/3)) + (1/18)*log(1 - (1 - x^2)^(1/3)) - log(2^(2/3) - (1 - x^2)^(1/3))/(36*2^(2/3))]

    @test_int [x^4/((1 - x^2)^(1/3)*(3 + x^2)), x, 7, (-(3/7))*x*(1 - x^2)^(2/3) + (54*x)/(7*(1 - sqrt(3) - (1 - x^2)^(1/3))) + (3*sqrt(3)*atan(sqrt(3)/x))/(2*2^(2/3)) + (3*sqrt(3)*atan((sqrt(3)*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/x))/(2*2^(2/3)) - (3*atanh(x))/(2*2^(2/3)) + (9*atanh(x/(1 + 2^(1/3)*(1 - x^2)^(1/3))))/(2*2^(2/3)) + (27*3^(1/4)*sqrt(2 + sqrt(3))*(1 - (1 - x^2)^(1/3))*sqrt((1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))), -7 + 4*sqrt(3)))/(7*x*sqrt(-((1 - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2))) - (18*sqrt(2)*3^(3/4)*(1 - (1 - x^2)^(1/3))*sqrt((1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))), -7 + 4*sqrt(3)))/(7*x*sqrt(-((1 - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)))]
    @test_int [x^2/((1 - x^2)^(1/3)*(3 + x^2)), x, 6, -((3*x)/(1 - sqrt(3) - (1 - x^2)^(1/3))) - (sqrt(3)*atan(sqrt(3)/x))/(2*2^(2/3)) - (sqrt(3)*atan((sqrt(3)*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/x))/(2*2^(2/3)) + atanh(x)/(2*2^(2/3)) - (3*atanh(x/(1 + 2^(1/3)*(1 - x^2)^(1/3))))/(2*2^(2/3)) - (3*3^(1/4)*sqrt(2 + sqrt(3))*(1 - (1 - x^2)^(1/3))*sqrt((1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))), -7 + 4*sqrt(3)))/(2*x*sqrt(-((1 - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2))) + (sqrt(2)*3^(3/4)*(1 - (1 - x^2)^(1/3))*sqrt((1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))), -7 + 4*sqrt(3)))/(x*sqrt(-((1 - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)))]
    @test_int [x^0/((1 - x^2)^(1/3)*(3 + x^2)), x, 1, atan(sqrt(3)/x)/(2*2^(2/3)*sqrt(3)) + atan((sqrt(3)*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/x)/(2*2^(2/3)*sqrt(3)) - atanh(x)/(6*2^(2/3)) + atanh(x/(1 + 2^(1/3)*(1 - x^2)^(1/3)))/(2*2^(2/3))]
    @test_int [1/(x^2*(1 - x^2)^(1/3)*(3 + x^2)), x, 7, -((1 - x^2)^(2/3)/(3*x)) + x/(3*(1 - sqrt(3) - (1 - x^2)^(1/3))) - atan(sqrt(3)/x)/(6*2^(2/3)*sqrt(3)) - atan((sqrt(3)*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/x)/(6*2^(2/3)*sqrt(3)) + atanh(x)/(18*2^(2/3)) - atanh(x/(1 + 2^(1/3)*(1 - x^2)^(1/3)))/(6*2^(2/3)) + (sqrt(2 + sqrt(3))*(1 - (1 - x^2)^(1/3))*sqrt((1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))), -7 + 4*sqrt(3)))/(2*3^(3/4)*x*sqrt(-((1 - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2))) - (sqrt(2)*(1 - (1 - x^2)^(1/3))*sqrt((1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))), -7 + 4*sqrt(3)))/(3*3^(1/4)*x*sqrt(-((1 - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)))]
    @test_int [1/(x^4*(1 - x^2)^(1/3)*(3 + x^2)), x, 8, -((1 - x^2)^(2/3)/(9*x^3)) - (2*(1 - x^2)^(2/3))/(27*x) + (2*x)/(27*(1 - sqrt(3) - (1 - x^2)^(1/3))) + atan(sqrt(3)/x)/(18*2^(2/3)*sqrt(3)) + atan((sqrt(3)*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/x)/(18*2^(2/3)*sqrt(3)) - atanh(x)/(54*2^(2/3)) + atanh(x/(1 + 2^(1/3)*(1 - x^2)^(1/3)))/(18*2^(2/3)) + (sqrt(2 + sqrt(3))*(1 - (1 - x^2)^(1/3))*sqrt((1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))), -7 + 4*sqrt(3)))/(9*3^(3/4)*x*sqrt(-((1 - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2))) - (2*sqrt(2)*(1 - (1 - x^2)^(1/3))*sqrt((1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))), -7 + 4*sqrt(3)))/(27*3^(1/4)*x*sqrt(-((1 - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)))]

    @test_int [x^7/((1 - x^2)^(1/3)*(3 + x^2)^2), x, 7, -((3*x^4*(1 - x^2)^(2/3))/(10*(3 + x^2))) + (9*(1 - x^2)^(2/3)*(69 + 14*x^2))/(40*(3 + x^2)) + (99*sqrt(3)*atan((1 + (2 - 2*x^2)^(1/3))/sqrt(3)))/(8*2^(2/3)) - (99*log(3 + x^2))/(16*2^(2/3)) + (297*log(2^(2/3) - (1 - x^2)^(1/3)))/(16*2^(2/3))]
    @test_int [x^5/((1 - x^2)^(1/3)*(3 + x^2)^2), x, 7, (-(3/4))*(1 - x^2)^(2/3) - (9*(1 - x^2)^(2/3))/(8*(3 + x^2)) - (21*sqrt(3)*atan((1 + (2 - 2*x^2)^(1/3))/sqrt(3)))/(8*2^(2/3)) + (21*log(3 + x^2))/(16*2^(2/3)) - (63*log(2^(2/3) - (1 - x^2)^(1/3)))/(16*2^(2/3))]
    @test_int [x^3/((1 - x^2)^(1/3)*(3 + x^2)^2), x, 6, (3*(1 - x^2)^(2/3))/(8*(3 + x^2)) + (3*sqrt(3)*atan((1 + (2 - 2*x^2)^(1/3))/sqrt(3)))/(8*2^(2/3)) - (3*log(3 + x^2))/(16*2^(2/3)) + (9*log(2^(2/3) - (1 - x^2)^(1/3)))/(16*2^(2/3))]
    @test_int [x^1/((1 - x^2)^(1/3)*(3 + x^2)^2), x, 6, -((1 - x^2)^(2/3)/(8*(3 + x^2))) + atan((1 + (2 - 2*x^2)^(1/3))/sqrt(3))/(8*2^(2/3)*sqrt(3)) - log(3 + x^2)/(48*2^(2/3)) + log(2^(2/3) - (1 - x^2)^(1/3))/(16*2^(2/3))]
    @test_int [1/(x^1*(1 - x^2)^(1/3)*(3 + x^2)^2), x, 11, (1 - x^2)^(2/3)/(24*(3 + x^2)) - (5*atan((1 + (2 - 2*x^2)^(1/3))/sqrt(3)))/(24*2^(2/3)*sqrt(3)) + atan((1 + 2*(1 - x^2)^(1/3))/sqrt(3))/(6*sqrt(3)) - log(x)/18 + (5*log(3 + x^2))/(144*2^(2/3)) + (1/12)*log(1 - (1 - x^2)^(1/3)) - (5*log(2^(2/3) - (1 - x^2)^(1/3)))/(48*2^(2/3))]
    @test_int [1/(x^3*(1 - x^2)^(1/3)*(3 + x^2)^2), x, 12, -((5*(1 - x^2)^(2/3))/(72*(3 + x^2))) - (1 - x^2)^(2/3)/(6*x^2*(3 + x^2)) + atan((1 + (2 - 2*x^2)^(1/3))/sqrt(3))/(8*2^(2/3)*sqrt(3)) - atan((1 + 2*(1 - x^2)^(1/3))/sqrt(3))/(18*sqrt(3)) + log(x)/54 - log(3 + x^2)/(48*2^(2/3)) - (1/36)*log(1 - (1 - x^2)^(1/3)) + log(2^(2/3) - (1 - x^2)^(1/3))/(16*2^(2/3))]
    @test_int [1/(x^5*(1 - x^2)^(1/3)*(3 + x^2)^2), x, 13, (1 - x^2)^(2/3)/(216*(3 + x^2)) - (1 - x^2)^(2/3)/(12*x^4*(3 + x^2)) - (1 - x^2)^(2/3)/(36*x^2*(3 + x^2)) - (13*atan((1 + (2 - 2*x^2)^(1/3))/sqrt(3)))/(216*2^(2/3)*sqrt(3)) + atan((1 + 2*(1 - x^2)^(1/3))/sqrt(3))/(18*sqrt(3)) - log(x)/54 + (13*log(3 + x^2))/(1296*2^(2/3)) + (1/36)*log(1 - (1 - x^2)^(1/3)) - (13*log(2^(2/3) - (1 - x^2)^(1/3)))/(432*2^(2/3))]

    @test_int [x^4/((1 - x^2)^(1/3)*(3 + x^2)^2), x, 7, (3*x*(1 - x^2)^(2/3))/(8*(3 + x^2)) - (27*x)/(8*(1 - sqrt(3) - (1 - x^2)^(1/3))) - (5*sqrt(3)*atan(sqrt(3)/x))/(8*2^(2/3)) - (5*sqrt(3)*atan((sqrt(3)*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/x))/(8*2^(2/3)) + (5*atanh(x))/(8*2^(2/3)) - (15*atanh(x/(1 + 2^(1/3)*(1 - x^2)^(1/3))))/(8*2^(2/3)) - (27*3^(1/4)*sqrt(2 + sqrt(3))*(1 - (1 - x^2)^(1/3))*sqrt((1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))), -7 + 4*sqrt(3)))/(16*x*sqrt(-((1 - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2))) + (9*3^(3/4)*(1 - (1 - x^2)^(1/3))*sqrt((1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))), -7 + 4*sqrt(3)))/(4*sqrt(2)*x*sqrt(-((1 - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)))]
    @test_int [x^2/((1 - x^2)^(1/3)*(3 + x^2)^2), x, 7, -((x*(1 - x^2)^(2/3))/(8*(3 + x^2))) + x/(8*(1 - sqrt(3) - (1 - x^2)^(1/3))) + atan(sqrt(3)/x)/(8*2^(2/3)*sqrt(3)) + atan((sqrt(3)*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/x)/(8*2^(2/3)*sqrt(3)) - atanh(x)/(24*2^(2/3)) + atanh(x/(1 + 2^(1/3)*(1 - x^2)^(1/3)))/(8*2^(2/3)) + (3^(1/4)*sqrt(2 + sqrt(3))*(1 - (1 - x^2)^(1/3))*sqrt((1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))), -7 + 4*sqrt(3)))/(16*x*sqrt(-((1 - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2))) - ((1 - (1 - x^2)^(1/3))*sqrt((1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))), -7 + 4*sqrt(3)))/(4*sqrt(2)*3^(1/4)*x*sqrt(-((1 - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)))]
    @test_int [x^0/((1 - x^2)^(1/3)*(3 + x^2)^2), x, 7, (x*(1 - x^2)^(2/3))/(24*(3 + x^2)) - x/(24*(1 - sqrt(3) - (1 - x^2)^(1/3))) + atan(sqrt(3)/x)/(8*2^(2/3)*sqrt(3)) + atan((sqrt(3)*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/x)/(8*2^(2/3)*sqrt(3)) - atanh(x)/(24*2^(2/3)) + atanh(x/(1 + 2^(1/3)*(1 - x^2)^(1/3)))/(8*2^(2/3)) - (sqrt(2 + sqrt(3))*(1 - (1 - x^2)^(1/3))*sqrt((1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))), -7 + 4*sqrt(3)))/(16*3^(3/4)*x*sqrt(-((1 - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2))) + ((1 - (1 - x^2)^(1/3))*sqrt((1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))), -7 + 4*sqrt(3)))/(12*sqrt(2)*3^(1/4)*x*sqrt(-((1 - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)))]
    @test_int [1/(x^2*(1 - x^2)^(1/3)*(3 + x^2)^2), x, 8, -((1 - x^2)^(2/3)/(8*x)) + (1 - x^2)^(2/3)/(24*x*(3 + x^2)) + x/(8*(1 - sqrt(3) - (1 - x^2)^(1/3))) - (7*atan(sqrt(3)/x))/(72*2^(2/3)*sqrt(3)) - (7*atan((sqrt(3)*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/x))/(72*2^(2/3)*sqrt(3)) + (7*atanh(x))/(216*2^(2/3)) - (7*atanh(x/(1 + 2^(1/3)*(1 - x^2)^(1/3))))/(72*2^(2/3)) + (3^(1/4)*sqrt(2 + sqrt(3))*(1 - (1 - x^2)^(1/3))*sqrt((1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))), -7 + 4*sqrt(3)))/(16*x*sqrt(-((1 - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2))) - ((1 - (1 - x^2)^(1/3))*sqrt((1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))), -7 + 4*sqrt(3)))/(4*sqrt(2)*3^(1/4)*x*sqrt(-((1 - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)))]
    @test_int [1/(x^4*(1 - x^2)^(1/3)*(3 + x^2)^2), x, 9, -((11*(1 - x^2)^(2/3))/(216*x^3)) + (11*(1 - x^2)^(2/3))/(648*x) + (1 - x^2)^(2/3)/(24*x^3*(3 + x^2)) - (11*x)/(648*(1 - sqrt(3) - (1 - x^2)^(1/3))) + (11*atan(sqrt(3)/x))/(216*2^(2/3)*sqrt(3)) + (11*atan((sqrt(3)*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/x))/(216*2^(2/3)*sqrt(3)) - (11*atanh(x))/(648*2^(2/3)) + (11*atanh(x/(1 + 2^(1/3)*(1 - x^2)^(1/3))))/(216*2^(2/3)) - (11*sqrt(2 + sqrt(3))*(1 - (1 - x^2)^(1/3))*sqrt((1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))), -7 + 4*sqrt(3)))/(432*3^(3/4)*x*sqrt(-((1 - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2))) + (11*(1 - (1 - x^2)^(1/3))*sqrt((1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))), -7 + 4*sqrt(3)))/(324*sqrt(2)*3^(1/4)*x*sqrt(-((1 - (1 - x^2)^(1/3))/(1 - sqrt(3) - (1 - x^2)^(1/3))^2)))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(e*x)^m*(a+b*x^2)^(p/4)*(c+d*x^2)^q=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*e^m*(a+b*x^2)^(1/4)*(c+d*x^2)^q*when*b*c-2*a*d=0=#

    #= ::Subsubsection::Closed:: =#
    #=a>0=#

    @test_int [x^7/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)), x, 10, (56/243)*(2 - 3*x^2)^(3/4) - (16/567)*(2 - 3*x^2)^(7/4) + (2/891)*(2 - 3*x^2)^(11/4) + (32/81)*2^(1/4)*atan((sqrt(2) - sqrt(2 - 3*x^2))/(2^(3/4)*(2 - 3*x^2)^(1/4))) + (32/81)*2^(1/4)*atanh((sqrt(2) + sqrt(2 - 3*x^2))/(2^(3/4)*(2 - 3*x^2)^(1/4)))]
    @test_int [x^5/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)), x, 7, (4/27)*(2 - 3*x^2)^(3/4) - (2/189)*(2 - 3*x^2)^(7/4) + (8/27)*2^(1/4)*atan((sqrt(2) - sqrt(2 - 3*x^2))/(2^(3/4)*(2 - 3*x^2)^(1/4))) + (8/27)*2^(1/4)*atanh((sqrt(2) + sqrt(2 - 3*x^2))/(2^(3/4)*(2 - 3*x^2)^(1/4)))]
    @test_int [x^3/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)), x, 4, (2/27)*(2 - 3*x^2)^(3/4) + (2/9)*2^(1/4)*atan((sqrt(2) - sqrt(2 - 3*x^2))/(2^(3/4)*(2 - 3*x^2)^(1/4))) + (2/9)*2^(1/4)*atanh((sqrt(2) + sqrt(2 - 3*x^2))/(2^(3/4)*(2 - 3*x^2)^(1/4)))]
    @test_int [x^1/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)), x, 1, atan((sqrt(2) - sqrt(2 - 3*x^2))/(2^(3/4)*(2 - 3*x^2)^(1/4)))/(3*2^(3/4)) + atanh((sqrt(2) + sqrt(2 - 3*x^2))/(2^(3/4)*(2 - 3*x^2)^(1/4)))/(3*2^(3/4))]
    @test_int [1/(x^1*(2 - 3*x^2)^(1/4)*(4 - 3*x^2)), x, 8, atan((2 - 3*x^2)^(1/4)/2^(1/4))/(4*2^(1/4)) + atan((sqrt(2) - sqrt(2 - 3*x^2))/(2^(3/4)*(2 - 3*x^2)^(1/4)))/(4*2^(3/4)) - atanh((2 - 3*x^2)^(1/4)/2^(1/4))/(4*2^(1/4)) + atanh((sqrt(2) + sqrt(2 - 3*x^2))/(2^(3/4)*(2 - 3*x^2)^(1/4)))/(4*2^(3/4))]
    @test_int [1/(x^3*(2 - 3*x^2)^(1/4)*(4 - 3*x^2)), x, 14, -((2 - 3*x^2)^(3/4)/(16*x^2)) + (9*atan((2 - 3*x^2)^(1/4)/2^(1/4)))/(32*2^(1/4)) + (3*atan((sqrt(2) - sqrt(2 - 3*x^2))/(2^(3/4)*(2 - 3*x^2)^(1/4))))/(16*2^(3/4)) - (9*atanh((2 - 3*x^2)^(1/4)/2^(1/4)))/(32*2^(1/4)) + (3*atanh((sqrt(2) + sqrt(2 - 3*x^2))/(2^(3/4)*(2 - 3*x^2)^(1/4))))/(16*2^(3/4))]

    @test_int [x^4/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)), x, 6, (2/45)*x*(2 - 3*x^2)^(3/4) + (4*2^(1/4)*atan((2^(3/4) - 2^(1/4)*sqrt(2 - 3*x^2))/(sqrt(3)*x*(2 - 3*x^2)^(1/4))))/(9*sqrt(3)) + (4*2^(1/4)*atanh((2^(3/4) + 2^(1/4)*sqrt(2 - 3*x^2))/(sqrt(3)*x*(2 - 3*x^2)^(1/4))))/(9*sqrt(3)) - (16*2^(1/4)*Elliptic.E((1/2)*asin(sqrt(3/2)*x), 2))/(15*sqrt(3))]
    @test_int [x^2/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)), x, 4, (2^(1/4)*atan((2^(3/4) - 2^(1/4)*sqrt(2 - 3*x^2))/(sqrt(3)*x*(2 - 3*x^2)^(1/4))))/(3*sqrt(3)) + (2^(1/4)*atanh((2^(3/4) + 2^(1/4)*sqrt(2 - 3*x^2))/(sqrt(3)*x*(2 - 3*x^2)^(1/4))))/(3*sqrt(3)) - (2*2^(1/4)*Elliptic.E((1/2)*asin(sqrt(3/2)*x), 2))/(3*sqrt(3))]
    @test_int [x^0/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)), x, 1, atan((2 - sqrt(2)*sqrt(2 - 3*x^2))/(2^(1/4)*sqrt(3)*x*(2 - 3*x^2)^(1/4)))/(2*2^(3/4)*sqrt(3)) + atanh((2 + sqrt(2)*sqrt(2 - 3*x^2))/(2^(1/4)*sqrt(3)*x*(2 - 3*x^2)^(1/4)))/(2*2^(3/4)*sqrt(3))]
    @test_int [1/(x^2*(2 - 3*x^2)^(1/4)*(4 - 3*x^2)), x, 5, -((2 - 3*x^2)^(3/4)/(8*x)) + (sqrt(3)*atan((2^(3/4) - 2^(1/4)*sqrt(2 - 3*x^2))/(sqrt(3)*x*(2 - 3*x^2)^(1/4))))/(8*2^(3/4)) + (sqrt(3)*atanh((2^(3/4) + 2^(1/4)*sqrt(2 - 3*x^2))/(sqrt(3)*x*(2 - 3*x^2)^(1/4))))/(8*2^(3/4)) - (sqrt(3)*Elliptic.E((1/2)*asin(sqrt(3/2)*x), 2))/(4*2^(3/4))]
    @test_int [1/(x^4*(2 - 3*x^2)^(1/4)*(4 - 3*x^2)), x, 8, -((2 - 3*x^2)^(3/4)/(24*x^3)) - (3*(2 - 3*x^2)^(3/4))/(16*x) + (3*sqrt(3)*atan((2^(3/4) - 2^(1/4)*sqrt(2 - 3*x^2))/(sqrt(3)*x*(2 - 3*x^2)^(1/4))))/(32*2^(3/4)) + (3*sqrt(3)*atanh((2^(3/4) + 2^(1/4)*sqrt(2 - 3*x^2))/(sqrt(3)*x*(2 - 3*x^2)^(1/4))))/(32*2^(3/4)) - (3*sqrt(3)*Elliptic.E((1/2)*asin(sqrt(3/2)*x), 2))/(8*2^(3/4))]

    #= ::Subsubsection::Closed:: =#
    #=a<0=#

    @test_int [x^7/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)), x, 7, (14/243)*(-1 + 3*x^2)^(3/4) + (8/567)*(-1 + 3*x^2)^(7/4) + (2/891)*(-1 + 3*x^2)^(11/4) + (8/81)*atan((-1 + 3*x^2)^(1/4)) - (8/81)*atanh((-1 + 3*x^2)^(1/4))]
    @test_int [x^5/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)), x, 7, (2/27)*(-1 + 3*x^2)^(3/4) + (2/189)*(-1 + 3*x^2)^(7/4) + (4/27)*atan((-1 + 3*x^2)^(1/4)) - (4/27)*atanh((-1 + 3*x^2)^(1/4))]
    @test_int [x^3/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)), x, 6, (2/27)*(-1 + 3*x^2)^(3/4) + (2/9)*atan((-1 + 3*x^2)^(1/4)) - (2/9)*atanh((-1 + 3*x^2)^(1/4))]
    @test_int [x^1/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)), x, 5, (1/3)*atan((-1 + 3*x^2)^(1/4)) - (1/3)*atanh((-1 + 3*x^2)^(1/4))]
    @test_int [1/(x^1*(-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)), x, 16, (1/2)*atan((-1 + 3*x^2)^(1/4)) + atan(1 - sqrt(2)*(-1 + 3*x^2)^(1/4))/(2*sqrt(2)) - atan(1 + sqrt(2)*(-1 + 3*x^2)^(1/4))/(2*sqrt(2)) - (1/2)*atanh((-1 + 3*x^2)^(1/4)) - log(1 - sqrt(2)*(-1 + 3*x^2)^(1/4) + sqrt(-1 + 3*x^2))/(4*sqrt(2)) + log(1 + sqrt(2)*(-1 + 3*x^2)^(1/4) + sqrt(-1 + 3*x^2))/(4*sqrt(2))]
    @test_int [1/(x^3*(-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)), x, 17, -((-1 + 3*x^2)^(3/4)/(4*x^2)) + (3/4)*atan((-1 + 3*x^2)^(1/4)) + (9*atan(1 - sqrt(2)*(-1 + 3*x^2)^(1/4)))/(8*sqrt(2)) - (9*atan(1 + sqrt(2)*(-1 + 3*x^2)^(1/4)))/(8*sqrt(2)) - (3/4)*atanh((-1 + 3*x^2)^(1/4)) - (9*log(1 - sqrt(2)*(-1 + 3*x^2)^(1/4) + sqrt(-1 + 3*x^2)))/(16*sqrt(2)) + (9*log(1 + sqrt(2)*(-1 + 3*x^2)^(1/4) + sqrt(-1 + 3*x^2)))/(16*sqrt(2))]

    @test_int [x^4/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)), x, 12, (2/45)*x*(-1 + 3*x^2)^(3/4) + (8*x*(-1 + 3*x^2)^(1/4))/(15*(1 + sqrt(-1 + 3*x^2))) - (1/9)*sqrt(2/3)*atan((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4)) - (1/9)*sqrt(2/3)*atanh((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4)) - (8*sqrt(x^2/(1 + sqrt(-1 + 3*x^2))^2)*(1 + sqrt(-1 + 3*x^2))*Elliptic.E(2*atan((-1 + 3*x^2)^(1/4)), 1/2))/(15*sqrt(3)*x) + (4*sqrt(x^2/(1 + sqrt(-1 + 3*x^2))^2)*(1 + sqrt(-1 + 3*x^2))*Elliptic.F(2*atan((-1 + 3*x^2)^(1/4)), 1/2))/(15*sqrt(3)*x)]
    @test_int [x^2/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)), x, 7, (2*x*(-1 + 3*x^2)^(1/4))/(3*(1 + sqrt(-1 + 3*x^2))) - atan((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4))/(3*sqrt(6)) - atanh((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4))/(3*sqrt(6)) - (2*sqrt(x^2/(1 + sqrt(-1 + 3*x^2))^2)*(1 + sqrt(-1 + 3*x^2))*Elliptic.E(2*atan((-1 + 3*x^2)^(1/4)), 1/2))/(3*sqrt(3)*x) + (sqrt(x^2/(1 + sqrt(-1 + 3*x^2))^2)*(1 + sqrt(-1 + 3*x^2))*Elliptic.F(2*atan((-1 + 3*x^2)^(1/4)), 1/2))/(3*sqrt(3)*x)]
    @test_int [x^0/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)), x, 1, -(atan((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4))/(2*sqrt(6))) - atanh((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4))/(2*sqrt(6))]
    @test_int [1/(x^2*(-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)), x, 8, -((-1 + 3*x^2)^(3/4)/(2*x)) + (3*x*(-1 + 3*x^2)^(1/4))/(2*(1 + sqrt(-1 + 3*x^2))) - (1/4)*sqrt(3/2)*atan((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4)) - (1/4)*sqrt(3/2)*atanh((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4)) - (sqrt(3)*sqrt(x^2/(1 + sqrt(-1 + 3*x^2))^2)*(1 + sqrt(-1 + 3*x^2))*Elliptic.E(2*atan((-1 + 3*x^2)^(1/4)), 1/2))/(2*x) + (sqrt(3)*sqrt(x^2/(1 + sqrt(-1 + 3*x^2))^2)*(1 + sqrt(-1 + 3*x^2))*Elliptic.F(2*atan((-1 + 3*x^2)^(1/4)), 1/2))/(4*x)]
    @test_int [1/(x^4*(-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)), x, 14, -((-1 + 3*x^2)^(3/4)/(6*x^3)) - (3*(-1 + 3*x^2)^(3/4))/(2*x) + (9*x*(-1 + 3*x^2)^(1/4))/(2*(1 + sqrt(-1 + 3*x^2))) - (3/8)*sqrt(3/2)*atan((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4)) - (3/8)*sqrt(3/2)*atanh((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4)) - (3*sqrt(3)*sqrt(x^2/(1 + sqrt(-1 + 3*x^2))^2)*(1 + sqrt(-1 + 3*x^2))*Elliptic.E(2*atan((-1 + 3*x^2)^(1/4)), 1/2))/(2*x) + (3*sqrt(3)*sqrt(x^2/(1 + sqrt(-1 + 3*x^2))^2)*(1 + sqrt(-1 + 3*x^2))*Elliptic.F(2*atan((-1 + 3*x^2)^(1/4)), 1/2))/(4*x)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*e^m*(a+b*x^2)^(3/4)*(c+d*x^2)^q*when*b*c-2*a*d=0=#

    #= ::Subsubsection::Closed:: =#
    #=a>0=#

    @test_int [x^2/((2 + 3*x^2)^(3/4)*(4 + 3*x^2)), x, 1, -(atan((2*2^(3/4) + 2*2^(1/4)*sqrt(2 + 3*x^2))/(2*sqrt(3)*x*(2 + 3*x^2)^(1/4)))/(3*2^(1/4)*sqrt(3))) + atanh((2*2^(3/4) - 2*2^(1/4)*sqrt(2 + 3*x^2))/(2*sqrt(3)*x*(2 + 3*x^2)^(1/4)))/(3*2^(1/4)*sqrt(3))]
    @test_int [x^2/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)), x, 1, atan((2 - sqrt(2)*sqrt(2 - 3*x^2))/(2^(1/4)*sqrt(3)*x*(2 - 3*x^2)^(1/4)))/(3*2^(1/4)*sqrt(3)) - atanh((2 + sqrt(2)*sqrt(2 - 3*x^2))/(2^(1/4)*sqrt(3)*x*(2 - 3*x^2)^(1/4)))/(3*2^(1/4)*sqrt(3))]

    @test_int [x^2/((2 + b*x^2)^(3/4)*(4 + b*x^2)), x, 1, -(atan((2*2^(3/4) + 2*2^(1/4)*sqrt(2 + b*x^2))/(2*sqrt(b)*x*(2 + b*x^2)^(1/4)))/(2^(1/4)*b^(3/2))) + atanh((2*2^(3/4) - 2*2^(1/4)*sqrt(2 + b*x^2))/(2*sqrt(b)*x*(2 + b*x^2)^(1/4)))/(2^(1/4)*b^(3/2))]
    @test_int [x^2/((2 - b*x^2)^(3/4)*(4 - b*x^2)), x, 1, atan((2 - sqrt(2)*sqrt(2 - b*x^2))/(2^(1/4)*sqrt(b)*x*(2 - b*x^2)^(1/4)))/(2^(1/4)*b^(3/2)) - atanh((2 + sqrt(2)*sqrt(2 - b*x^2))/(2^(1/4)*sqrt(b)*x*(2 - b*x^2)^(1/4)))/(2^(1/4)*b^(3/2))]

    @test_int [x^2/((a + 3*x^2)^(3/4)*(2*a + 3*x^2)), x, 1, -(atan((a^(3/4)*(1 + sqrt(a + 3*x^2)/sqrt(a)))/(sqrt(3)*x*(a + 3*x^2)^(1/4)))/(3*sqrt(3)*a^(1/4))) + atanh((a^(3/4)*(1 - sqrt(a + 3*x^2)/sqrt(a)))/(sqrt(3)*x*(a + 3*x^2)^(1/4)))/(3*sqrt(3)*a^(1/4))]
    @test_int [x^2/((a - 3*x^2)^(3/4)*(2*a - 3*x^2)), x, 1, atan((a^(3/4)*(1 - sqrt(a - 3*x^2)/sqrt(a)))/(sqrt(3)*x*(a - 3*x^2)^(1/4)))/(3*sqrt(3)*a^(1/4)) - atanh((a^(3/4)*(1 + sqrt(a - 3*x^2)/sqrt(a)))/(sqrt(3)*x*(a - 3*x^2)^(1/4)))/(3*sqrt(3)*a^(1/4))]

    @test_int [x^2/((a + b*x^2)^(3/4)*(2*a + b*x^2)), x, 1, -(atan((a^(3/4)*(1 + sqrt(a + b*x^2)/sqrt(a)))/(sqrt(b)*x*(a + b*x^2)^(1/4)))/(a^(1/4)*b^(3/2))) + atanh((a^(3/4)*(1 - sqrt(a + b*x^2)/sqrt(a)))/(sqrt(b)*x*(a + b*x^2)^(1/4)))/(a^(1/4)*b^(3/2))]
    @test_int [x^2/((a - b*x^2)^(3/4)*(2*a - b*x^2)), x, 1, atan((a^(3/4)*(1 - sqrt(a - b*x^2)/sqrt(a)))/(sqrt(b)*x*(a - b*x^2)^(1/4)))/(a^(1/4)*b^(3/2)) - atanh((a^(3/4)*(1 + sqrt(a - b*x^2)/sqrt(a)))/(sqrt(b)*x*(a - b*x^2)^(1/4)))/(a^(1/4)*b^(3/2))]

    @test_int [x^7/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)), x, 20, (56/81)*(2 - 3*x^2)^(1/4) - (16/405)*(2 - 3*x^2)^(5/4) + (2/729)*(2 - 3*x^2)^(9/4) - (16/81)*2^(3/4)*atan(1 + (4 - 6*x^2)^(1/4)) + (16/81)*2^(3/4)*atan(1 - 2^(1/4)*(2 - 3*x^2)^(1/4)) + (8/81)*2^(3/4)*log(sqrt(2) - 2^(3/4)*(2 - 3*x^2)^(1/4) + sqrt(2 - 3*x^2)) - (8/81)*2^(3/4)*log(sqrt(2) + 2^(3/4)*(2 - 3*x^2)^(1/4) + sqrt(2 - 3*x^2))]
    @test_int [x^5/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)), x, 17, (4/9)*(2 - 3*x^2)^(1/4) - (2/135)*(2 - 3*x^2)^(5/4) - (4/27)*2^(3/4)*atan(1 + (4 - 6*x^2)^(1/4)) + (4/27)*2^(3/4)*atan(1 - 2^(1/4)*(2 - 3*x^2)^(1/4)) + (2/27)*2^(3/4)*log(sqrt(2) - 2^(3/4)*(2 - 3*x^2)^(1/4) + sqrt(2 - 3*x^2)) - (2/27)*2^(3/4)*log(sqrt(2) + 2^(3/4)*(2 - 3*x^2)^(1/4) + sqrt(2 - 3*x^2))]
    @test_int [x^3/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)), x, 14, (2/9)*(2 - 3*x^2)^(1/4) - (1/9)*2^(3/4)*atan(1 + (4 - 6*x^2)^(1/4)) + (1/9)*2^(3/4)*atan(1 - 2^(1/4)*(2 - 3*x^2)^(1/4)) + log(sqrt(2) - 2^(3/4)*(2 - 3*x^2)^(1/4) + sqrt(2 - 3*x^2))/(9*2^(1/4)) - log(sqrt(2) + 2^(3/4)*(2 - 3*x^2)^(1/4) + sqrt(2 - 3*x^2))/(9*2^(1/4))]
    @test_int [x^1/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)), x, 11, -(atan(1 + (4 - 6*x^2)^(1/4))/(6*2^(1/4))) + atan(1 - 2^(1/4)*(2 - 3*x^2)^(1/4))/(6*2^(1/4)) + log(sqrt(2) - 2^(3/4)*(2 - 3*x^2)^(1/4) + sqrt(2 - 3*x^2))/(12*2^(1/4)) - log(sqrt(2) + 2^(3/4)*(2 - 3*x^2)^(1/4) + sqrt(2 - 3*x^2))/(12*2^(1/4))]
    @test_int [1/(x^1*(2 - 3*x^2)^(3/4)*(4 - 3*x^2)), x, 18, -(atan((2 - 3*x^2)^(1/4)/2^(1/4))/(4*2^(3/4))) - atan(1 + (4 - 6*x^2)^(1/4))/(8*2^(1/4)) + atan(1 - 2^(1/4)*(2 - 3*x^2)^(1/4))/(8*2^(1/4)) - atanh((2 - 3*x^2)^(1/4)/2^(1/4))/(4*2^(3/4)) + log(sqrt(2) - 2^(3/4)*(2 - 3*x^2)^(1/4) + sqrt(2 - 3*x^2))/(16*2^(1/4)) - log(sqrt(2) + 2^(3/4)*(2 - 3*x^2)^(1/4) + sqrt(2 - 3*x^2))/(16*2^(1/4))]
    @test_int [1/(x^3*(2 - 3*x^2)^(3/4)*(4 - 3*x^2)), x, 24, -((2 - 3*x^2)^(1/4)/(16*x^2)) - (15*atan((2 - 3*x^2)^(1/4)/2^(1/4)))/(32*2^(3/4)) - (3*atan(1 + (4 - 6*x^2)^(1/4)))/(32*2^(1/4)) + (3*atan(1 - 2^(1/4)*(2 - 3*x^2)^(1/4)))/(32*2^(1/4)) - (15*atanh((2 - 3*x^2)^(1/4)/2^(1/4)))/(32*2^(3/4)) + (3*log(sqrt(2) - 2^(3/4)*(2 - 3*x^2)^(1/4) + sqrt(2 - 3*x^2)))/(64*2^(1/4)) - (3*log(sqrt(2) + 2^(3/4)*(2 - 3*x^2)^(1/4) + sqrt(2 - 3*x^2)))/(64*2^(1/4))]

    @test_int [x^6/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)), x, 11, (80/567)*x*(2 - 3*x^2)^(1/4) + (2/63)*x^3*(2 - 3*x^2)^(1/4) + (8*2^(3/4)*atan((2^(3/4) - 2^(1/4)*sqrt(2 - 3*x^2))/(sqrt(3)*x*(2 - 3*x^2)^(1/4))))/(27*sqrt(3)) - (8*2^(3/4)*atanh((2^(3/4) + 2^(1/4)*sqrt(2 - 3*x^2))/(sqrt(3)*x*(2 - 3*x^2)^(1/4))))/(27*sqrt(3)) - (160*2^(3/4)*Elliptic.F((1/2)*asin(sqrt(3/2)*x), 2))/(567*sqrt(3))]
    @test_int [x^4/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)), x, 8, (2/27)*x*(2 - 3*x^2)^(1/4) + (2*2^(3/4)*atan((2^(3/4) - 2^(1/4)*sqrt(2 - 3*x^2))/(sqrt(3)*x*(2 - 3*x^2)^(1/4))))/(9*sqrt(3)) - (2*2^(3/4)*atanh((2^(3/4) + 2^(1/4)*sqrt(2 - 3*x^2))/(sqrt(3)*x*(2 - 3*x^2)^(1/4))))/(9*sqrt(3)) - (4*2^(3/4)*Elliptic.F((1/2)*asin(sqrt(3/2)*x), 2))/(27*sqrt(3))]
    @test_int [x^2/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)), x, 1, atan((2 - sqrt(2)*sqrt(2 - 3*x^2))/(2^(1/4)*sqrt(3)*x*(2 - 3*x^2)^(1/4)))/(3*2^(1/4)*sqrt(3)) - atanh((2 + sqrt(2)*sqrt(2 - 3*x^2))/(2^(1/4)*sqrt(3)*x*(2 - 3*x^2)^(1/4)))/(3*2^(1/4)*sqrt(3))]
    @test_int [x^0/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)), x, 3, atan((2^(3/4) - 2^(1/4)*sqrt(2 - 3*x^2))/(sqrt(3)*x*(2 - 3*x^2)^(1/4)))/(4*2^(1/4)*sqrt(3)) - atanh((2^(3/4) + 2^(1/4)*sqrt(2 - 3*x^2))/(sqrt(3)*x*(2 - 3*x^2)^(1/4)))/(4*2^(1/4)*sqrt(3)) + Elliptic.F((1/2)*asin(sqrt(3/2)*x), 2)/(2*2^(1/4)*sqrt(3))]
    @test_int [1/(x^2*(2 - 3*x^2)^(3/4)*(4 - 3*x^2)), x, 7, -((2 - 3*x^2)^(1/4)/(8*x)) + (sqrt(3)*atan((2^(3/4) - 2^(1/4)*sqrt(2 - 3*x^2))/(sqrt(3)*x*(2 - 3*x^2)^(1/4))))/(16*2^(1/4)) - (sqrt(3)*atanh((2^(3/4) + 2^(1/4)*sqrt(2 - 3*x^2))/(sqrt(3)*x*(2 - 3*x^2)^(1/4))))/(16*2^(1/4)) + (sqrt(3)*Elliptic.F((1/2)*asin(sqrt(3/2)*x), 2))/(4*2^(1/4))]
    @test_int [1/(x^4*(2 - 3*x^2)^(3/4)*(4 - 3*x^2)), x, 10, -((2 - 3*x^2)^(1/4)/(24*x^3)) - (2 - 3*x^2)^(1/4)/(4*x) + (3*sqrt(3)*atan((2^(3/4) - 2^(1/4)*sqrt(2 - 3*x^2))/(sqrt(3)*x*(2 - 3*x^2)^(1/4))))/(64*2^(1/4)) - (3*sqrt(3)*atanh((2^(3/4) + 2^(1/4)*sqrt(2 - 3*x^2))/(sqrt(3)*x*(2 - 3*x^2)^(1/4))))/(64*2^(1/4)) + (11*sqrt(3)*Elliptic.F((1/2)*asin(sqrt(3/2)*x), 2))/(32*2^(1/4))]

    #= ::Subsubsection::Closed:: =#
    #=a<0=#

    @test_int [x^2/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)), x, 1, atan((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4))/(3*sqrt(6)) - atanh((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4))/(3*sqrt(6))]
    @test_int [x^2/((-2 - 3*x^2)*(-1 - 3*x^2)^(3/4)), x, 1, atan((sqrt(3/2)*x)/(-1 - 3*x^2)^(1/4))/(3*sqrt(6)) - atanh((sqrt(3/2)*x)/(-1 - 3*x^2)^(1/4))/(3*sqrt(6))]

    @test_int [x^2/((-2 + b*x^2)*(-1 + b*x^2)^(3/4)), x, 1, atan((sqrt(b)*x)/(sqrt(2)*(-1 + b*x^2)^(1/4)))/(sqrt(2)*b^(3/2)) - atanh((sqrt(b)*x)/(sqrt(2)*(-1 + b*x^2)^(1/4)))/(sqrt(2)*b^(3/2))]
    @test_int [x^2/((-2 - b*x^2)*(-1 - b*x^2)^(3/4)), x, 1, atan((sqrt(b)*x)/(sqrt(2)*(-1 - b*x^2)^(1/4)))/(sqrt(2)*b^(3/2)) - atanh((sqrt(b)*x)/(sqrt(2)*(-1 - b*x^2)^(1/4)))/(sqrt(2)*b^(3/2))]

    @test_int [x^2/((-a + 3*x^2)^(3/4)*(-2*a + 3*x^2)), x, 1, atan((sqrt(3/2)*x)/(a^(1/4)*(-a + 3*x^2)^(1/4)))/(3*sqrt(6)*a^(1/4)) - atanh((sqrt(3/2)*x)/(a^(1/4)*(-a + 3*x^2)^(1/4)))/(3*sqrt(6)*a^(1/4))]
    @test_int [x^2/((-a - 3*x^2)^(3/4)*(-2*a - 3*x^2)), x, 1, atan((sqrt(3/2)*x)/(a^(1/4)*(-a - 3*x^2)^(1/4)))/(3*sqrt(6)*a^(1/4)) - atanh((sqrt(3/2)*x)/(a^(1/4)*(-a - 3*x^2)^(1/4)))/(3*sqrt(6)*a^(1/4))]

    @test_int [x^2/((-a + b*x^2)^(3/4)*(-2*a + b*x^2)), x, 1, atan((sqrt(b)*x)/(sqrt(2)*a^(1/4)*(-a + b*x^2)^(1/4)))/(sqrt(2)*a^(1/4)*b^(3/2)) - atanh((sqrt(b)*x)/(sqrt(2)*a^(1/4)*(-a + b*x^2)^(1/4)))/(sqrt(2)*a^(1/4)*b^(3/2))]
    @test_int [x^2/((-a - b*x^2)^(3/4)*(-2*a - b*x^2)), x, 1, atan((sqrt(b)*x)/(sqrt(2)*a^(1/4)*(-a - b*x^2)^(1/4)))/(sqrt(2)*a^(1/4)*b^(3/2)) - atanh((sqrt(b)*x)/(sqrt(2)*a^(1/4)*(-a - b*x^2)^(1/4)))/(sqrt(2)*a^(1/4)*b^(3/2))]

    @test_int [x^7/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)), x, 7, (14/81)*(-1 + 3*x^2)^(1/4) + (8/405)*(-1 + 3*x^2)^(5/4) + (2/729)*(-1 + 3*x^2)^(9/4) - (8/81)*atan((-1 + 3*x^2)^(1/4)) - (8/81)*atanh((-1 + 3*x^2)^(1/4))]
    @test_int [x^5/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)), x, 7, (2/9)*(-1 + 3*x^2)^(1/4) + (2/135)*(-1 + 3*x^2)^(5/4) - (4/27)*atan((-1 + 3*x^2)^(1/4)) - (4/27)*atanh((-1 + 3*x^2)^(1/4))]
    @test_int [x^3/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)), x, 6, (2/9)*(-1 + 3*x^2)^(1/4) - (2/9)*atan((-1 + 3*x^2)^(1/4)) - (2/9)*atanh((-1 + 3*x^2)^(1/4))]
    @test_int [x^1/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)), x, 5, (-(1/3))*atan((-1 + 3*x^2)^(1/4)) - (1/3)*atanh((-1 + 3*x^2)^(1/4))]
    @test_int [1/(x^1*(-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)), x, 16, (-(1/2))*atan((-1 + 3*x^2)^(1/4)) + atan(1 - sqrt(2)*(-1 + 3*x^2)^(1/4))/(2*sqrt(2)) - atan(1 + sqrt(2)*(-1 + 3*x^2)^(1/4))/(2*sqrt(2)) - (1/2)*atanh((-1 + 3*x^2)^(1/4)) + log(1 - sqrt(2)*(-1 + 3*x^2)^(1/4) + sqrt(-1 + 3*x^2))/(4*sqrt(2)) - log(1 + sqrt(2)*(-1 + 3*x^2)^(1/4) + sqrt(-1 + 3*x^2))/(4*sqrt(2))]
    @test_int [1/(x^3*(-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)), x, 17, -((-1 + 3*x^2)^(1/4)/(4*x^2)) - (3/4)*atan((-1 + 3*x^2)^(1/4)) + (15*atan(1 - sqrt(2)*(-1 + 3*x^2)^(1/4)))/(8*sqrt(2)) - (15*atan(1 + sqrt(2)*(-1 + 3*x^2)^(1/4)))/(8*sqrt(2)) - (3/4)*atanh((-1 + 3*x^2)^(1/4)) + (15*log(1 - sqrt(2)*(-1 + 3*x^2)^(1/4) + sqrt(-1 + 3*x^2)))/(16*sqrt(2)) - (15*log(1 + sqrt(2)*(-1 + 3*x^2)^(1/4) + sqrt(-1 + 3*x^2)))/(16*sqrt(2))]

    @test_int [x^6/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)), x, 15, (40/567)*x*(-1 + 3*x^2)^(1/4) + (2/63)*x^3*(-1 + 3*x^2)^(1/4) + (2/27)*sqrt(2/3)*atan((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4)) - (2/27)*sqrt(2/3)*atanh((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4)) + (40*sqrt(x^2/(1 + sqrt(-1 + 3*x^2))^2)*(1 + sqrt(-1 + 3*x^2))*Elliptic.F(2*atan((-1 + 3*x^2)^(1/4)), 1/2))/(567*sqrt(3)*x)]
    @test_int [x^4/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)), x, 11, (2/27)*x*(-1 + 3*x^2)^(1/4) + (1/9)*sqrt(2/3)*atan((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4)) - (1/9)*sqrt(2/3)*atanh((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4)) + (2*sqrt(x^2/(1 + sqrt(-1 + 3*x^2))^2)*(1 + sqrt(-1 + 3*x^2))*Elliptic.F(2*atan((-1 + 3*x^2)^(1/4)), 1/2))/(27*sqrt(3)*x)]
    @test_int [x^2/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)), x, 1, atan((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4))/(3*sqrt(6)) - atanh((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4))/(3*sqrt(6))]
    @test_int [x^0/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)), x, 4, atan((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4))/(2*sqrt(6)) - atanh((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4))/(2*sqrt(6)) - (sqrt(x^2/(1 + sqrt(-1 + 3*x^2))^2)*(1 + sqrt(-1 + 3*x^2))*Elliptic.F(2*atan((-1 + 3*x^2)^(1/4)), 1/2))/(2*sqrt(3)*x)]
    @test_int [1/(x^2*(-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)), x, 9, -((-1 + 3*x^2)^(1/4)/(2*x)) + (1/4)*sqrt(3/2)*atan((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4)) - (1/4)*sqrt(3/2)*atanh((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4)) - (sqrt(3)*sqrt(x^2/(1 + sqrt(-1 + 3*x^2))^2)*(1 + sqrt(-1 + 3*x^2))*Elliptic.F(2*atan((-1 + 3*x^2)^(1/4)), 1/2))/(2*x)]
    @test_int [1/(x^4*(-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)), x, 13, -((-1 + 3*x^2)^(1/4)/(6*x^3)) - (2*(-1 + 3*x^2)^(1/4))/x + (3/8)*sqrt(3/2)*atan((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4)) - (3/8)*sqrt(3/2)*atanh((sqrt(3/2)*x)/(-1 + 3*x^2)^(1/4)) - (11*sqrt(3)*sqrt(x^2/(1 + sqrt(-1 + 3*x^2))^2)*(1 + sqrt(-1 + 3*x^2))*Elliptic.F(2*atan((-1 + 3*x^2)^(1/4)), 1/2))/(8*x)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(e*x)^(m/2)*(a+b*x^2)^(p/4)*(c+d*x^2)=#

    #= ::Subsubsection:: =#
    #=p>0=#

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(e*x)^(5/2)*(c + d*x^2)/(a + b*x^2)^(3/4), x, 7, ((8*b*c - 7*a*d)*e*(e*x)^(3/2)*(a + b*x^2)^(1/4))/(16*b^2) + (d*(e*x)^(7/2)*(a + b*x^2)^(1/4))/(4*b*e) + (3*a*(8*b*c - 7*a*d)*e^(5/2)*atan((b^(1/4)*sqrt(e*x))/(sqrt(e)*(a + b*x^2)^(1/4))))/(32*b^(11/4)) - (3*a*(8*b*c - 7*a*d)*e^(5/2)*atanh((b^(1/4)*sqrt(e*x))/(sqrt(e)*(a + b*x^2)^(1/4))))/(32*b^(11/4))]
    @test_int [(e*x)^(1/2)*(c + d*x^2)/(a + b*x^2)^(3/4), x, 6, (d*(e*x)^(3/2)*(a + b*x^2)^(1/4))/(2*b*e) - ((4*b*c - 3*a*d)*sqrt(e)*atan((b^(1/4)*sqrt(e*x))/(sqrt(e)*(a + b*x^2)^(1/4))))/(4*b^(7/4)) + ((4*b*c - 3*a*d)*sqrt(e)*atanh((b^(1/4)*sqrt(e*x))/(sqrt(e)*(a + b*x^2)^(1/4))))/(4*b^(7/4))]
    @test_int [(c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(3/4)), x, 6, -((2*c*(a + b*x^2)^(1/4))/(a*e*sqrt(e*x))) - (d*atan((b^(1/4)*sqrt(e*x))/(sqrt(e)*(a + b*x^2)^(1/4))))/(b^(3/4)*e^(3/2)) + (d*atanh((b^(1/4)*sqrt(e*x))/(sqrt(e)*(a + b*x^2)^(1/4))))/(b^(3/4)*e^(3/2))]
    @test_int [(c + d*x^2)/((e*x)^(7/2)*(a + b*x^2)^(3/4)), x, 2, -((2*c*(a + b*x^2)^(1/4))/(5*a*e*(e*x)^(5/2))) + (2*(4*b*c - 5*a*d)*(a + b*x^2)^(1/4))/(5*a^2*e^3*sqrt(e*x))]
    @test_int [(c + d*x^2)/((e*x)^(11/2)*(a + b*x^2)^(3/4)), x, 3, -((2*c*(a + b*x^2)^(1/4))/(9*a*e*(e*x)^(9/2))) + (2*(8*b*c - 9*a*d)*(a + b*x^2)^(1/4))/(9*a^2*e^3*(e*x)^(5/2)) - (8*(8*b*c - 9*a*d)*(a + b*x^2)^(5/4))/(45*a^3*e^3*(e*x)^(5/2))]
    @test_int [(c + d*x^2)/((e*x)^(15/2)*(a + b*x^2)^(3/4)), x, 4, -((2*c*(a + b*x^2)^(1/4))/(13*a*e*(e*x)^(13/2))) + (2*(12*b*c - 13*a*d)*(a + b*x^2)^(1/4))/(13*a^2*e^3*(e*x)^(9/2)) - (16*(12*b*c - 13*a*d)*(a + b*x^2)^(5/4))/(65*a^3*e^3*(e*x)^(9/2)) + (64*(12*b*c - 13*a*d)*(a + b*x^2)^(9/4))/(585*a^4*e^3*(e*x)^(9/2))]

    @test_int [(e*x)^(7/2)*(c + d*x^2)/(a + b*x^2)^(3/4), x, 8, -((a*(10*b*c - 9*a*d)*e^3*sqrt(e*x)*(a + b*x^2)^(1/4))/(12*b^3)) + ((10*b*c - 9*a*d)*e*(e*x)^(5/2)*(a + b*x^2)^(1/4))/(30*b^2) + (d*(e*x)^(9/2)*(a + b*x^2)^(1/4))/(5*b*e) - (a^(3/2)*(10*b*c - 9*a*d)*e^2*(1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*Elliptic.F((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(12*b^(5/2)*(a + b*x^2)^(3/4))]
    @test_int [(e*x)^(3/2)*(c + d*x^2)/(a + b*x^2)^(3/4), x, 7, ((6*b*c - 5*a*d)*e*sqrt(e*x)*(a + b*x^2)^(1/4))/(6*b^2) + (d*(e*x)^(5/2)*(a + b*x^2)^(1/4))/(3*b*e) + (sqrt(a)*(6*b*c - 5*a*d)*(1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*Elliptic.F((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(6*b^(3/2)*(a + b*x^2)^(3/4))]
    @test_int [(c + d*x^2)/((e*x)^(1/2)*(a + b*x^2)^(3/4)), x, 6, (d*sqrt(e*x)*(a + b*x^2)^(1/4))/(b*e) - ((2*b*c - a*d)*(1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*Elliptic.F((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(sqrt(a)*sqrt(b)*e^2*(a + b*x^2)^(3/4))]
    @test_int [(c + d*x^2)/((e*x)^(5/2)*(a + b*x^2)^(3/4)), x, 6, -((2*c*(a + b*x^2)^(1/4))/(3*a*e*(e*x)^(3/2))) + (2*sqrt(b)*(2*b*c - 3*a*d)*(1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*Elliptic.F((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(3*a^(3/2)*e^4*(a + b*x^2)^(3/4))]
    @test_int [(c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(3/4)), x, 7, -((2*c*(a + b*x^2)^(1/4))/(7*a*e*(e*x)^(7/2))) + (2*(6*b*c - 7*a*d)*(a + b*x^2)^(1/4))/(21*a^2*e^3*(e*x)^(3/2)) - (4*b^(3/2)*(6*b*c - 7*a*d)*(1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*Elliptic.F((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(21*a^(5/2)*e^6*(a + b*x^2)^(3/4))]
    @test_int [(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(3/4)), x, 8, -((2*c*(a + b*x^2)^(1/4))/(11*a*e*(e*x)^(11/2))) + (2*(10*b*c - 11*a*d)*(a + b*x^2)^(1/4))/(77*a^2*e^3*(e*x)^(7/2)) - (4*b*(10*b*c - 11*a*d)*(a + b*x^2)^(1/4))/(77*a^3*e^5*(e*x)^(3/2)) + (8*b^(5/2)*(10*b*c - 11*a*d)*(1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*Elliptic.F((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(77*a^(7/2)*e^8*(a + b*x^2)^(3/4))]

    @test_int [(e*x)^(3/2)*(c + d*x^2)/(a + b*x^2)^(5/4), x, 7, -(((4*b*c - 5*a*d)*e*sqrt(e*x))/(2*b^2*(a + b*x^2)^(1/4))) + (d*(e*x)^(5/2))/(2*b*e*(a + b*x^2)^(1/4)) + ((4*b*c - 5*a*d)*e^(3/2)*atan((b^(1/4)*sqrt(e*x))/(sqrt(e)*(a + b*x^2)^(1/4))))/(4*b^(9/4)) + ((4*b*c - 5*a*d)*e^(3/2)*atanh((b^(1/4)*sqrt(e*x))/(sqrt(e)*(a + b*x^2)^(1/4))))/(4*b^(9/4))]
    @test_int [(c + d*x^2)/((e*x)^(1/2)*(a + b*x^2)^(5/4)), x, 6, (2*(b*c - a*d)*sqrt(e*x))/(a*b*e*(a + b*x^2)^(1/4)) + (d*atan((b^(1/4)*sqrt(e*x))/(sqrt(e)*(a + b*x^2)^(1/4))))/(b^(5/4)*sqrt(e)) + (d*atanh((b^(1/4)*sqrt(e*x))/(sqrt(e)*(a + b*x^2)^(1/4))))/(b^(5/4)*sqrt(e))]
    @test_int [(c + d*x^2)/((e*x)^(5/2)*(a + b*x^2)^(5/4)), x, 2, -((2*c)/(3*a*e*(e*x)^(3/2)*(a + b*x^2)^(1/4))) - (2*(4*b*c - 3*a*d)*sqrt(e*x))/(3*a^2*e^3*(a + b*x^2)^(1/4))]
    @test_int [(c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(5/4)), x, 3, -((2*c)/(7*a*e*(e*x)^(7/2)*(a + b*x^2)^(1/4))) - (2*(8*b*c - 7*a*d))/(7*a^2*e^3*(e*x)^(3/2)*(a + b*x^2)^(1/4)) + (8*(8*b*c - 7*a*d)*(a + b*x^2)^(3/4))/(21*a^3*e^3*(e*x)^(3/2))]
    @test_int [(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(5/4)), x, 4, -((2*c)/(11*a*e*(e*x)^(11/2)*(a + b*x^2)^(1/4))) - (2*(12*b*c - 11*a*d))/(11*a^2*e^3*(e*x)^(7/2)*(a + b*x^2)^(1/4)) + (16*(12*b*c - 11*a*d)*(a + b*x^2)^(3/4))/(33*a^3*e^3*(e*x)^(7/2)) - (64*(12*b*c - 11*a*d)*(a + b*x^2)^(7/4))/(231*a^4*e^3*(e*x)^(7/2))]

    @test_int [(e*x)^(9/2)*(c + d*x^2)/(a + b*x^2)^(5/4), x, 6, -((7*a*(10*b*c - 11*a*d)*e^3*(e*x)^(3/2))/(60*b^3*(a + b*x^2)^(1/4))) + ((10*b*c - 11*a*d)*e*(e*x)^(7/2))/(30*b^2*(a + b*x^2)^(1/4)) + (d*(e*x)^(11/2))/(5*b*e*(a + b*x^2)^(1/4)) - (7*a^(3/2)*(10*b*c - 11*a*d)*e^4*(1 + a/(b*x^2))^(1/4)*sqrt(e*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(20*b^(7/2)*(a + b*x^2)^(1/4))]
    @test_int [(e*x)^(5/2)*(c + d*x^2)/(a + b*x^2)^(5/4), x, 5, ((6*b*c - 7*a*d)*e*(e*x)^(3/2))/(6*b^2*(a + b*x^2)^(1/4)) + (d*(e*x)^(7/2))/(3*b*e*(a + b*x^2)^(1/4)) + (sqrt(a)*(6*b*c - 7*a*d)*e^2*(1 + a/(b*x^2))^(1/4)*sqrt(e*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(2*b^(5/2)*(a + b*x^2)^(1/4))]
    @test_int [(e*x)^(1/2)*(c + d*x^2)/(a + b*x^2)^(5/4), x, 4, (d*(e*x)^(3/2))/(b*e*(a + b*x^2)^(1/4)) - ((2*b*c - 3*a*d)*(1 + a/(b*x^2))^(1/4)*sqrt(e*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(sqrt(a)*b^(3/2)*(a + b*x^2)^(1/4))]
    @test_int [(c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(5/4)), x, 4, -((2*c)/(a*e*sqrt(e*x)*(a + b*x^2)^(1/4))) + (2*(2*b*c - a*d)*(1 + a/(b*x^2))^(1/4)*sqrt(e*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(a^(3/2)*sqrt(b)*e^2*(a + b*x^2)^(1/4))]
    @test_int [(c + d*x^2)/((e*x)^(7/2)*(a + b*x^2)^(5/4)), x, 5, -((2*c)/(5*a*e*(e*x)^(5/2)*(a + b*x^2)^(1/4))) + (2*(6*b*c - 5*a*d))/(5*a^2*e^3*sqrt(e*x)*(a + b*x^2)^(1/4)) - (4*sqrt(b)*(6*b*c - 5*a*d)*(1 + a/(b*x^2))^(1/4)*sqrt(e*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(5*a^(5/2)*e^4*(a + b*x^2)^(1/4))]
    @test_int [(c + d*x^2)/((e*x)^(11/2)*(a + b*x^2)^(5/4)), x, 6, -((2*c)/(9*a*e*(e*x)^(9/2)*(a + b*x^2)^(1/4))) + (2*(10*b*c - 9*a*d))/(45*a^2*e^3*(e*x)^(5/2)*(a + b*x^2)^(1/4)) - (4*b*(10*b*c - 9*a*d))/(15*a^3*e^5*sqrt(e*x)*(a + b*x^2)^(1/4)) + (8*b^(3/2)*(10*b*c - 9*a*d)*(1 + a/(b*x^2))^(1/4)*sqrt(e*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(15*a^(7/2)*e^6*(a + b*x^2)^(1/4))]

    @test_int [(e*x)^(5/2)*(c + d*x^2)/(a + b*x^2)^(7/4), x, 7, (2*(b*c - a*d)*(e*x)^(7/2))/(3*a*b*e*(a + b*x^2)^(3/4)) - ((4*b*c - 7*a*d)*e*(e*x)^(3/2)*(a + b*x^2)^(1/4))/(6*a*b^2) - ((4*b*c - 7*a*d)*e^(5/2)*atan((b^(1/4)*sqrt(e*x))/(sqrt(e)*(a + b*x^2)^(1/4))))/(4*b^(11/4)) + ((4*b*c - 7*a*d)*e^(5/2)*atanh((b^(1/4)*sqrt(e*x))/(sqrt(e)*(a + b*x^2)^(1/4))))/(4*b^(11/4))]
    @test_int [(e*x)^(1/2)*(c + d*x^2)/(a + b*x^2)^(7/4), x, 6, (2*(b*c - a*d)*(e*x)^(3/2))/(3*a*b*e*(a + b*x^2)^(3/4)) - (d*sqrt(e)*atan((b^(1/4)*sqrt(e*x))/(sqrt(e)*(a + b*x^2)^(1/4))))/b^(7/4) + (d*sqrt(e)*atanh((b^(1/4)*sqrt(e*x))/(sqrt(e)*(a + b*x^2)^(1/4))))/b^(7/4)]
    @test_int [(c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(7/4)), x, 2, -((2*c)/(a*e*sqrt(e*x)*(a + b*x^2)^(3/4))) - (2*(4*b*c - a*d)*(e*x)^(3/2))/(3*a^2*e^3*(a + b*x^2)^(3/4))]
    @test_int [(c + d*x^2)/((e*x)^(7/2)*(a + b*x^2)^(7/4)), x, 3, -((2*c)/(5*a*e*(e*x)^(5/2)*(a + b*x^2)^(3/4))) - (2*(8*b*c - 5*a*d))/(15*a^2*e^3*sqrt(e*x)*(a + b*x^2)^(3/4)) + (8*(8*b*c - 5*a*d)*(a + b*x^2)^(1/4))/(15*a^3*e^3*sqrt(e*x))]
    @test_int [(c + d*x^2)/((e*x)^(11/2)*(a + b*x^2)^(7/4)), x, 4, -((2*c)/(9*a*e*(e*x)^(9/2)*(a + b*x^2)^(3/4))) - (2*(4*b*c - 3*a*d))/(9*a^2*e^3*(e*x)^(5/2)*(a + b*x^2)^(3/4)) + (16*(4*b*c - 3*a*d)*(a + b*x^2)^(1/4))/(9*a^3*e^3*(e*x)^(5/2)) - (64*(4*b*c - 3*a*d)*(a + b*x^2)^(5/4))/(45*a^4*e^3*(e*x)^(5/2))]

    @test_int [(e*x)^(7/2)*(c + d*x^2)/(a + b*x^2)^(7/4), x, 8, (2*(b*c - a*d)*(e*x)^(9/2))/(3*a*b*e*(a + b*x^2)^(3/4)) + (5*(2*b*c - 3*a*d)*e^3*sqrt(e*x)*(a + b*x^2)^(1/4))/(6*b^3) - ((2*b*c - 3*a*d)*e*(e*x)^(5/2)*(a + b*x^2)^(1/4))/(3*a*b^2) + (5*sqrt(a)*(2*b*c - 3*a*d)*e^2*(1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*Elliptic.F((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(6*b^(5/2)*(a + b*x^2)^(3/4))]
    @test_int [(e*x)^(3/2)*(c + d*x^2)/(a + b*x^2)^(7/4), x, 7, (2*(b*c - a*d)*(e*x)^(5/2))/(3*a*b*e*(a + b*x^2)^(3/4)) - ((2*b*c - 5*a*d)*e*sqrt(e*x)*(a + b*x^2)^(1/4))/(3*a*b^2) - ((2*b*c - 5*a*d)*(1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*Elliptic.F((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(3*sqrt(a)*b^(3/2)*(a + b*x^2)^(3/4))]
    @test_int [(c + d*x^2)/((e*x)^(1/2)*(a + b*x^2)^(7/4)), x, 6, (2*(b*c - a*d)*sqrt(e*x))/(3*a*b*e*(a + b*x^2)^(3/4)) - (2*(2*b*c + a*d)*(1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*Elliptic.F((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(3*a^(3/2)*sqrt(b)*e^2*(a + b*x^2)^(3/4))]
    @test_int [(c + d*x^2)/((e*x)^(5/2)*(a + b*x^2)^(7/4)), x, 7, -((2*c)/(3*a*e*(e*x)^(3/2)*(a + b*x^2)^(3/4))) - (2*(2*b*c - a*d)*sqrt(e*x))/(3*a^2*e^3*(a + b*x^2)^(3/4)) + (4*sqrt(b)*(2*b*c - a*d)*(1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*Elliptic.F((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(3*a^(5/2)*e^4*(a + b*x^2)^(3/4))]
    @test_int [(c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(7/4)), x, 8, -((2*c)/(7*a*e*(e*x)^(7/2)*(a + b*x^2)^(3/4))) - (2*(10*b*c - 7*a*d))/(21*a^2*e^3*(e*x)^(3/2)*(a + b*x^2)^(3/4)) + (4*(10*b*c - 7*a*d)*(a + b*x^2)^(1/4))/(21*a^3*e^3*(e*x)^(3/2)) - (8*b^(3/2)*(10*b*c - 7*a*d)*(1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*Elliptic.F((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(21*a^(7/2)*e^6*(a + b*x^2)^(3/4))]

    @test_int [(e*x)^(7/2)*(c + d*x^2)/(a + b*x^2)^(9/4), x, 8, (2*(b*c - a*d)*(e*x)^(9/2))/(5*a*b*e*(a + b*x^2)^(5/4)) - ((4*b*c - 9*a*d)*e^3*sqrt(e*x))/(2*b^3*(a + b*x^2)^(1/4)) - ((4*b*c - 9*a*d)*e*(e*x)^(5/2))/(10*a*b^2*(a + b*x^2)^(1/4)) + ((4*b*c - 9*a*d)*e^(7/2)*atan((b^(1/4)*sqrt(e*x))/(sqrt(e)*(a + b*x^2)^(1/4))))/(4*b^(13/4)) + ((4*b*c - 9*a*d)*e^(7/2)*atanh((b^(1/4)*sqrt(e*x))/(sqrt(e)*(a + b*x^2)^(1/4))))/(4*b^(13/4))]
    @test_int [(e*x)^(3/2)*(c + d*x^2)/(a + b*x^2)^(9/4), x, 7, (2*(b*c - a*d)*(e*x)^(5/2))/(5*a*b*e*(a + b*x^2)^(5/4)) - (2*d*e*sqrt(e*x))/(b^2*(a + b*x^2)^(1/4)) + (d*e^(3/2)*atan((b^(1/4)*sqrt(e*x))/(sqrt(e)*(a + b*x^2)^(1/4))))/b^(9/4) + (d*e^(3/2)*atanh((b^(1/4)*sqrt(e*x))/(sqrt(e)*(a + b*x^2)^(1/4))))/b^(9/4)]
    @test_int [(c + d*x^2)/((e*x)^(1/2)*(a + b*x^2)^(9/4)), x, 2, (2*(b*c - a*d)*sqrt(e*x))/(5*a*b*e*(a + b*x^2)^(5/4)) + (2*(4*b*c + a*d)*sqrt(e*x))/(5*a^2*b*e*(a + b*x^2)^(1/4))]
    @test_int [(c + d*x^2)/((e*x)^(5/2)*(a + b*x^2)^(9/4)), x, 3, -((2*c)/(3*a*e*(e*x)^(3/2)*(a + b*x^2)^(5/4))) - (2*(8*b*c - 3*a*d)*sqrt(e*x))/(15*a^2*e^3*(a + b*x^2)^(5/4)) - (8*(8*b*c - 3*a*d)*sqrt(e*x))/(15*a^3*e^3*(a + b*x^2)^(1/4))]
    @test_int [(c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(9/4)), x, 4, -((2*c)/(7*a*e*(e*x)^(7/2)*(a + b*x^2)^(5/4))) - (2*(12*b*c - 7*a*d))/(35*a^2*e^3*(e*x)^(3/2)*(a + b*x^2)^(5/4)) - (16*(12*b*c - 7*a*d))/(35*a^3*e^3*(e*x)^(3/2)*(a + b*x^2)^(1/4)) + (64*(12*b*c - 7*a*d)*(a + b*x^2)^(3/4))/(105*a^4*e^3*(e*x)^(3/2))]
    @test_int [(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(9/4)), x, 5, -((2*c)/(11*a*e*(e*x)^(11/2)*(a + b*x^2)^(5/4))) - (2*(16*b*c - 11*a*d))/(55*a^2*e^3*(e*x)^(7/2)*(a + b*x^2)^(5/4)) - (24*(16*b*c - 11*a*d))/(55*a^3*e^3*(e*x)^(7/2)*(a + b*x^2)^(1/4)) + (64*(16*b*c - 11*a*d)*(a + b*x^2)^(3/4))/(55*a^4*e^3*(e*x)^(7/2)) - (256*(16*b*c - 11*a*d)*(a + b*x^2)^(7/4))/(385*a^5*e^3*(e*x)^(7/2))]

    @test_int [(e*x)^(13/2)*(c + d*x^2)/(a + b*x^2)^(9/4), x, 7, (2*(b*c - a*d)*(e*x)^(15/2))/(5*a*b*e*(a + b*x^2)^(5/4)) - (77*a*(2*b*c - 3*a*d)*e^5*(e*x)^(3/2))/(60*b^4*(a + b*x^2)^(1/4)) + (11*(2*b*c - 3*a*d)*e^3*(e*x)^(7/2))/(30*b^3*(a + b*x^2)^(1/4)) - ((2*b*c - 3*a*d)*e*(e*x)^(11/2))/(5*a*b^2*(a + b*x^2)^(1/4)) - (77*a^(3/2)*(2*b*c - 3*a*d)*e^6*(1 + a/(b*x^2))^(1/4)*sqrt(e*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(20*b^(9/2)*(a + b*x^2)^(1/4))]
    @test_int [(e*x)^(9/2)*(c + d*x^2)/(a + b*x^2)^(9/4), x, 6, (2*(b*c - a*d)*(e*x)^(11/2))/(5*a*b*e*(a + b*x^2)^(5/4)) + (7*(6*b*c - 11*a*d)*e^3*(e*x)^(3/2))/(30*b^3*(a + b*x^2)^(1/4)) - ((6*b*c - 11*a*d)*e*(e*x)^(7/2))/(15*a*b^2*(a + b*x^2)^(1/4)) + (7*sqrt(a)*(6*b*c - 11*a*d)*e^4*(1 + a/(b*x^2))^(1/4)*sqrt(e*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(10*b^(7/2)*(a + b*x^2)^(1/4))]
    @test_int [(e*x)^(5/2)*(c + d*x^2)/(a + b*x^2)^(9/4), x, 5, (2*(b*c - a*d)*(e*x)^(7/2))/(5*a*b*e*(a + b*x^2)^(5/4)) - ((2*b*c - 7*a*d)*e*(e*x)^(3/2))/(5*a*b^2*(a + b*x^2)^(1/4)) - (3*(2*b*c - 7*a*d)*e^2*(1 + a/(b*x^2))^(1/4)*sqrt(e*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(5*sqrt(a)*b^(5/2)*(a + b*x^2)^(1/4))]
    @test_int [(e*x)^(1/2)*(c + d*x^2)/(a + b*x^2)^(9/4), x, 4, (2*(b*c - a*d)*(e*x)^(3/2))/(5*a*b*e*(a + b*x^2)^(5/4)) - (2*(2*b*c + 3*a*d)*(1 + a/(b*x^2))^(1/4)*sqrt(e*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(5*a^(3/2)*b^(3/2)*(a + b*x^2)^(1/4))]
    @test_int [(c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(9/4)), x, 5, -((2*c)/(a*e*sqrt(e*x)*(a + b*x^2)^(5/4))) - (2*(6*b*c - a*d)*(e*x)^(3/2))/(5*a^2*e^3*(a + b*x^2)^(5/4)) + (4*(6*b*c - a*d)*(1 + a/(b*x^2))^(1/4)*sqrt(e*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(5*a^(5/2)*sqrt(b)*e^2*(a + b*x^2)^(1/4))]
    @test_int [(c + d*x^2)/((e*x)^(7/2)*(a + b*x^2)^(9/4)), x, 6, -((2*c)/(5*a*e*(e*x)^(5/2)*(a + b*x^2)^(5/4))) - (2*(2*b*c - a*d))/(5*a^2*e^3*sqrt(e*x)*(a + b*x^2)^(5/4)) + (12*(2*b*c - a*d))/(5*a^3*e^3*sqrt(e*x)*(a + b*x^2)^(1/4)) - (24*sqrt(b)*(2*b*c - a*d)*(1 + a/(b*x^2))^(1/4)*sqrt(e*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(5*a^(7/2)*e^4*(a + b*x^2)^(1/4))]
    @test_int [(c + d*x^2)/((e*x)^(11/2)*(a + b*x^2)^(9/4)), x, 7, -((2*c)/(9*a*e*(e*x)^(9/2)*(a + b*x^2)^(5/4))) - (2*(14*b*c - 9*a*d))/(45*a^2*e^3*(e*x)^(5/2)*(a + b*x^2)^(5/4)) + (4*(14*b*c - 9*a*d))/(45*a^3*e^3*(e*x)^(5/2)*(a + b*x^2)^(1/4)) - (8*b*(14*b*c - 9*a*d))/(15*a^4*e^5*sqrt(e*x)*(a + b*x^2)^(1/4)) + (16*b^(3/2)*(14*b*c - 9*a*d)*(1 + a/(b*x^2))^(1/4)*sqrt(e*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(15*a^(9/2)*e^6*(a + b*x^2)^(1/4))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(e*x)^m*(a+b*x^2)^p*(c+d*x^2)^q*with*p*and/or*q*symbolic=#

    @test_int [(a + b*x^2)^p*(c + d*x^2)^q*(e*x)^m, x, 3, ((e*x)^(1 + m)*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1((1 + m)/2, -p, -q, (3 + m)/2, -((b*x^2)/a), -((d*x^2)/c)))/((1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q*(e*(1 + m)))]

    @test_int [(a + b*x^2)^p*(c + d*x^2)^q*x^4, x, 3, ((1/5)*x^5*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1(5/2, -p, -q, 7/2, -((b*x^2)/a), -((d*x^2)/c)))/((1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q)]
    @test_int [(a + b*x^2)^p*(c + d*x^2)^q*x^2, x, 3, ((1/3)*x^3*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1(3/2, -p, -q, 5/2, -((b*x^2)/a), -((d*x^2)/c)))/((1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q)]
    @test_int [(a + b*x^2)^p*(c + d*x^2)^q*x^0, x, 3, (x*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1(1/2, -p, -q, 3/2, -((b*x^2)/a), -((d*x^2)/c)))/((1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q)]
    @test_int [(a + b*x^2)^p*(c + d*x^2)^q/x^2, x, 3, -(((a + b*x^2)^p*(c + d*x^2)^q*AppellF1(-(1/2), -p, -q, 1/2, -((b*x^2)/a), -((d*x^2)/c)))/((1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q*x))]
    @test_int [(a + b*x^2)^p*(c + d*x^2)^q/x^4, x, 3, -(((a + b*x^2)^p*(c + d*x^2)^q*AppellF1(-(3/2), -p, -q, -(1/2), -((b*x^2)/a), -((d*x^2)/c)))/((1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q*(3*x^3)))]

    @test_int [(a + b*x^2)^p*(c + d*x^2)^q*x^5, x, 5, -(((b*c*(2 + p) + a*d*(2 + q))*(a + b*x^2)^(1 + p)*(c + d*x^2)^(1 + q))/(2*b^2*d^2*(2 + p + q)*(3 + p + q))) + (x^2*(a + b*x^2)^(1 + p)*(c + d*x^2)^(1 + q))/(2*b*d*(3 + p + q)) + ((b^2*c^2*(2 + 3*p + p^2) + 2*a*b*c*d*(1 + p)*(1 + q) + a^2*d^2*(2 + 3*q + q^2))*(a + b*x^2)^(1 + p)*(c + d*x^2)^q*HypergeometricFunctions._₂F₁(1 + p, -q, 2 + p, -((d*(a + b*x^2))/(b*c - a*d))))/(((b*(c + d*x^2))/(b*c - a*d))^q*(2*b^3*d^2*(1 + p)*(2 + p + q)*(3 + p + q)))]
    @test_int [(a + b*x^2)^p*(c + d*x^2)^q*x^3, x, 4, ((a + b*x^2)^(1 + p)*(c + d*x^2)^(1 + q))/(2*b*d*(2 + p + q)) - ((b*c*(1 + p) + a*d*(1 + q))*(a + b*x^2)^(1 + p)*(c + d*x^2)^q*HypergeometricFunctions._₂F₁(1 + p, -q, 2 + p, -((d*(a + b*x^2))/(b*c - a*d))))/(((b*(c + d*x^2))/(b*c - a*d))^q*(2*b^2*d*(1 + p)*(2 + p + q)))]
    @test_int [(a + b*x^2)^p*(c + d*x^2)^q*x^1, x, 3, ((a + b*x^2)^(1 + p)*(c + d*x^2)^q*HypergeometricFunctions._₂F₁(1 + p, -q, 2 + p, -((d*(a + b*x^2))/(b*c - a*d))))/(((b*(c + d*x^2))/(b*c - a*d))^q*(2*b*(1 + p)))]
    @test_int [(a + b*x^2)^p*(c + d*x^2)^q/x^1, x, 3, -(((a + b*x^2)^(1 + p)*(c + d*x^2)^q*AppellF1(1 + p, -q, 1, 2 + p, -((d*(a + b*x^2))/(b*c - a*d)), (a + b*x^2)/a))/(((b*(c + d*x^2))/(b*c - a*d))^q*(2*a*(1 + p))))]
    @test_int [(a + b*x^2)^p*(c + d*x^2)^q/x^3, x, 3, (b*(a + b*x^2)^(1 + p)*(c + d*x^2)^q*AppellF1(1 + p, -q, 2, 2 + p, -((d*(a + b*x^2))/(b*c - a*d)), (a + b*x^2)/a))/(((b*(c + d*x^2))/(b*c - a*d))^q*(2*a^2*(1 + p)))]
    @test_int [(a + b*x^2)^p*(c + d*x^2)^q/x^5, x, 3, -((b^2*(a + b*x^2)^(1 + p)*(c + d*x^2)^q*AppellF1(1 + p, -q, 3, 2 + p, -((d*(a + b*x^2))/(b*c - a*d)), (a + b*x^2)/a))/(((b*(c + d*x^2))/(b*c - a*d))^q*(2*a^3*(1 + p))))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(e*x)^(m/2)*(a+b*x^2)^p*(c+d*x^2)^q*with*p*and/or*q*symbolic=#

    @test_int [(a + b*x^2)^p*(c + d*x^2)^q*(e*x)^(5/2), x, 3, (2*(e*x)^(7/2)*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1(7/4, -p, -q, 11/4, -((b*x^2)/a), -((d*x^2)/c)))/((1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q*(7*e))]
    @test_int [(a + b*x^2)^p*(c + d*x^2)^q*(e*x)^(3/2), x, 3, (2*(e*x)^(5/2)*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1(5/4, -p, -q, 9/4, -((b*x^2)/a), -((d*x^2)/c)))/((1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q*(5*e))]
    @test_int [(a + b*x^2)^p*(c + d*x^2)^q*(e*x)^(1/2), x, 3, (2*(e*x)^(3/2)*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1(3/4, -p, -q, 7/4, -((b*x^2)/a), -((d*x^2)/c)))/((1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q*(3*e))]
    @test_int [(a + b*x^2)^p*(c + d*x^2)^q/(e*x)^(1/2), x, 3, (2*sqrt(e*x)*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1(1/4, -p, -q, 5/4, -((b*x^2)/a), -((d*x^2)/c)))/((1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q*e)]
    @test_int [(a + b*x^2)^p*(c + d*x^2)^q/(e*x)^(3/2), x, 3, -((2*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1(-(1/4), -p, -q, 3/4, -((b*x^2)/a), -((d*x^2)/c)))/((1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q*(e*sqrt(e*x))))]
    @test_int [(a + b*x^2)^p*(c + d*x^2)^q/(e*x)^(5/2), x, 3, -((2*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1(-(3/4), -p, -q, 1/4, -((b*x^2)/a), -((d*x^2)/c)))/((1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q*(3*e*(e*x)^(3/2))))]
    end

max_stars_repo_path: test/1 Algebraic functions/1.1 Binomial products/1.1.2 Quadratic/1.1.2.4 (e x)^m (a+b x^2)^p (c+d x^2)^q.jl