int_rules_1_2_1_3 = @theory begin

    #= ::Subsection::Closed:: =#
    #= 1.2.1.3*(d+e*x)^m*(f+g*x)*(a+b*x+c*x^2)^p =#
    @apply_utils Antiderivative((~(e') * ~x) ^ ~(m') * (~f + ~(g') * ~x) * (~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => (~g * (~e * ~x) ^ ~m * (~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1)) / (~c * (~m + 2 * ~p + 2)) <-- FreeQ([~b, ~c, ~e, ~f, ~g, ~m, ~p], ~x) && (EqQ(~b * ~g * (~m + ~p + 1) - ~c * ~f * (~m + 2 * ~p + 2), 0) && NeQ(~m + 2 * ~p + 2, 0))
    @apply_utils Antiderivative((~x) ^ ~(m') * (~f + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => ~f * Antiderivative((~x) ^ ~m * (~a + ~c * (~x) ^ 2) ^ ~p, ~x) + ~g * Antiderivative((~x) ^ (~m + 1) * (~a + ~c * (~x) ^ 2) ^ ~p, ~x) <-- FreeQ([~a, ~c, ~f, ~g, ~p], ~x) && (IntegerQ(~m) && Not(IntegerQ(2 * ~p)))
    @apply_utils Antiderivative((~(e') * ~x) ^ ~(m') * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => Antiderivative(ExpandIntegrand((~e * ~x) ^ ~m * (~f + ~g * ~x) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ ~p, ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~e, ~f, ~g, ~m], ~x) && (IntegerQ(~p) && (GtQ(~p, 0) || EqQ(~a, 0) && IntegerQ(~m)))
    @apply_utils Antiderivative((~(e') * ~x) ^ ~(m') * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => Antiderivative(ExpandIntegrand((~e * ~x) ^ ~m * (~f + ~g * ~x) * (~a + ~c * (~x) ^ 2) ^ ~p, ~x), ~x) <-- FreeQ([~a, ~c, ~e, ~f, ~g, ~m], ~x) && IGtQ(~p, 0)
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~(m') * (~f + ~(g') * ~x) * (~a + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => (-(~f) * ~g * (~d + ~e * ~x) ^ (~m + 1) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1)) / (~b * (~p + 1) * (~e * ~f - ~d * ~g)) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~m, ~p], ~x) && (EqQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (EqQ(~m + 2 * ~p + 3, 0) && EqQ(2 * ~c * ~f - ~b * ~g, 0)))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~(m') * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~p, ~x) => (~g * (~d + ~e * ~x) ^ ~m * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1)) / (2 * ~c * (~p + 1)) - ((~e * ~g * ~m) / (2 * ~c * (~p + 1))) * Antiderivative((~d + ~e * ~x) ^ (~m - 1) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g], ~x) && (EqQ(2 * ~c * ~f - ~b * ~g, 0) && (LtQ(~p, -1) && GtQ(~m, 0)))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~(m') * (~(f') + ~(g') * ~x) * (~a + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~p, ~x) => (-2 * ~c * (~e * ~f - ~d * ~g) * (~d + ~e * ~x) ^ (~m + 1) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1)) / ((~p + 1) * (2 * ~c * ~d - ~b * ~e) ^ 2) + ((2 * ~c * ~f - ~b * ~g) / (2 * ~c * ~d - ~b * ~e)) * Antiderivative((~d + ~e * ~x) ^ (~m + 1) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ ~p, ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~m, ~p], ~x) && (EqQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (EqQ(~m + 2 * ~p + 3, 0) && (NeQ(2 * ~c * ~f - ~b * ~g, 0) && NeQ(2 * ~c * ~d - ~b * ~e, 0))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~(m') * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~p, ~x) => ((~a + ~b * ~x + ~c * (~x) ^ 2) ^ FracPart(~p) / ((~c) ^ IntPart(~p) * (~b / 2 + ~c * ~x) ^ (2 * FracPart(~p)))) * Antiderivative((~d + ~e * ~x) ^ ~m * (~f + ~g * ~x) * (~b / 2 + ~c * ~x) ^ (2 * ~p), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~m], ~x) && EqQ((~b) ^ 2 - 4 * ~a * ~c, 0)
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~(m') * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => Antiderivative(ExpandIntegrand((~d + ~e * ~x) ^ ~m * (~f + ~g * ~x) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ ~p, ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~m], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (IntegerQ(~p) && (GtQ(~p, 0) || EqQ(~a, 0) && IntegerQ(~m))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~(m') * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => Antiderivative(ExpandIntegrand((~d + ~e * ~x) ^ ~m * (~f + ~g * ~x) * (~a + ~c * (~x) ^ 2) ^ ~p, ~x), ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~m], ~x) && IGtQ(~p, 0)
    @apply_utils Antiderivative(((~(d') + ~(e') * ~x) * (~f + ~(g') * ~x)) / (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2), ~x) => (~e * ~g * ~x) / ~c + (1 / ~c) * Antiderivative(((~c * ~d * ~f - ~a * ~e * ~g) + ((~c * ~e * ~f + ~c * ~d * ~g) - ~b * ~e * ~g) * ~x) / (~a + ~b * ~x + ~c * (~x) ^ 2), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g], ~x) && NeQ((~b) ^ 2 - 4 * ~a * ~c, 0)
    @apply_utils Antiderivative(((~(d') + ~(e') * ~x) * (~f + ~(g') * ~x)) / (~a + ~(c') * (~x) ^ 2), ~x) => (~e * ~g * ~x) / ~c + (1 / ~c) * Antiderivative(((~c * ~d * ~f - ~a * ~e * ~g) + ~c * (~e * ~f + ~d * ~g) * ~x) / (~a + ~c * (~x) ^ 2), ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g], ~x)
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~p, ~x) => (-(((~b * ~e * ~g * (~p + 2) - ~c * (~e * ~f + ~d * ~g) * (2 * ~p + 3)) - 2 * ~c * ~e * ~g * (~p + 1) * ~x)) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1)) / (2 * (~c) ^ 2 * (~p + 1) * (2 * ~p + 3)) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~p], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (EqQ(((~b) ^ 2 * ~e * ~g * (~p + 2) - 2 * ~a * ~c * ~e * ~g) + ~c * (2 * ~c * ~d * ~f - ~b * (~e * ~f + ~d * ~g)) * (2 * ~p + 3), 0) && NeQ(~p, -1)))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~p, ~x) => (((~e * ~f + ~d * ~g) * (2 * ~p + 3) + 2 * ~e * ~g * (~p + 1) * ~x) * (~a + ~c * (~x) ^ 2) ^ (~p + 1)) / (2 * ~c * (~p + 1) * (2 * ~p + 3)) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~p], ~x) && (EqQ(~a * ~e * ~g - ~c * ~d * ~f * (2 * ~p + 3), 0) && NeQ(~p, -1))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~p, ~x) => (-(((2 * ~a * ~c * (~e * ~f + ~d * ~g) - ~b * (~c * ~d * ~f + ~a * ~e * ~g)) - (((~b) ^ 2 * ~e * ~g - ~b * ~c * (~e * ~f + ~d * ~g)) + 2 * ~c * (~c * ~d * ~f - ~a * ~e * ~g)) * ~x)) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1)) / (~c * (~p + 1) * ((~b) ^ 2 - 4 * ~a * ~c)) - ((((~b) ^ 2 * ~e * ~g * (~p + 2) - 2 * ~a * ~c * ~e * ~g) + ~c * (2 * ~c * ~d * ~f - ~b * (~e * ~f + ~d * ~g)) * (2 * ~p + 3)) / (~c * (~p + 1) * ((~b) ^ 2 - 4 * ~a * ~c))) * Antiderivative((~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && LtQ(~p, -1))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~p, ~x) => ((~a * (~e * ~f + ~d * ~g) - (~c * ~d * ~f - ~a * ~e * ~g) * ~x) * (~a + ~c * (~x) ^ 2) ^ (~p + 1)) / (2 * ~a * ~c * (~p + 1)) - ((~a * ~e * ~g - ~c * ~d * ~f * (2 * ~p + 3)) / (2 * ~a * ~c * (~p + 1))) * Antiderivative((~a + ~c * (~x) ^ 2) ^ (~p + 1), ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g], ~x) && LtQ(~p, -1)
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~p, ~x) => (-(((~b * ~e * ~g * (~p + 2) - ~c * (~e * ~f + ~d * ~g) * (2 * ~p + 3)) - 2 * ~c * ~e * ~g * (~p + 1) * ~x)) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1)) / (2 * (~c) ^ 2 * (~p + 1) * (2 * ~p + 3)) + ((((~b) ^ 2 * ~e * ~g * (~p + 2) - 2 * ~a * ~c * ~e * ~g) + ~c * (2 * ~c * ~d * ~f - ~b * (~e * ~f + ~d * ~g)) * (2 * ~p + 3)) / (2 * (~c) ^ 2 * (2 * ~p + 3))) * Antiderivative((~a + ~b * ~x + ~c * (~x) ^ 2) ^ ~p, ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~p], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && Not(LeQ(~p, -1)))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~p, ~x) => (((~e * ~f + ~d * ~g) * (2 * ~p + 3) + 2 * ~e * ~g * (~p + 1) * ~x) * (~a + ~c * (~x) ^ 2) ^ (~p + 1)) / (2 * ~c * (~p + 1) * (2 * ~p + 3)) - ((~a * ~e * ~g - ~c * ~d * ~f * (2 * ~p + 3)) / (~c * (2 * ~p + 3))) * Antiderivative((~a + ~c * (~x) ^ 2) ^ ~p, ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~p], ~x) && Not(LeQ(~p, -1))
    @apply_utils Antiderivative((~(e') * ~x) ^ ~(m') * (~(f') + ~(g') * ~x) * (~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => (1 / (~e) ^ ~p) * Antiderivative((~e * ~x) ^ (~m + ~p) * (~f + ~g * ~x) * (~b + ~c * ~x) ^ ~p, ~x) <-- FreeQ([~b, ~c, ~e, ~f, ~g, ~m], ~x) && IntegerQ(~p)
    @apply_utils Antiderivative((~d + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => Antiderivative((~d + ~e * ~x) ^ (~m + ~p) * (~f + ~g * ~x) * (~a / ~d + (~c / ~e) * ~x) ^ ~p, ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~m], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (EqQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && IntegerQ(~p)))
    @apply_utils Antiderivative((~d + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => Antiderivative((~d + ~e * ~x) ^ (~m + ~p) * (~f + ~g * ~x) * (~a / ~d + (~c / ~e) * ~x) ^ ~p, ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~m], ~x) && (EqQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && (IntegerQ(~p) || GtQ(~a, 0) && (GtQ(~d, 0) && EqQ(~m + ~p, 0))))
    @apply_utils Antiderivative((~d + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~p, ~x) => (~d) ^ ~m * (~e) ^ ~m * Antiderivative(((~f + ~g * ~x) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~m + ~p)) / (~a * ~e + ~c * ~d * ~x) ^ ~m, ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~p], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (EqQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && (Not(IntegerQ(2 * ~p)) && ILtQ(~m, 0))))
    @apply_utils Antiderivative(~x * (~d + ~(e') * ~x) ^ ~m * (~a + ~(c') * (~x) ^ 2) ^ ~p, ~x) => (~d) ^ ~m * (~e) ^ ~m * Antiderivative((~x * (~a + ~c * (~x) ^ 2) ^ (~m + ~p)) / (~a * ~e + ~c * ~d * ~x) ^ ~m, ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~p], ~x) && (EqQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && (Not(IntegerQ(~p)) && (ILtQ(~m, 0) && (EqQ(~m, -1) && Not(ILtQ(~p - 1 / 2, 0))))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~p, ~x) => (~g * (~d + ~e * ~x) ^ ~m * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1)) / (~c * (~m + 2 * ~p + 2)) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~m, ~p], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (EqQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && EqQ(~m * (~g * (~c * ~d - ~b * ~e) + ~c * ~e * ~f) + ~e * (~p + 1) * (2 * ~c * ~f - ~b * ~g), 0)))
    @apply_utils Antiderivative((~d + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~p, ~x) => (~g * (~d + ~e * ~x) ^ ~m * (~a + ~c * (~x) ^ 2) ^ (~p + 1)) / (~c * (~m + 2 * ~p + 2)) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~m, ~p], ~x) && (EqQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && EqQ(~m * (~d * ~g + ~e * ~f) + 2 * ~e * ~f * (~p + 1), 0))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~p, ~x) => ((~g * (~c * ~d - ~b * ~e) + ~c * ~e * ~f) * (~d + ~e * ~x) ^ ~m * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1)) / (~c * (~p + 1) * (2 * ~c * ~d - ~b * ~e)) - ((~e * (~m * (~g * (~c * ~d - ~b * ~e) + ~c * ~e * ~f) + ~e * (~p + 1) * (2 * ~c * ~f - ~b * ~g))) / (~c * (~p + 1) * (2 * ~c * ~d - ~b * ~e))) * Antiderivative((~d + ~e * ~x) ^ (~m - 1) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (EqQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && (LtQ(~p, -1) && GtQ(~m, 0))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~p, ~x) => ((~d * ~g + ~e * ~f) * (~d + ~e * ~x) ^ ~m * (~a + ~c * (~x) ^ 2) ^ (~p + 1)) / (2 * ~c * ~d * (~p + 1)) - ((~e * (~m * (~d * ~g + ~e * ~f) + 2 * ~e * ~f * (~p + 1))) / (2 * ~c * ~d * (~p + 1))) * Antiderivative((~d + ~e * ~x) ^ (~m - 1) * (~a + ~c * (~x) ^ 2) ^ (~p + 1), ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g], ~x) && (EqQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && (LtQ(~p, -1) && GtQ(~m, 0)))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~p, ~x) => ((~g * (~c * ~d - ~b * ~e) + ~c * ~e * ~f) * (~d + ~e * ~x) ^ ~m * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1)) / (~c * (~p + 1) * (2 * ~c * ~d - ~b * ~e)) - ((~e * (~m * (~g * (~c * ~d - ~b * ~e) + ~c * ~e * ~f) + ~e * (~p + 1) * (2 * ~c * ~f - ~b * ~g))) / (~c * (~p + 1) * (2 * ~c * ~d - ~b * ~e))) * Antiderivative((~d + ~e * ~x) ^ Simplify(~m - 1) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ Simplify(~p + 1), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~m, ~p], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (EqQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && (SumSimplerQ(~p, 1) && (SumSimplerQ(~m, -1) && NeQ(~p, -1)))))
    @apply_utils Antiderivative((~d + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~p, ~x) => ((~d * ~g + ~e * ~f) * (~d + ~e * ~x) ^ ~m * (~a + ~c * (~x) ^ 2) ^ (~p + 1)) / (2 * ~c * ~d * (~p + 1)) - ((~e * (~m * (~d * ~g + ~e * ~f) + 2 * ~e * ~f * (~p + 1))) / (2 * ~c * ~d * (~p + 1))) * Antiderivative((~d + ~e * ~x) ^ Simplify(~m - 1) * (~a + ~c * (~x) ^ 2) ^ Simplify(~p + 1), ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~m, ~p], ~x) && (EqQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && (SumSimplerQ(~p, 1) && (SumSimplerQ(~m, -1) && (NeQ(~p, -1) && Not(IGtQ(~m, 0))))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~p, ~x) => ((~d * ~g - ~e * ~f) * (~d + ~e * ~x) ^ ~m * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1)) / ((2 * ~c * ~d - ~b * ~e) * (~m + ~p + 1)) + ((~m * (~g * (~c * ~d - ~b * ~e) + ~c * ~e * ~f) + ~e * (~p + 1) * (2 * ~c * ~f - ~b * ~g)) / (~e * (2 * ~c * ~d - ~b * ~e) * (~m + ~p + 1))) * Antiderivative((~d + ~e * ~x) ^ (~m + 1) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ ~p, ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~m, ~p], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (EqQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && ((LtQ(~m, -1) && Not(IGtQ(~m + ~p + 1, 0)) || (LtQ(~m, 0) && LtQ(~p, -1) || EqQ(~m + 2 * ~p + 2, 0))) && NeQ(~m + ~p + 1, 0))))
    @apply_utils Antiderivative((~d + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~p, ~x) => ((~d * ~g - ~e * ~f) * (~d + ~e * ~x) ^ ~m * (~a + ~c * (~x) ^ 2) ^ (~p + 1)) / (2 * ~c * ~d * (~m + ~p + 1)) + ((~m * (~g * ~c * ~d + ~c * ~e * ~f) + 2 * ~e * ~c * ~f * (~p + 1)) / (~e * (2 * ~c * ~d) * (~m + ~p + 1))) * Antiderivative((~d + ~e * ~x) ^ (~m + 1) * (~a + ~c * (~x) ^ 2) ^ ~p, ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~m, ~p], ~x) && (EqQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && ((LtQ(~m, -1) && Not(IGtQ(~m + ~p + 1, 0)) || (LtQ(~m, 0) && LtQ(~p, -1) || EqQ(~m + 2 * ~p + 2, 0))) && NeQ(~m + ~p + 1, 0)))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~p, ~x) => (~g * (~d + ~e * ~x) ^ ~m * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1)) / (~c * (~m + 2 * ~p + 2)) + ((~m * (~g * (~c * ~d - ~b * ~e) + ~c * ~e * ~f) + ~e * (~p + 1) * (2 * ~c * ~f - ~b * ~g)) / (~c * ~e * (~m + 2 * ~p + 2))) * Antiderivative((~d + ~e * ~x) ^ ~m * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ ~p, ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~m, ~p], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (EqQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && (NeQ(~m + 2 * ~p + 2, 0) && (NeQ(~m, 2) || EqQ(~d, 0)))))
    @apply_utils Antiderivative((~d + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~p, ~x) => (~g * (~d + ~e * ~x) ^ ~m * (~a + ~c * (~x) ^ 2) ^ (~p + 1)) / (~c * (~m + 2 * ~p + 2)) + ((~m * (~d * ~g + ~e * ~f) + 2 * ~e * ~f * (~p + 1)) / (~e * (~m + 2 * ~p + 2))) * Antiderivative((~d + ~e * ~x) ^ ~m * (~a + ~c * (~x) ^ 2) ^ ~p, ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~m, ~p], ~x) && (EqQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && (NeQ(~m + 2 * ~p + 2, 0) && NeQ(~m, 2)))
    @apply_utils Antiderivative((~x) ^ 2 * (~f + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~p, ~x) => ((~x) ^ 2 * (~a * ~g - ~c * ~f * ~x) * (~a + ~c * (~x) ^ 2) ^ (~p + 1)) / (2 * ~a * ~c * (~p + 1)) - (1 / (2 * ~a * ~c * (~p + 1))) * Antiderivative(~x * Simp(2 * ~a * ~g - ~c * ~f * (2 * ~p + 5) * ~x, ~x) * (~a + ~c * (~x) ^ 2) ^ (~p + 1), ~x) <-- FreeQ([~a, ~c, ~f, ~g], ~x) && (EqQ(~a * (~g) ^ 2 + (~f) ^ 2 * ~c, 0) && LtQ(~p, -2))
    @apply_utils Antiderivative((~x) ^ 2 * (~f + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~p, ~x) => (1 / ~c) * Antiderivative((~f + ~g * ~x) * (~a + ~c * (~x) ^ 2) ^ (~p + 1), ~x) - (~a / ~c) * Antiderivative((~f + ~g * ~x) * (~a + ~c * (~x) ^ 2) ^ ~p, ~x) <-- FreeQ([~a, ~c, ~f, ~g, ~p], ~x) && EqQ(~a * (~g) ^ 2 + (~f) ^ 2 * ~c, 0)
    @apply_utils Antiderivative((~d + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => Antiderivative((~d + ~e * ~x) ^ ~m * (~f + ~g * ~x) ^ (~p + 1) * (~a / ~f + (~c / ~g) * ~x) ^ ~p, ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~m], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (EqQ((~c * (~f) ^ 2 - ~b * ~f * ~g) + ~a * (~g) ^ 2, 0) && IntegerQ(~p)))
    @apply_utils Antiderivative((~d + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => Antiderivative((~d + ~e * ~x) ^ ~m * (~f + ~g * ~x) ^ (~p + 1) * (~a / ~f + (~c / ~g) * ~x) ^ ~p, ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~m], ~x) && (EqQ(~c * (~f) ^ 2 + ~a * (~g) ^ 2, 0) && (IntegerQ(~p) || GtQ(~a, 0) && (GtQ(~f, 0) && EqQ(~p, -1))))
    @apply_utils Antiderivative(((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x)) / (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2), ~x) => Antiderivative(ExpandIntegrand(((~d + ~e * ~x) ^ ~m * (~f + ~g * ~x)) / (~a + ~b * ~x + ~c * (~x) ^ 2), ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (NeQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && IntegerQ(~m)))
    @apply_utils Antiderivative(((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x)) / (~a + ~(c') * (~x) ^ 2), ~x) => Antiderivative(ExpandIntegrand(((~d + ~e * ~x) ^ ~m * (~f + ~g * ~x)) / (~a + ~c * (~x) ^ 2), ~x), ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g], ~x) && (NeQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && IntegerQ(~m))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => (-((~e * ~f - ~d * ~g)) * (~d + ~e * ~x) ^ (~m + 1) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1)) / (2 * (~p + 1) * ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2)) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~m, ~p], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (NeQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && (EqQ(Simplify(~m + 2 * ~p + 3), 0) && EqQ(~b * (~e * ~f + ~d * ~g) - 2 * (~c * ~d * ~f + ~a * ~e * ~g), 0))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => (-((~e * ~f - ~d * ~g)) * (~d + ~e * ~x) ^ (~m + 1) * (~a + ~c * (~x) ^ 2) ^ (~p + 1)) / (2 * (~p + 1) * (~c * (~d) ^ 2 + ~a * (~e) ^ 2)) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~m, ~p], ~x) && (NeQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && (EqQ(Simplify(~m + 2 * ~p + 3), 0) && EqQ(~c * ~d * ~f + ~a * ~e * ~g, 0)))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~p, ~x) => ((~d + ~e * ~x) ^ ~m * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1) * ((~b * ~f - 2 * ~a * ~g) + (2 * ~c * ~f - ~b * ~g) * ~x)) / ((~p + 1) * ((~b) ^ 2 - 4 * ~a * ~c)) - ((~m * (~b * (~e * ~f + ~d * ~g) - 2 * (~c * ~d * ~f + ~a * ~e * ~g))) / ((~p + 1) * ((~b) ^ 2 - 4 * ~a * ~c))) * Antiderivative((~d + ~e * ~x) ^ (~m - 1) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (NeQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && (EqQ(Simplify(~m + 2 * ~p + 3), 0) && LtQ(~p, -1))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~p, ~x) => ((~d + ~e * ~x) ^ ~m * (~a + ~c * (~x) ^ 2) ^ (~p + 1) * (~a * ~g - ~c * ~f * ~x)) / (2 * ~a * ~c * (~p + 1)) - ((~m * (~c * ~d * ~f + ~a * ~e * ~g)) / (2 * ~a * ~c * (~p + 1))) * Antiderivative((~d + ~e * ~x) ^ (~m - 1) * (~a + ~c * (~x) ^ 2) ^ (~p + 1), ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g], ~x) && (NeQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && (EqQ(Simplify(~m + 2 * ~p + 3), 0) && LtQ(~p, -1)))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => (-((~e * ~f - ~d * ~g)) * (~d + ~e * ~x) ^ (~m + 1) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1)) / (2 * (~p + 1) * ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2)) - ((~b * (~e * ~f + ~d * ~g) - 2 * (~c * ~d * ~f + ~a * ~e * ~g)) / (2 * ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2))) * Antiderivative((~d + ~e * ~x) ^ (~m + 1) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ ~p, ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~m, ~p], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (NeQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && EqQ(Simplify(~m + 2 * ~p + 3), 0)))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => (-((~e * ~f - ~d * ~g)) * (~d + ~e * ~x) ^ (~m + 1) * (~a + ~c * (~x) ^ 2) ^ (~p + 1)) / (2 * (~p + 1) * (~c * (~d) ^ 2 + ~a * (~e) ^ 2)) + ((~c * ~d * ~f + ~a * ~e * ~g) / (~c * (~d) ^ 2 + ~a * (~e) ^ 2)) * Antiderivative((~d + ~e * ~x) ^ (~m + 1) * (~a + ~c * (~x) ^ 2) ^ ~p, ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~m, ~p], ~x) && (NeQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && EqQ(Simplify(~m + 2 * ~p + 3), 0))
    @apply_utils Antiderivative((~(e') * ~x) ^ ~m * (~f + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~p, ~x) => ~f * Antiderivative((~e * ~x) ^ ~m * (~a + ~c * (~x) ^ 2) ^ ~p, ~x) + (~g / ~e) * Antiderivative((~e * ~x) ^ (~m + 1) * (~a + ~c * (~x) ^ 2) ^ ~p, ~x) <-- FreeQ([~a, ~c, ~e, ~f, ~g, ~p], ~x) && (Not(RationalQ(~m)) && Not(IGtQ(~p, 0)))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~a + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~p, ~x) => (((~d + ~e * ~x) ^ FracPart(~p) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ FracPart(~p)) / (~a * ~d + ~c * ~e * (~x) ^ 3) ^ FracPart(~p)) * Antiderivative((~f + ~g * ~x) * (~a * ~d + ~c * ~e * (~x) ^ 3) ^ ~p, ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~m, ~p], ~x) && (EqQ(~m, ~p) && (EqQ(~b * ~d + ~a * ~e, 0) && EqQ(~c * ~d + ~b * ~e, 0)))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => ((-((~d + ~e * ~x) ^ (~m + 1)) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ ~p) / ((~e) ^ 2 * (~m + 1) * (~m + 2) * ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2))) * (((~d * ~g - ~e * ~f * (~m + 2)) * ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2) - ~d * ~p * (2 * ~c * ~d - ~b * ~e) * (~e * ~f - ~d * ~g)) - ~e * (~g * (~m + 1) * ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2) + ~p * (2 * ~c * ~d - ~b * ~e) * (~e * ~f - ~d * ~g)) * ~x) - (~p / ((~e) ^ 2 * (~m + 1) * (~m + 2) * ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2))) * Antiderivative((~d + ~e * ~x) ^ (~m + 2) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p - 1) * Simp((2 * ~a * ~c * ~e * (~e * ~f - ~d * ~g) * (~m + 2) + (~b) ^ 2 * ~e * (~d * ~g * (~p + 1) - ~e * ~f * (~m + ~p + 2)) + ~b * (~a * (~e) ^ 2 * ~g * (~m + 1) - ~c * ~d * (~d * ~g * (2 * ~p + 1) - ~e * ~f * (~m + 2 * ~p + 2)))) - ~c * (2 * ~c * ~d * (~d * ~g * (2 * ~p + 1) - ~e * ~f * (~m + 2 * ~p + 2)) - ~e * (2 * ~a * ~e * ~g * (~m + 1) - ~b * (~d * ~g * (~m - 2 * ~p) + ~e * ~f * (~m + 2 * ~p + 2)))) * ~x, ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (NeQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && (GtQ(~p, 0) && (LtQ(~m, -2) && (LtQ(~m + 2 * ~p, 0) && Not(ILtQ(~m + 2 * ~p + 3, 0)))))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => ((-((~d + ~e * ~x) ^ (~m + 1)) * (~a + ~c * (~x) ^ 2) ^ ~p) / ((~e) ^ 2 * (~m + 1) * (~m + 2) * (~c * (~d) ^ 2 + ~a * (~e) ^ 2))) * (((~d * ~g - ~e * ~f * (~m + 2)) * (~c * (~d) ^ 2 + ~a * (~e) ^ 2) - 2 * ~c * (~d) ^ 2 * ~p * (~e * ~f - ~d * ~g)) - ~e * (~g * (~m + 1) * (~c * (~d) ^ 2 + ~a * (~e) ^ 2) + 2 * ~c * ~d * ~p * (~e * ~f - ~d * ~g)) * ~x) - (~p / ((~e) ^ 2 * (~m + 1) * (~m + 2) * (~c * (~d) ^ 2 + ~a * (~e) ^ 2))) * Antiderivative((~d + ~e * ~x) ^ (~m + 2) * (~a + ~c * (~x) ^ 2) ^ (~p - 1) * Simp(2 * ~a * ~c * ~e * (~e * ~f - ~d * ~g) * (~m + 2) - ~c * (2 * ~c * ~d * (~d * ~g * (2 * ~p + 1) - ~e * ~f * (~m + 2 * ~p + 2)) - 2 * ~a * (~e) ^ 2 * ~g * (~m + 1)) * ~x, ~x), ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g], ~x) && (NeQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && (GtQ(~p, 0) && (LtQ(~m, -2) && (LtQ(~m + 2 * ~p, 0) && Not(ILtQ(~m + 2 * ~p + 3, 0))))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => ((~d + ~e * ~x) ^ (~m + 1) * ((~e * ~f * (~m + 2 * ~p + 2) - ~d * ~g * (2 * ~p + 1)) + ~e * ~g * (~m + 1) * ~x) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ ~p) / ((~e) ^ 2 * (~m + 1) * (~m + 2 * ~p + 2)) + (~p / ((~e) ^ 2 * (~m + 1) * (~m + 2 * ~p + 2))) * Antiderivative((~d + ~e * ~x) ^ (~m + 1) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p - 1) * Simp((~g * (~b * ~d + 2 * ~a * ~e + 2 * ~a * ~e * ~m + 2 * ~b * ~d * ~p) - ~f * ~b * ~e * (~m + 2 * ~p + 2)) + (~g * (2 * ~c * ~d + ~b * ~e + ~b * ~e * ~m + 4 * ~c * ~d * ~p) - 2 * ~c * ~e * ~f * (~m + 2 * ~p + 2)) * ~x, ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~m], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (NeQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && (RationalQ(~p) && (~p > 0 && ((LtQ(~m, -1) || (EqQ(~p, 1) || IntegerQ(~p) && Not(RationalQ(~m)))) && (NeQ(~m, -1) && (Not(ILtQ(~m + 2 * ~p + 1, 0)) && (IntegerQ(~m) || (IntegerQ(~p) || IntegersQ(2 * ~m, 2 * ~p))))))))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => ((~d + ~e * ~x) ^ (~m + 1) * ((~e * ~f * (~m + 2 * ~p + 2) - ~d * ~g * (2 * ~p + 1)) + ~e * ~g * (~m + 1) * ~x) * (~a + ~c * (~x) ^ 2) ^ ~p) / ((~e) ^ 2 * (~m + 1) * (~m + 2 * ~p + 2)) + (~p / ((~e) ^ 2 * (~m + 1) * (~m + 2 * ~p + 2))) * Antiderivative((~d + ~e * ~x) ^ (~m + 1) * (~a + ~c * (~x) ^ 2) ^ (~p - 1) * Simp(~g * (2 * ~a * ~e + 2 * ~a * ~e * ~m) + (~g * (2 * ~c * ~d + 4 * ~c * ~d * ~p) - 2 * ~c * ~e * ~f * (~m + 2 * ~p + 2)) * ~x, ~x), ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~m], ~x) && (NeQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && (RationalQ(~p) && (~p > 0 && ((LtQ(~m, -1) || (EqQ(~p, 1) || IntegerQ(~p) && Not(RationalQ(~m)))) && (NeQ(~m, -1) && (Not(ILtQ(~m + 2 * ~p + 1, 0)) && (IntegerQ(~m) || (IntegerQ(~p) || IntegersQ(2 * ~m, 2 * ~p)))))))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => ((~d + ~e * ~x) ^ (~m + 1) * ((~c * ~e * ~f * (~m + 2 * ~p + 2) - ~g * ((~c * ~d + 2 * ~c * ~d * ~p) - ~b * ~e * ~p)) + ~g * ~c * ~e * (~m + 2 * ~p + 1) * ~x) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ ~p) / (~c * (~e) ^ 2 * (~m + 2 * ~p + 1) * (~m + 2 * ~p + 2)) - (~p / (~c * (~e) ^ 2 * (~m + 2 * ~p + 1) * (~m + 2 * ~p + 2))) * Antiderivative((~d + ~e * ~x) ^ ~m * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p - 1) * Simp(~c * ~e * ~f * (~b * ~d - 2 * ~a * ~e) * (~m + 2 * ~p + 2) + ~g * (~a * ~e * ((~b * ~e - 2 * ~c * ~d * ~m) + ~b * ~e * ~m) + ~b * ~d * ((~b * ~e * ~p - ~c * ~d) - 2 * ~c * ~d * ~p)) + (~c * ~e * ~f * (2 * ~c * ~d - ~b * ~e) * (~m + 2 * ~p + 2) + ~g * (((~b) ^ 2 * (~e) ^ 2 * (~p + ~m + 1) - 2 * (~c) ^ 2 * (~d) ^ 2 * (1 + 2 * ~p)) - ~c * ~e * (~b * ~d * (~m - 2 * ~p) + 2 * ~a * ~e * (~m + 2 * ~p + 1)))) * ~x, ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~m], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (NeQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && (GtQ(~p, 0) && ((IntegerQ(~p) || (Not(RationalQ(~m)) || GeQ(~m, -1) && LtQ(~m, 0))) && (Not(ILtQ(~m + 2 * ~p, 0)) && (IntegerQ(~m) || (IntegerQ(~p) || IntegersQ(2 * ~m, 2 * ~p))))))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => ((~d + ~e * ~x) ^ (~m + 1) * ((~c * ~e * ~f * (~m + 2 * ~p + 2) - ~g * ~c * ~d * (2 * ~p + 1)) + ~g * ~c * ~e * (~m + 2 * ~p + 1) * ~x) * (~a + ~c * (~x) ^ 2) ^ ~p) / (~c * (~e) ^ 2 * (~m + 2 * ~p + 1) * (~m + 2 * ~p + 2)) + ((2 * ~p) / (~c * (~e) ^ 2 * (~m + 2 * ~p + 1) * (~m + 2 * ~p + 2))) * Antiderivative((~d + ~e * ~x) ^ ~m * (~a + ~c * (~x) ^ 2) ^ (~p - 1) * Simp((~f * ~a * ~c * (~e) ^ 2 * (~m + 2 * ~p + 2) + ~a * ~c * ~d * ~e * ~g * ~m) - ((~c) ^ 2 * ~f * ~d * ~e * (~m + 2 * ~p + 2) - ~g * ((~c) ^ 2 * (~d) ^ 2 * (2 * ~p + 1) + ~a * ~c * (~e) ^ 2 * (~m + 2 * ~p + 1))) * ~x, ~x), ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~m], ~x) && (NeQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && (GtQ(~p, 0) && ((IntegerQ(~p) || (Not(RationalQ(~m)) || GeQ(~m, -1) && LtQ(~m, 0))) && (Not(ILtQ(~m + 2 * ~p, 0)) && (IntegerQ(~m) || (IntegerQ(~p) || IntegersQ(2 * ~m, 2 * ~p)))))))
    @apply_utils Antiderivative((~d + ~(e') * ~x) ^ ~m * (~f + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~p, ~x) => Antiderivative((~a + ~b * ~x + ~c * (~x) ^ 2) ^ ~p * ExpandIntegrand((~d + ~e * ~x) ^ ~m * (~f + ~g * ~x), ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (NeQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && (ILtQ(~p, -1) && (IGtQ(~m, 0) && RationalQ(~a, ~b, ~c, ~d, ~e, ~f, ~g)))))
    @apply_utils Antiderivative((~d + ~(e') * ~x) ^ ~m * (~f + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~p, ~x) => Antiderivative((~a + ~c * (~x) ^ 2) ^ ~p * ExpandIntegrand((~d + ~e * ~x) ^ ~m * (~f + ~g * ~x), ~x), ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g], ~x) && (NeQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && (ILtQ(~p, -1) && (IGtQ(~m, 0) && RationalQ(~a, ~c, ~d, ~e, ~f, ~g))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => (-((~d + ~e * ~x) ^ (~m - 1)) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1) * ((2 * ~a * ~c * (~e * ~f + ~d * ~g) - ~b * (~c * ~d * ~f + ~a * ~e * ~g)) - ((2 * (~c) ^ 2 * ~d * ~f + (~b) ^ 2 * ~e * ~g) - ~c * (~b * ~e * ~f + ~b * ~d * ~g + 2 * ~a * ~e * ~g)) * ~x)) / (~c * (~p + 1) * ((~b) ^ 2 - 4 * ~a * ~c)) - (1 / (~c * (~p + 1) * ((~b) ^ 2 - 4 * ~a * ~c))) * Antiderivative((~d + ~e * ~x) ^ (~m - 2) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1) * Simp(((2 * (~c) ^ 2 * (~d) ^ 2 * ~f * (2 * ~p + 3) + ~b * ~e * ~g * (~a * ~e * (~m - 1) + ~b * ~d * (~p + 2))) - ~c * (2 * ~a * ~e * (~e * ~f * (~m - 1) + ~d * ~g * ~m) + ~b * ~d * (~d * ~g * (2 * ~p + 3) - ~e * ~f * ((~m - 2 * ~p) - 4)))) + ~e * (((~b) ^ 2 * ~e * ~g * (~m + ~p + 1) + 2 * (~c) ^ 2 * ~d * ~f * (~m + 2 * ~p + 2)) - ~c * (2 * ~a * ~e * ~g * ~m + ~b * (~e * ~f + ~d * ~g) * (~m + 2 * ~p + 2))) * ~x, ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (NeQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && (LtQ(~p, -1) && (GtQ(~m, 1) && (EqQ(~m, 2) && (EqQ(~p, -3) && RationalQ(~a, ~b, ~c, ~d, ~e, ~f, ~g)) || Not(ILtQ(~m + 2 * ~p + 3, 0)))))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => ((~d + ~e * ~x) ^ (~m - 1) * (~a + ~c * (~x) ^ 2) ^ (~p + 1) * (~a * (~e * ~f + ~d * ~g) - (~c * ~d * ~f - ~a * ~e * ~g) * ~x)) / (2 * ~a * ~c * (~p + 1)) - (1 / (2 * ~a * ~c * (~p + 1))) * Antiderivative((~d + ~e * ~x) ^ (~m - 2) * (~a + ~c * (~x) ^ 2) ^ (~p + 1) * Simp((~a * ~e * (~e * ~f * (~m - 1) + ~d * ~g * ~m) - ~c * (~d) ^ 2 * ~f * (2 * ~p + 3)) + ~e * (~a * ~e * ~g * ~m - ~c * ~d * ~f * (~m + 2 * ~p + 2)) * ~x, ~x), ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g], ~x) && (NeQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && (LtQ(~p, -1) && (GtQ(~m, 1) && (EqQ(~d, 0) || (EqQ(~m, 2) && (EqQ(~p, -3) && RationalQ(~a, ~c, ~d, ~e, ~f, ~g)) || Not(ILtQ(~m + 2 * ~p + 3, 0)))))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~p, ~x) => ((~d + ~e * ~x) ^ ~m * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1) * ((~f * ~b - 2 * ~a * ~g) + (2 * ~c * ~f - ~b * ~g) * ~x)) / ((~p + 1) * ((~b) ^ 2 - 4 * ~a * ~c)) + (1 / ((~p + 1) * ((~b) ^ 2 - 4 * ~a * ~c))) * Antiderivative((~d + ~e * ~x) ^ (~m - 1) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1) * Simp((~g * (2 * ~a * ~e * ~m + ~b * ~d * (2 * ~p + 3)) - ~f * (~b * ~e * ~m + 2 * ~c * ~d * (2 * ~p + 3))) - ~e * (2 * ~c * ~f - ~b * ~g) * (~m + 2 * ~p + 3) * ~x, ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (NeQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && (LtQ(~p, -1) && (GtQ(~m, 0) && (IntegerQ(~m) || (IntegerQ(~p) || IntegersQ(2 * ~m, 2 * ~p)))))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~p, ~x) => ((~d + ~e * ~x) ^ ~m * (~a + ~c * (~x) ^ 2) ^ (~p + 1) * (~a * ~g - ~c * ~f * ~x)) / (2 * ~a * ~c * (~p + 1)) - (1 / (2 * ~a * ~c * (~p + 1))) * Antiderivative((~d + ~e * ~x) ^ (~m - 1) * (~a + ~c * (~x) ^ 2) ^ (~p + 1) * Simp((~a * ~e * ~g * ~m - ~c * ~d * ~f * (2 * ~p + 3)) - ~c * ~e * ~f * (~m + 2 * ~p + 3) * ~x, ~x), ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g], ~x) && (NeQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && (LtQ(~p, -1) && (GtQ(~m, 0) && (IntegerQ(~m) || (IntegerQ(~p) || IntegersQ(2 * ~m, 2 * ~p))))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~p, ~x) => ((~d + ~e * ~x) ^ (~m + 1) * ((~f * ((~b * ~c * ~d - (~b) ^ 2 * ~e) + 2 * ~a * ~c * ~e) - ~a * ~g * (2 * ~c * ~d - ~b * ~e)) + ~c * (~f * (2 * ~c * ~d - ~b * ~e) - ~g * (~b * ~d - 2 * ~a * ~e)) * ~x) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1)) / ((~p + 1) * ((~b) ^ 2 - 4 * ~a * ~c) * ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2)) + (1 / ((~p + 1) * ((~b) ^ 2 - 4 * ~a * ~c) * ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2))) * Antiderivative((~d + ~e * ~x) ^ ~m * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1) * Simp((~f * (((~b * ~c * ~d * ~e * ((2 * ~p - ~m) + 2) + (~b) ^ 2 * (~e) ^ 2 * (~p + ~m + 2)) - 2 * (~c) ^ 2 * (~d) ^ 2 * (2 * ~p + 3)) - 2 * ~a * ~c * (~e) ^ 2 * (~m + 2 * ~p + 3)) - ~g * (~a * ~e * ((~b * ~e - 2 * ~c * ~d * ~m) + ~b * ~e * ~m) - ~b * ~d * (((3 * ~c * ~d - ~b * ~e) + 2 * ~c * ~d * ~p) - ~b * ~e * ~p))) + ~c * ~e * (~g * (~b * ~d - 2 * ~a * ~e) - ~f * (2 * ~c * ~d - ~b * ~e)) * (~m + 2 * ~p + 4) * ~x, ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~m], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (NeQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && (LtQ(~p, -1) && (IntegerQ(~m) || (IntegerQ(~p) || IntegersQ(2 * ~m, 2 * ~p))))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~p, ~x) => (-((~d + ~e * ~x) ^ (~m + 1)) * ((~f * ~a * ~c * ~e - ~a * ~g * ~c * ~d) + ~c * (~c * ~d * ~f + ~a * ~e * ~g) * ~x) * (~a + ~c * (~x) ^ 2) ^ (~p + 1)) / (2 * ~a * ~c * (~p + 1) * (~c * (~d) ^ 2 + ~a * (~e) ^ 2)) + (1 / (2 * ~a * ~c * (~p + 1) * (~c * (~d) ^ 2 + ~a * (~e) ^ 2))) * Antiderivative((~d + ~e * ~x) ^ ~m * (~a + ~c * (~x) ^ 2) ^ (~p + 1) * Simp((~f * ((~c) ^ 2 * (~d) ^ 2 * (2 * ~p + 3) + ~a * ~c * (~e) ^ 2 * (~m + 2 * ~p + 3)) - ~a * ~c * ~d * ~e * ~g * ~m) + ~c * ~e * (~c * ~d * ~f + ~a * ~e * ~g) * (~m + 2 * ~p + 4) * ~x, ~x), ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g], ~x) && (NeQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && (LtQ(~p, -1) && (IntegerQ(~m) || (IntegerQ(~p) || IntegersQ(2 * ~m, 2 * ~p)))))
    @apply_utils Antiderivative(((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x)) / (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2), ~x) => (~g * (~d + ~e * ~x) ^ ~m) / (~c * ~m) + (1 / ~c) * Antiderivative(((~d + ~e * ~x) ^ (~m - 1) * Simp((~c * ~d * ~f - ~a * ~e * ~g) + ((~g * ~c * ~d - ~b * ~e * ~g) + ~c * ~e * ~f) * ~x, ~x)) / (~a + ~b * ~x + ~c * (~x) ^ 2), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (NeQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && (FractionQ(~m) && GtQ(~m, 0))))
    @apply_utils Antiderivative(((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x)) / (~a + ~(c') * (~x) ^ 2), ~x) => (~g * (~d + ~e * ~x) ^ ~m) / (~c * ~m) + (1 / ~c) * Antiderivative(((~d + ~e * ~x) ^ (~m - 1) * Simp((~c * ~d * ~f - ~a * ~e * ~g) + (~g * ~c * ~d + ~c * ~e * ~f) * ~x, ~x)) / (~a + ~c * (~x) ^ 2), ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g], ~x) && (NeQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && (FractionQ(~m) && GtQ(~m, 0)))
    @apply_utils Antiderivative((~(f') + ~(g') * ~x) / (sqrt(~(d') + ~(e') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2)), ~x) => 2 * Subst(Antiderivative(((~e * ~f - ~d * ~g) + ~g * (~x) ^ 2) / ((((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2) - (2 * ~c * ~d - ~b * ~e) * (~x) ^ 2) + ~c * (~x) ^ 4), ~x), ~x, sqrt(~d + ~e * ~x)) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && NeQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0))
    @apply_utils Antiderivative((~(f') + ~(g') * ~x) / (sqrt(~(d') + ~(e') * ~x) * (~a + ~(c') * (~x) ^ 2)), ~x) => 2 * Subst(Antiderivative(((~e * ~f - ~d * ~g) + ~g * (~x) ^ 2) / (((~c * (~d) ^ 2 + ~a * (~e) ^ 2) - 2 * ~c * ~d * (~x) ^ 2) + ~c * (~x) ^ 4), ~x), ~x, sqrt(~d + ~e * ~x)) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g], ~x) && NeQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0)
    @apply_utils Antiderivative(((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x)) / (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2), ~x) => ((~e * ~f - ~d * ~g) * (~d + ~e * ~x) ^ (~m + 1)) / ((~m + 1) * ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2)) + (1 / ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2)) * Antiderivative(((~d + ~e * ~x) ^ (~m + 1) * Simp(((~c * ~d * ~f - ~f * ~b * ~e) + ~a * ~e * ~g) - ~c * (~e * ~f - ~d * ~g) * ~x, ~x)) / (~a + ~b * ~x + ~c * (~x) ^ 2), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~m], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (NeQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && (FractionQ(~m) && LtQ(~m, -1))))
    @apply_utils Antiderivative(((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x)) / (~a + ~(c') * (~x) ^ 2), ~x) => ((~e * ~f - ~d * ~g) * (~d + ~e * ~x) ^ (~m + 1)) / ((~m + 1) * (~c * (~d) ^ 2 + ~a * (~e) ^ 2)) + (1 / (~c * (~d) ^ 2 + ~a * (~e) ^ 2)) * Antiderivative(((~d + ~e * ~x) ^ (~m + 1) * Simp((~c * ~d * ~f + ~a * ~e * ~g) - ~c * (~e * ~f - ~d * ~g) * ~x, ~x)) / (~a + ~c * (~x) ^ 2), ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~m], ~x) && (NeQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && (FractionQ(~m) && LtQ(~m, -1)))
    @apply_utils Antiderivative(((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x)) / (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2), ~x) => Antiderivative(ExpandIntegrand((~d + ~e * ~x) ^ ~m, (~f + ~g * ~x) / (~a + ~b * ~x + ~c * (~x) ^ 2), ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (NeQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && Not(RationalQ(~m))))
    @apply_utils Antiderivative(((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x)) / (~a + ~(c') * (~x) ^ 2), ~x) => Antiderivative(ExpandIntegrand((~d + ~e * ~x) ^ ~m, (~f + ~g * ~x) / (~a + ~c * (~x) ^ 2), ~x), ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g], ~x) && (NeQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && Not(RationalQ(~m)))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => (~g * (~d + ~e * ~x) ^ ~m * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1)) / (~c * (~m + 2 * ~p + 2)) + (1 / (~c * (~m + 2 * ~p + 2))) * Antiderivative((~d + ~e * ~x) ^ (~m - 1) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ ~p * Simp(~m * (~c * ~d * ~f - ~a * ~e * ~g) + ~d * (2 * ~c * ~f - ~b * ~g) * (~p + 1) + (~m * ((~c * ~e * ~f + ~c * ~d * ~g) - ~b * ~e * ~g) + ~e * (~p + 1) * (2 * ~c * ~f - ~b * ~g)) * ~x, ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~p], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (NeQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && (GtQ(~m, 0) && (NeQ(~m + 2 * ~p + 2, 0) && ((IntegerQ(~m) || (IntegerQ(~p) || IntegersQ(2 * ~m, 2 * ~p))) && Not(IGtQ(~m, 0) && EqQ(~f, 0)))))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => (~g * (~d + ~e * ~x) ^ ~m * (~a + ~c * (~x) ^ 2) ^ (~p + 1)) / (~c * (~m + 2 * ~p + 2)) + (1 / (~c * (~m + 2 * ~p + 2))) * Antiderivative((~d + ~e * ~x) ^ (~m - 1) * (~a + ~c * (~x) ^ 2) ^ ~p * Simp((~c * ~d * ~f * (~m + 2 * ~p + 2) - ~a * ~e * ~g * ~m) + ~c * (~e * ~f * (~m + 2 * ~p + 2) + ~d * ~g * ~m) * ~x, ~x), ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~p], ~x) && (NeQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && (GtQ(~m, 0) && (NeQ(~m + 2 * ~p + 2, 0) && ((IntegerQ(~m) || (IntegerQ(~p) || IntegersQ(2 * ~m, 2 * ~p))) && Not(IGtQ(~m, 0) && EqQ(~f, 0))))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => ((~e * ~f - ~d * ~g) * (~d + ~e * ~x) ^ (~m + 1) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1)) / ((~m + 1) * ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2)) + (1 / ((~m + 1) * ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2))) * Antiderivative((~d + ~e * ~x) ^ (~m + 1) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ ~p * Simp((((~c * ~d * ~f - ~f * ~b * ~e) + ~a * ~e * ~g) * (~m + 1) + ~b * (~d * ~g - ~e * ~f) * (~p + 1)) - ~c * (~e * ~f - ~d * ~g) * (~m + 2 * ~p + 3) * ~x, ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~p], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (NeQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && (LtQ(~m, -1) && (IntegerQ(~m) || (IntegerQ(~p) || IntegersQ(2 * ~m, 2 * ~p))))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => ((~e * ~f - ~d * ~g) * (~d + ~e * ~x) ^ (~m + 1) * (~a + ~c * (~x) ^ 2) ^ (~p + 1)) / ((~m + 1) * (~c * (~d) ^ 2 + ~a * (~e) ^ 2)) + (1 / ((~m + 1) * (~c * (~d) ^ 2 + ~a * (~e) ^ 2))) * Antiderivative((~d + ~e * ~x) ^ (~m + 1) * (~a + ~c * (~x) ^ 2) ^ ~p * Simp((~c * ~d * ~f + ~a * ~e * ~g) * (~m + 1) - ~c * (~e * ~f - ~d * ~g) * (~m + 2 * ~p + 3) * ~x, ~x), ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~p], ~x) && (NeQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && (LtQ(~m, -1) && (IntegerQ(~m) || (IntegerQ(~p) || IntegersQ(2 * ~m, 2 * ~p)))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => ((~e * ~f - ~d * ~g) * (~d + ~e * ~x) ^ (~m + 1) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ (~p + 1)) / ((~m + 1) * ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2)) + (1 / ((~m + 1) * ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2))) * Antiderivative((~d + ~e * ~x) ^ (~m + 1) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ ~p * Simp((((~c * ~d * ~f - ~f * ~b * ~e) + ~a * ~e * ~g) * (~m + 1) + ~b * (~d * ~g - ~e * ~f) * (~p + 1)) - ~c * (~e * ~f - ~d * ~g) * (~m + 2 * ~p + 3) * ~x, ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~m, ~p], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (NeQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && (ILtQ(Simplify(~m + 2 * ~p + 3), 0) && NeQ(~m, -1))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => ((~e * ~f - ~d * ~g) * (~d + ~e * ~x) ^ (~m + 1) * (~a + ~c * (~x) ^ 2) ^ (~p + 1)) / ((~m + 1) * (~c * (~d) ^ 2 + ~a * (~e) ^ 2)) + (1 / ((~m + 1) * (~c * (~d) ^ 2 + ~a * (~e) ^ 2))) * Antiderivative((~d + ~e * ~x) ^ (~m + 1) * (~a + ~c * (~x) ^ 2) ^ ~p * Simp((~c * ~d * ~f + ~a * ~e * ~g) * (~m + 1) - ~c * (~e * ~f - ~d * ~g) * (~m + 2 * ~p + 3) * ~x, ~x), ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~m, ~p], ~x) && (NeQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && (ILtQ(Simplify(~m + 2 * ~p + 3), 0) && NeQ(~m, -1)))
    @apply_utils Antiderivative((~f + ~(g') * ~x) / ((~(d') + ~(e') * ~x) * sqrt(~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2)), ~x) => ((4 * ~f * (~a - ~d)) / (~b * ~d - ~a * ~e)) * Subst(Antiderivative(1 / (4 * (~a - ~d) - (~x) ^ 2), ~x), ~x, (2 * (~a - ~d) + (~b - ~e) * ~x) / sqrt(~a + ~b * ~x + ~c * (~x) ^ 2)) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g], ~x) && (EqQ(4 * ~c * (~a - ~d) - (~b - ~e) ^ 2, 0) && (EqQ(~e * ~f * (~b - ~e) - 2 * ~g * (~b * ~d - ~a * ~e), 0) && NeQ(~b * ~d - ~a * ~e, 0)))
    @apply_utils Antiderivative((~f + ~(g') * ~x) / (sqrt(~x) * sqrt(~a + ~(b') * ~x + ~(c') * (~x) ^ 2)), ~x) => 2 * Subst(Antiderivative((~f + ~g * (~x) ^ 2) / sqrt(~a + ~b * (~x) ^ 2 + ~c * (~x) ^ 4), ~x), ~x, sqrt(~x)) <-- FreeQ([~a, ~b, ~c, ~f, ~g], ~x) && NeQ((~b) ^ 2 - 4 * ~a * ~c, 0)
    @apply_utils Antiderivative((~f + ~(g') * ~x) / (sqrt(~x) * sqrt(~a + ~(c') * (~x) ^ 2)), ~x) => 2 * Subst(Antiderivative((~f + ~g * (~x) ^ 2) / sqrt(~a + ~c * (~x) ^ 4), ~x), ~x, sqrt(~x)) <-- FreeQ([~a, ~c, ~f, ~g], ~x)
    @apply_utils Antiderivative((~f + ~(g') * ~x) / (sqrt(~e * ~x) * sqrt(~a + ~(b') * ~x + ~(c') * (~x) ^ 2)), ~x) => (sqrt(~x) / sqrt(~e * ~x)) * Antiderivative((~f + ~g * ~x) / (sqrt(~x) * sqrt(~a + ~b * ~x + ~c * (~x) ^ 2)), ~x) <-- FreeQ([~a, ~b, ~c, ~e, ~f, ~g], ~x) && NeQ((~b) ^ 2 - 4 * ~a * ~c, 0)
    @apply_utils Antiderivative((~f + ~(g') * ~x) / (sqrt(~e * ~x) * sqrt(~a + ~(c') * (~x) ^ 2)), ~x) => (sqrt(~x) / sqrt(~e * ~x)) * Antiderivative((~f + ~g * ~x) / (sqrt(~x) * sqrt(~a + ~c * (~x) ^ 2)), ~x) <-- FreeQ([~a, ~c, ~e, ~f, ~g], ~x)
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => (~g / ~e) * Antiderivative((~d + ~e * ~x) ^ (~m + 1) * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ ~p, ~x) + ((~e * ~f - ~d * ~g) / ~e) * Antiderivative((~d + ~e * ~x) ^ ~m * (~a + ~b * ~x + ~c * (~x) ^ 2) ^ ~p, ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~m, ~p], ~x) && (NeQ((~b) ^ 2 - 4 * ~a * ~c, 0) && (NeQ((~c * (~d) ^ 2 - ~b * ~d * ~e) + ~a * (~e) ^ 2, 0) && Not(IGtQ(~m, 0))))
    @apply_utils Antiderivative((~(d') + ~(e') * ~x) ^ ~m * (~(f') + ~(g') * ~x) * (~a + ~(c') * (~x) ^ 2) ^ ~(p'), ~x) => (~g / ~e) * Antiderivative((~d + ~e * ~x) ^ (~m + 1) * (~a + ~c * (~x) ^ 2) ^ ~p, ~x) + ((~e * ~f - ~d * ~g) / ~e) * Antiderivative((~d + ~e * ~x) ^ ~m * (~a + ~c * (~x) ^ 2) ^ ~p, ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~m, ~p], ~x) && (NeQ(~c * (~d) ^ 2 + ~a * (~e) ^ 2, 0) && Not(IGtQ(~m, 0)))
end
