How find this $5xy\sqrt{(x^2+y^2)^3}$ can write the sum of Four 5-th powers of positive integers. 
Find  all positive integer $x,y$ such
$$5xy\sqrt{(x^2+y^2)^3}$$ can write
the sum of Four  5-th powers of positive integers.In other words: there exst $a,b,c,d\in N^{+}$ such
$$5xy\sqrt{(x^2+y^2)^3}=a^5+b^5+c^5+d^5$$

This problem is from Math competition simulation test.I seach this problem and  found  this problem background is Euler's sum of powers conjecture.can see link
maybe this problem is not hard.because is from competition.
since
$$5xy\sqrt{(x^2+y^2)^3}=5xy(x^2+y^2)\sqrt{x^2+y^2}$$
so $$x^2+y^2=m^2$$
$$x=3,y=4,m=5$$
and $$x=(a'^2-b'^2),y=2a'b',m=a'^2+b'^2$$
then I can't it
Thank you for you help .

Something that might help. Using what you said, namely:
$x^2+y^2=k^2$, with $x=m^2-n^2$, $y=2mn$ then the left hand side of the equation becomes:
$LHS=10(m^9n-mn^9+2m^7n^3-2m^3n^7)$
Since $RHS\equiv 0$ modulo 2 and modulo 5 then we have $a+b+c+d\equiv 0 \mod{2}$ and $a+b+c+d\equiv 0 \mod{5}$
