\[ \int \frac {x^4}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {3 x}{128 a^{2} b^{2} \sqrt {\left (b \,x^{2}+a \right )^{2}}}-\frac {x^{3}}{8 b \left (b \,x^{2}+a \right )^{3} \sqrt {\left (b \,x^{2}+a \right )^{2}}}-\frac {x}{16 b^{2} \left (b \,x^{2}+a \right )^{2} \sqrt {\left (b \,x^{2}+a \right )^{2}}}+\frac {x}{64 a \,b^{2} \left (b \,x^{2}+a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}+\frac {3 \left (b \,x^{2}+a \right ) \arctan \left (\frac {x \sqrt {b}}{\sqrt {a}}\right )}{128 a^{\frac {5}{2}} b^{\frac {5}{2}} \sqrt {\left (b \,x^{2}+a \right )^{2}}} \]
command
integrate(x^4/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {3 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{2} b^{2} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {3 \, b^{3} x^{7} + 11 \, a b^{2} x^{5} - 11 \, a^{2} b x^{3} - 3 \, a^{3} x}{128 \, {\left (b x^{2} + a\right )}^{4} a^{2} b^{2} \mathrm {sgn}\left (b x^{2} + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________