\[ \int \frac {x^5}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {a}{12 b^{2} \left (b \,x^{3}+a \right )^{3} \sqrt {\left (b \,x^{3}+a \right )^{2}}}-\frac {1}{9 b^{2} \left (b \,x^{3}+a \right )^{2} \sqrt {\left (b \,x^{3}+a \right )^{2}}} \]
command
integrate(x^5/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {4 \, b x^{3} + a}{36 \, {\left (b x^{3} + a\right )}^{4} b^{2} \mathrm {sgn}\left (b x^{3} + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________