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