\[ \int \frac {x^2 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {A \,x^{3}}{3 a^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {3}{2}}}-\frac {\left (A b -a B \right ) x^{4}}{4 a^{2} \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate(x^2*(B*x+A)/(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
\[ -\frac {12 \, B b^{3} x^{3} + 18 \, B a b^{2} x^{2} + 6 \, A b^{3} x^{2} + 12 \, B a^{2} b x + 4 \, A a b^{2} x + 3 \, B a^{3} + A a^{2} b}{12 \, {\left (b x + a\right )}^{4} b^{4} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________