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