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