\[ \int \frac {x^3 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {3 a B}{b^{5} \sqrt {\left (b x +a \right )^{2}}}+\frac {\left (A b -a B \right ) x^{4}}{4 a b \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}}}+\frac {a^{3} B}{3 b^{5} \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}}}-\frac {3 a^{2} B}{2 b^{5} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {B \left (b x +a \right ) \ln \left (b x +a \right )}{b^{5} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate(x^3*(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
\[ \frac {B \log \left ({\left | b x + a \right |}\right )}{b^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {12 \, {\left (4 \, B a b^{2} - A b^{3}\right )} x^{3} + 18 \, {\left (6 \, B a^{2} b - A a b^{2}\right )} x^{2} + 4 \, {\left (22 \, B a^{3} - 3 \, A a^{2} b\right )} x + \frac {25 \, B a^{4} - 3 \, A a^{3} b}{b}}{12 \, {\left (b x + a\right )}^{4} b^{4} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________