\[ \int \frac {x^3 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {a^{2} \left (3 A b -4 a B \right )}{b^{5} \sqrt {\left (b x +a \right )^{2}}}+\frac {a^{3} \left (A b -a B \right )}{2 b^{5} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {\left (A b -3 a B \right ) x \left (b x +a \right )}{b^{4} \sqrt {\left (b x +a \right )^{2}}}+\frac {B \,x^{2} \left (b x +a \right )}{2 b^{3} \sqrt {\left (b x +a \right )^{2}}}-\frac {3 a \left (A b -2 a B \right ) \left (b x +a \right ) \ln \left (b x +a \right )}{b^{5} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate(x^3*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {3 \, {\left (2 \, B a^{2} - A a b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {B b^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) - 6 \, B a b^{2} x \mathrm {sgn}\left (b x + a\right ) + 2 \, A b^{3} x \mathrm {sgn}\left (b x + a\right )}{2 \, b^{6}} + \frac {7 \, B a^{4} - 5 \, A a^{3} b + 2 \, {\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{5} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________