\[ \int \frac {x^4 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {a^{3} \left (4 A b -5 a B \right )}{b^{6} \sqrt {\left (b x +a \right )^{2}}}-\frac {a^{4} \left (A b -a B \right )}{2 b^{6} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}-\frac {3 a \left (A b -2 a B \right ) x \left (b x +a \right )}{b^{5} \sqrt {\left (b x +a \right )^{2}}}+\frac {\left (A b -3 a B \right ) x^{2} \left (b x +a \right )}{2 b^{4} \sqrt {\left (b x +a \right )^{2}}}+\frac {B \,x^{3} \left (b x +a \right )}{3 b^{3} \sqrt {\left (b x +a \right )^{2}}}+\frac {2 a^{2} \left (3 A b -5 a B \right ) \left (b x +a \right ) \ln \left (b x +a \right )}{b^{6} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate(x^4*(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
\[ -\frac {2 \, {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {9 \, B a^{5} - 7 \, A a^{4} b + 2 \, {\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{6} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, B b^{6} x^{3} - 9 \, B a b^{5} x^{2} + 3 \, A b^{6} x^{2} + 36 \, B a^{2} b^{4} x - 18 \, A a b^{5} x}{6 \, b^{9} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________