\[ \int \frac {x^4 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 a \left (2 A b -5 a B \right )}{b^{6} \sqrt {\left (b x +a \right )^{2}}}-\frac {a^{4} \left (A b -a B \right )}{4 b^{6} \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}}}+\frac {a^{3} \left (4 A b -5 a B \right )}{3 b^{6} \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}}}-\frac {a^{2} \left (3 A b -5 a B \right )}{b^{6} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {B x \left (b x +a \right )}{b^{5} \sqrt {\left (b x +a \right )^{2}}}+\frac {\left (A b -5 a B \right ) \left (b x +a \right ) \ln \left (b x +a \right )}{b^{6} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate(x^4*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {B x}{b^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {{\left (5 \, B a - A b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {77 \, B a^{5} - 25 \, A a^{4} b + 24 \, {\left (5 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{3} + 12 \, {\left (25 \, B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{2} + 4 \, {\left (65 \, B a^{4} b - 22 \, A a^{3} b^{2}\right )} x}{12 \, {\left (b x + a\right )}^{4} b^{6} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________