\[ \int \frac {x^5}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {10 a^{2}}{b^{6} \sqrt {\left (b x +a \right )^{2}}}+\frac {a^{5}}{4 b^{6} \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}}}-\frac {5 a^{4}}{3 b^{6} \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}}}+\frac {5 a^{3}}{b^{6} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {x \left (b x +a \right )}{b^{5} \sqrt {\left (b x +a \right )^{2}}}-\frac {5 a \left (b x +a \right ) \ln \left (b x +a \right )}{b^{6} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate(x^5/(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
\[ \frac {x}{b^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {5 \, a \log \left ({\left | b x + a \right |}\right )}{b^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {120 \, a^{2} b^{3} x^{3} + 300 \, a^{3} b^{2} x^{2} + 260 \, a^{4} b x + 77 \, a^{5}}{12 \, {\left (b x + a\right )}^{4} b^{6} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________