\[ \int \frac {x^4}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {4 a}{b^{5} \sqrt {\left (b x +a \right )^{2}}}-\frac {a^{4}}{4 b^{5} \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}}}+\frac {4 a^{3}}{3 b^{5} \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}}}-\frac {3 a^{2}}{b^{5} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {\left (b x +a \right ) \ln \left (b x +a \right )}{b^{5} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate(x^4/(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
\[ \frac {\log \left ({\left | b x + a \right |}\right )}{b^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {48 \, a b^{2} x^{3} + 108 \, a^{2} b x^{2} + 88 \, a^{3} x + \frac {25 \, a^{4}}{b}}{12 \, {\left (b x + a\right )}^{4} b^{4} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________