\[ \int \frac {x^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {4 a^{3}}{b^{5} \sqrt {\left (b x +a \right )^{2}}}-\frac {a^{4}}{2 b^{5} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}-\frac {3 a x \left (b x +a \right )}{b^{4} \sqrt {\left (b x +a \right )^{2}}}+\frac {x^{2} \left (b x +a \right )}{2 b^{3} \sqrt {\left (b x +a \right )^{2}}}+\frac {6 a^{2} \left (b x +a \right ) \ln \left (b x +a \right )}{b^{5} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate(x^4/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {6 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {b^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{2} x \mathrm {sgn}\left (b x + a\right )}{2 \, b^{6}} + \frac {8 \, a^{3} b x + 7 \, a^{4}}{2 \, {\left (b x + a\right )}^{2} b^{5} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________