\[ \int \frac {x^6}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {20 a^{3}}{b^{7} \sqrt {\left (b x +a \right )^{2}}}-\frac {a^{6}}{4 b^{7} \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}}}+\frac {2 a^{5}}{b^{7} \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}}}-\frac {15 a^{4}}{2 b^{7} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}-\frac {5 a x \left (b x +a \right )}{b^{6} \sqrt {\left (b x +a \right )^{2}}}+\frac {x^{2} \left (b x +a \right )}{2 b^{5} \sqrt {\left (b x +a \right )^{2}}}+\frac {15 a^{2} \left (b x +a \right ) \ln \left (b x +a \right )}{b^{7} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate(x^6/(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
\[ \frac {15 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{7} \mathrm {sgn}\left (b x + a\right )} + \frac {b^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a b^{4} x \mathrm {sgn}\left (b x + a\right )}{2 \, b^{10}} + \frac {80 \, a^{3} b^{3} x^{3} + 210 \, a^{4} b^{2} x^{2} + 188 \, a^{5} b x + 57 \, a^{6}}{4 \, {\left (b x + a\right )}^{4} b^{7} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________