\[ \int \frac {1}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {10 b^{2}}{a^{6} \sqrt {\left (b x +a \right )^{2}}}+\frac {b^{2}}{4 a^{3} \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}}}+\frac {b^{2}}{a^{4} \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}}}+\frac {3 b^{2}}{a^{5} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {-b x -a}{2 a^{5} x^{2} \sqrt {\left (b x +a \right )^{2}}}+\frac {5 b \left (b x +a \right )}{a^{6} x \sqrt {\left (b x +a \right )^{2}}}+\frac {15 b^{2} \left (b x +a \right ) \ln \left (x \right )}{a^{7} \sqrt {\left (b x +a \right )^{2}}}-\frac {15 b^{2} \left (b x +a \right ) \ln \left (b x +a \right )}{a^{7} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate(1/x^3/(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
\[ -\frac {15 \, b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{7} \mathrm {sgn}\left (b x + a\right )} + \frac {15 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{7} \mathrm {sgn}\left (b x + a\right )} + \frac {60 \, a b^{5} x^{5} + 210 \, a^{2} b^{4} x^{4} + 260 \, a^{3} b^{3} x^{3} + 125 \, a^{4} b^{2} x^{2} + 12 \, a^{5} b x - 2 \, a^{6}}{4 \, {\left (b x + a\right )}^{4} a^{7} x^{2} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________