\[ \int \frac {1}{x \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {1}{a^{4} \sqrt {\left (b x +a \right )^{2}}}+\frac {1}{4 a \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}}}+\frac {1}{3 a^{2} \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}}}+\frac {1}{2 a^{3} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {\left (b x +a \right ) \ln \left (x \right )}{a^{5} \sqrt {\left (b x +a \right )^{2}}}-\frac {\left (b x +a \right ) \ln \left (b x +a \right )}{a^{5} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate(1/x/(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
\[ -\frac {\log \left ({\left | b x + a \right |}\right )}{a^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {\log \left ({\left | x \right |}\right )}{a^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {12 \, a b^{3} x^{3} + 42 \, a^{2} b^{2} x^{2} + 52 \, a^{3} b x + 25 \, a^{4}}{12 \, {\left (b x + a\right )}^{4} a^{5} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________