\[ \int \frac {1}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 b}{a^{3} \sqrt {\left (b x +a \right )^{2}}}-\frac {b}{2 a^{2} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {-b x -a}{a^{3} x \sqrt {\left (b x +a \right )^{2}}}-\frac {3 b \left (b x +a \right ) \ln \left (x \right )}{a^{4} \sqrt {\left (b x +a \right )^{2}}}+\frac {3 b \left (b x +a \right ) \ln \left (b x +a \right )}{a^{4} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate(1/x^2/(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
\[ \frac {3 \, b \log \left ({\left | b x + a \right |}\right )}{a^{4} \mathrm {sgn}\left (b x + a\right )} - \frac {3 \, b \log \left ({\left | x \right |}\right )}{a^{4} \mathrm {sgn}\left (b x + a\right )} - \frac {6 \, a b^{2} x^{2} + 9 \, a^{2} b x + 2 \, a^{3}}{2 \, {\left (b x + a\right )}^{2} a^{4} x \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________