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