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