\[ \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {b \left (3 A b -2 a B \right )}{a^{4} \sqrt {\left (b x +a \right )^{2}}}+\frac {b \left (A b -a B \right )}{2 a^{3} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}-\frac {A \left (b x +a \right )}{2 a^{3} x^{2} \sqrt {\left (b x +a \right )^{2}}}+\frac {\left (3 A b -a B \right ) \left (b x +a \right )}{a^{4} x \sqrt {\left (b x +a \right )^{2}}}+\frac {3 b \left (2 A b -a B \right ) \left (b x +a \right ) \ln \left (x \right )}{a^{5} \sqrt {\left (b x +a \right )^{2}}}-\frac {3 b \left (2 A b -a B \right ) \left (b x +a \right ) \ln \left (b x +a \right )}{a^{5} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((B*x+A)/x^3/(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
\[ -\frac {3 \, {\left (B a b - 2 \, A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {3 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{5} b \mathrm {sgn}\left (b x + a\right )} - \frac {6 \, B a b^{2} x^{3} - 12 \, A b^{3} x^{3} + 9 \, B a^{2} b x^{2} - 18 \, A a b^{2} x^{2} + 2 \, B a^{3} x - 4 \, A a^{2} b x + A a^{3}}{2 \, {\left (b x^{2} + a x\right )}^{2} a^{4} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________