\[ \int \frac {A+B x}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {-4 A b +a B}{a^{5} \sqrt {\left (b x +a \right )^{2}}}+\frac {-A b +a B}{4 a^{2} \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}}}+\frac {-2 A b +a B}{3 a^{3} \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}}}+\frac {-3 A b +a B}{2 a^{4} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}-\frac {A \left (b x +a \right )}{a^{5} x \sqrt {\left (b x +a \right )^{2}}}-\frac {\left (5 A b -a B \right ) \left (b x +a \right ) \ln \left (x \right )}{a^{6} \sqrt {\left (b x +a \right )^{2}}}+\frac {\left (5 A b -a B \right ) \left (b x +a \right ) \ln \left (b x +a \right )}{a^{6} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((B*x+A)/x^2/(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
\[ \frac {{\left (B a - 5 \, A b\right )} \log \left ({\left | x \right |}\right )}{a^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {{\left (B a b - 5 \, A b^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{6} b \mathrm {sgn}\left (b x + a\right )} - \frac {12 \, A a^{5} - 12 \, {\left (B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} - 42 \, {\left (B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} - 52 \, {\left (B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} - 25 \, {\left (B a^{5} - 5 \, A a^{4} b\right )} x}{12 \, {\left (b x + a\right )}^{4} a^{6} x \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________