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