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