\[ \int \frac {(A+B x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-A e +B d \right ) \left (e x +d \right )^{3}}{3 \left (-a e +b d \right )^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {3}{2}}}-\frac {\left (A b -a B \right ) \left (e x +d \right )^{4}}{4 \left (-a e +b d \right )^{2} \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((B*x+A)*(e*x+d)^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
\[ -\frac {12 \, B b^{3} x^{3} e^{2} + 12 \, B b^{3} d x^{2} e + 4 \, B b^{3} d^{2} x + 18 \, B a b^{2} x^{2} e^{2} + 6 \, A b^{3} x^{2} e^{2} + 8 \, B a b^{2} d x e + 8 \, A b^{3} d x e + B a b^{2} d^{2} + 3 \, A b^{3} d^{2} + 12 \, B a^{2} b x e^{2} + 4 \, A a b^{2} x e^{2} + 2 \, B a^{2} b d e + 2 \, A a b^{2} d e + 3 \, B a^{3} e^{2} + A a^{2} b e^{2}}{12 \, {\left (b x + a\right )}^{4} b^{4} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________