\[ \int \frac {(A+B x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-a e +b d \right ) \left (2 A b e -3 a B e +B b d \right )}{b^{4} \sqrt {\left (b x +a \right )^{2}}}-\frac {\left (A b -a B \right ) \left (-a e +b d \right )^{2}}{2 b^{4} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {B \,e^{2} x \left (b x +a \right )}{b^{3} \sqrt {\left (b x +a \right )^{2}}}+\frac {e \left (A b e -3 a B e +2 B b d \right ) \left (b x +a \right ) \ln \left (b x +a \right )}{b^{4} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((B*x+A)*(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {B x e^{2}}{b^{3} \mathrm {sgn}\left (b x + a\right )} + \frac {{\left (2 \, B b d e - 3 \, B a e^{2} + A b e^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4} \mathrm {sgn}\left (b x + a\right )} - \frac {B a b^{2} d^{2} + A b^{3} d^{2} - 6 \, B a^{2} b d e + 2 \, A a b^{2} d e + 5 \, B a^{3} e^{2} - 3 \, A a^{2} b e^{2} + 2 \, {\left (B b^{3} d^{2} - 4 \, B a b^{2} d e + 2 \, A b^{3} d e + 3 \, B a^{2} b e^{2} - 2 \, A a b^{2} e^{2}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{4} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________