\[ \int \frac {(A+B x) (d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {3 B \,e^{2} \left (-a e +b d \right )}{b^{5} \sqrt {\left (b x +a \right )^{2}}}-\frac {B \left (-a e +b d \right )^{3}}{3 b^{5} \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}}}-\frac {3 B e \left (-a e +b d \right )^{2}}{2 b^{5} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}-\frac {\left (A b -a B \right ) \left (e x +d \right )^{4}}{4 b \left (-a e +b d \right ) \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}}}+\frac {B \,e^{3} \left (b x +a \right ) \ln \left (b x +a \right )}{b^{5} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((B*x+A)*(e*x+d)^3/(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
\[ \frac {B e^{3} \log \left ({\left | b x + a \right |}\right )}{b^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {12 \, {\left (3 \, B b^{3} d e^{2} - 4 \, B a b^{2} e^{3} + A b^{3} e^{3}\right )} x^{3} + 18 \, {\left (B b^{3} d^{2} e + 3 \, B a b^{2} d e^{2} + A b^{3} d e^{2} - 6 \, B a^{2} b e^{3} + A a b^{2} e^{3}\right )} x^{2} + 4 \, {\left (B b^{3} d^{3} + 3 \, B a b^{2} d^{2} e + 3 \, A b^{3} d^{2} e + 9 \, B a^{2} b d e^{2} + 3 \, A a b^{2} d e^{2} - 22 \, B a^{3} e^{3} + 3 \, A a^{2} b e^{3}\right )} x + \frac {B a b^{3} d^{3} + 3 \, A b^{4} d^{3} + 3 \, B a^{2} b^{2} d^{2} e + 3 \, A a b^{3} d^{2} e + 9 \, B a^{3} b d e^{2} + 3 \, A a^{2} b^{2} d e^{2} - 25 \, B a^{4} e^{3} + 3 \, A a^{3} b e^{3}}{b}}{12 \, {\left (b x + a\right )}^{4} b^{4} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________