\[ \int \frac {(A+B x) (d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-a e +b d \right )^{2} \left (3 A b e -4 a B e +B b d \right )}{b^{5} \sqrt {\left (b x +a \right )^{2}}}-\frac {\left (A b -a B \right ) \left (-a e +b d \right )^{3}}{2 b^{5} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {e^{2} \left (A b e -3 a B e +3 B b d \right ) x \left (b x +a \right )}{b^{4} \sqrt {\left (b x +a \right )^{2}}}+\frac {B \,e^{3} x^{2} \left (b x +a \right )}{2 b^{3} \sqrt {\left (b x +a \right )^{2}}}+\frac {3 e \left (-a e +b d \right ) \left (A b e -2 a B e +B b d \right ) \left (b x +a \right ) \ln \left (b x +a \right )}{b^{5} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((B*x+A)*(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {3 \, {\left (B b^{2} d^{2} e - 3 \, B a b d e^{2} + A b^{2} d e^{2} + 2 \, B a^{2} e^{3} - A a b e^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {B b^{3} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, B b^{3} d x e^{2} \mathrm {sgn}\left (b x + a\right ) - 6 \, B a b^{2} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, A b^{3} x e^{3} \mathrm {sgn}\left (b x + a\right )}{2 \, b^{6}} - \frac {B a b^{3} d^{3} + A b^{4} d^{3} - 9 \, B a^{2} b^{2} d^{2} e + 3 \, A a b^{3} d^{2} e + 15 \, B a^{3} b d e^{2} - 9 \, A a^{2} b^{2} d e^{2} - 7 \, B a^{4} e^{3} + 5 \, A a^{3} b e^{3} + 2 \, {\left (B b^{4} d^{3} - 6 \, B a b^{3} d^{2} e + 3 \, A b^{4} d^{2} e + 9 \, B a^{2} b^{2} d e^{2} - 6 \, A a b^{3} d e^{2} - 4 \, B a^{3} b e^{3} + 3 \, A a^{2} b^{2} e^{3}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{5} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________