\[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-a e +b d \right )^{3} \left (4 A b e -5 a B e +B b d \right )}{b^{6} \sqrt {\left (b x +a \right )^{2}}}-\frac {\left (A b -a B \right ) \left (-a e +b d \right )^{4}}{2 b^{6} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {e^{2} \left (6 a^{2} B \,e^{2}-3 a b e \left (A e +4 B d \right )+2 b^{2} d \left (2 A e +3 B d \right )\right ) x \left (b x +a \right )}{b^{5} \sqrt {\left (b x +a \right )^{2}}}+\frac {e^{3} \left (A b e -3 a B e +4 B b d \right ) x^{2} \left (b x +a \right )}{2 b^{4} \sqrt {\left (b x +a \right )^{2}}}+\frac {B \,e^{4} x^{3} \left (b x +a \right )}{3 b^{3} \sqrt {\left (b x +a \right )^{2}}}+\frac {2 e \left (-a e +b d \right )^{2} \left (3 A b e -5 a B e +2 B b d \right ) \left (b x +a \right ) \ln \left (b x +a \right )}{b^{6} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((B*x+A)*(e*x+d)^4/(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
\[ \frac {2 \, {\left (2 \, B b^{3} d^{3} e - 9 \, B a b^{2} d^{2} e^{2} + 3 \, A b^{3} d^{2} e^{2} + 12 \, B a^{2} b d e^{3} - 6 \, A a b^{2} d e^{3} - 5 \, B a^{3} e^{4} + 3 \, A a^{2} b e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {B a b^{4} d^{4} + A b^{5} d^{4} - 12 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 30 \, B a^{3} b^{2} d^{2} e^{2} - 18 \, A a^{2} b^{3} d^{2} e^{2} - 28 \, B a^{4} b d e^{3} + 20 \, A a^{3} b^{2} d e^{3} + 9 \, B a^{5} e^{4} - 7 \, A a^{4} b e^{4} + 2 \, {\left (B b^{5} d^{4} - 8 \, B a b^{4} d^{3} e + 4 \, A b^{5} d^{3} e + 18 \, B a^{2} b^{3} d^{2} e^{2} - 12 \, A a b^{4} d^{2} e^{2} - 16 \, B a^{3} b^{2} d e^{3} + 12 \, A a^{2} b^{3} d e^{3} + 5 \, B a^{4} b e^{4} - 4 \, A a^{3} b^{2} e^{4}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{6} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, B b^{6} x^{3} e^{4} + 12 \, B b^{6} d x^{2} e^{3} + 36 \, B b^{6} d^{2} x e^{2} - 9 \, B a b^{5} x^{2} e^{4} + 3 \, A b^{6} x^{2} e^{4} - 72 \, B a b^{5} d x e^{3} + 24 \, A b^{6} d x e^{3} + 36 \, B a^{2} b^{4} x e^{4} - 18 \, A a b^{5} x e^{4}}{6 \, b^{9} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________