\[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 e^{2} \left (-a e +b d \right ) \left (2 A b e -5 a B e +3 B b d \right )}{b^{6} \sqrt {\left (b x +a \right )^{2}}}-\frac {\left (A b -a B \right ) \left (-a e +b d \right )^{4}}{4 b^{6} \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}}}-\frac {\left (-a e +b d \right )^{3} \left (4 A b e -5 a B e +B b d \right )}{3 b^{6} \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}}}-\frac {e \left (-a e +b d \right )^{2} \left (3 A b e -5 a B e +2 B b d \right )}{b^{6} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {B \,e^{4} x \left (b x +a \right )}{b^{5} \sqrt {\left (b x +a \right )^{2}}}+\frac {e^{3} \left (A b e -5 a B e +4 B b d \right ) \left (b x +a \right ) \ln \left (b x +a \right )}{b^{6} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((B*x+A)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {B x e^{4}}{b^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {{\left (4 \, B b d e^{3} - 5 \, B a e^{4} + A b e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {B a b^{4} d^{4} + 3 \, A b^{5} d^{4} + 4 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 18 \, B a^{3} b^{2} d^{2} e^{2} + 6 \, A a^{2} b^{3} d^{2} e^{2} - 100 \, B a^{4} b d e^{3} + 12 \, A a^{3} b^{2} d e^{3} + 77 \, B a^{5} e^{4} - 25 \, A a^{4} b e^{4} + 24 \, {\left (3 \, B b^{5} d^{2} e^{2} - 8 \, B a b^{4} d e^{3} + 2 \, A b^{5} d e^{3} + 5 \, B a^{2} b^{3} e^{4} - 2 \, A a b^{4} e^{4}\right )} x^{3} + 12 \, {\left (2 \, B b^{5} d^{3} e + 9 \, B a b^{4} d^{2} e^{2} + 3 \, A b^{5} d^{2} e^{2} - 36 \, B a^{2} b^{3} d e^{3} + 6 \, A a b^{4} d e^{3} + 25 \, B a^{3} b^{2} e^{4} - 9 \, A a^{2} b^{3} e^{4}\right )} x^{2} + 4 \, {\left (B b^{5} d^{4} + 4 \, B a b^{4} d^{3} e + 4 \, A b^{5} d^{3} e + 18 \, B a^{2} b^{3} d^{2} e^{2} + 6 \, A a b^{4} d^{2} e^{2} - 88 \, B a^{3} b^{2} d e^{3} + 12 \, A a^{2} b^{3} d e^{3} + 65 \, B a^{4} b e^{4} - 22 \, A a^{3} b^{2} e^{4}\right )} x}{12 \, {\left (b x + a\right )}^{4} b^{6} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________