\[ \int \frac {(A+B x) (d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {10 e^{2} \left (-a e +b d \right )^{2} \left (A b e -2 a B e +B b d \right )}{b^{7} \sqrt {\left (b x +a \right )^{2}}}-\frac {\left (A b -a B \right ) \left (-a e +b d \right )^{5}}{4 b^{7} \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}}}-\frac {\left (-a e +b d \right )^{4} \left (5 A b e -6 a B e +B b d \right )}{3 b^{7} \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}}}-\frac {5 e \left (-a e +b d \right )^{3} \left (2 A b e -3 a B e +B b d \right )}{2 b^{7} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {e^{4} \left (A b e -5 a B e +5 B b d \right ) x \left (b x +a \right )}{b^{6} \sqrt {\left (b x +a \right )^{2}}}+\frac {B \,e^{5} x^{2} \left (b x +a \right )}{2 b^{5} \sqrt {\left (b x +a \right )^{2}}}+\frac {5 e^{3} \left (-a e +b d \right ) \left (A b e -3 a B e +2 B b d \right ) \left (b x +a \right ) \ln \left (b x +a \right )}{b^{7} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((B*x+A)*(e*x+d)^5/(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
\[ \frac {5 \, {\left (2 \, B b^{2} d^{2} e^{3} - 5 \, B a b d e^{4} + A b^{2} d e^{4} + 3 \, B a^{2} e^{5} - A a b e^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7} \mathrm {sgn}\left (b x + a\right )} + \frac {B b^{5} x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 10 \, B b^{5} d x e^{4} \mathrm {sgn}\left (b x + a\right ) - 10 \, B a b^{4} x e^{5} \mathrm {sgn}\left (b x + a\right ) + 2 \, A b^{5} x e^{5} \mathrm {sgn}\left (b x + a\right )}{2 \, b^{10}} - \frac {B a b^{5} d^{5} + 3 \, A b^{6} d^{5} + 5 \, B a^{2} b^{4} d^{4} e + 5 \, A a b^{5} d^{4} e + 30 \, B a^{3} b^{3} d^{3} e^{2} + 10 \, A a^{2} b^{4} d^{3} e^{2} - 250 \, B a^{4} b^{2} d^{2} e^{3} + 30 \, A a^{3} b^{3} d^{2} e^{3} + 385 \, B a^{5} b d e^{4} - 125 \, A a^{4} b^{2} d e^{4} - 171 \, B a^{6} e^{5} + 77 \, A a^{5} b e^{5} + 120 \, {\left (B b^{6} d^{3} e^{2} - 4 \, B a b^{5} d^{2} e^{3} + A b^{6} d^{2} e^{3} + 5 \, B a^{2} b^{4} d e^{4} - 2 \, A a b^{5} d e^{4} - 2 \, B a^{3} b^{3} e^{5} + A a^{2} b^{4} e^{5}\right )} x^{3} + 30 \, {\left (B b^{6} d^{4} e + 6 \, B a b^{5} d^{3} e^{2} + 2 \, A b^{6} d^{3} e^{2} - 36 \, B a^{2} b^{4} d^{2} e^{3} + 6 \, A a b^{5} d^{2} e^{3} + 50 \, B a^{3} b^{3} d e^{4} - 18 \, A a^{2} b^{4} d e^{4} - 21 \, B a^{4} b^{2} e^{5} + 10 \, A a^{3} b^{3} e^{5}\right )} x^{2} + 4 \, {\left (B b^{6} d^{5} + 5 \, B a b^{5} d^{4} e + 5 \, A b^{6} d^{4} e + 30 \, B a^{2} b^{4} d^{3} e^{2} + 10 \, A a b^{5} d^{3} e^{2} - 220 \, B a^{3} b^{3} d^{2} e^{3} + 30 \, A a^{2} b^{4} d^{2} e^{3} + 325 \, B a^{4} b^{2} d e^{4} - 110 \, A a^{3} b^{3} d e^{4} - 141 \, B a^{5} b e^{5} + 65 \, A a^{4} b^{2} e^{5}\right )} x}{12 \, {\left (b x + a\right )}^{4} b^{7} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________