\[ \int \frac {(a+b x) (d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (e x +d \right )^{5}}{3 b \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {3}{2}}}-\frac {20 e^{2} \left (-a e +b d \right )^{3}}{3 b^{6} \sqrt {\left (b x +a \right )^{2}}}-\frac {5 e \left (-a e +b d \right )^{4}}{6 b^{6} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {5 e^{4} \left (-3 a e +4 b d \right ) x \left (b x +a \right )}{3 b^{5} \sqrt {\left (b x +a \right )^{2}}}+\frac {5 e^{5} x^{2} \left (b x +a \right )}{6 b^{4} \sqrt {\left (b x +a \right )^{2}}}+\frac {10 e^{3} \left (-a e +b d \right )^{2} \left (b x +a \right ) \ln \left (b x +a \right )}{b^{6} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((b*x+a)*(e*x+d)^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
\[ \frac {10 \, {\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} \mathrm {sgn}\left (b x + a\right )} + \frac {b^{4} x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 10 \, b^{4} d x e^{4} \mathrm {sgn}\left (b x + a\right ) - 8 \, a b^{3} x e^{5} \mathrm {sgn}\left (b x + a\right )}{2 \, b^{8}} - \frac {2 \, b^{5} d^{5} + 5 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} - 110 \, a^{3} b^{2} d^{2} e^{3} + 130 \, a^{4} b d e^{4} - 47 \, a^{5} e^{5} + 60 \, {\left (b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 15 \, {\left (b^{5} d^{4} e + 4 \, a b^{4} d^{3} e^{2} - 18 \, a^{2} b^{3} d^{2} e^{3} + 20 \, a^{3} b^{2} d e^{4} - 7 \, a^{4} b e^{5}\right )} x}{6 \, {\left (b x + a\right )}^{3} b^{6} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (b x + a\right )} {\left (e x + d\right )}^{5}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}\,{d x} \]________________________________________________________________________________________