\[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (e x +d \right )^{4}}{3 b \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {3}{2}}}-\frac {4 e^{2} \left (-a e +b d \right )^{2}}{b^{5} \sqrt {\left (b x +a \right )^{2}}}-\frac {2 e \left (-a e +b d \right )^{3}}{3 b^{5} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {4 e^{4} x \left (b x +a \right )}{3 b^{4} \sqrt {\left (b x +a \right )^{2}}}+\frac {4 e^{3} \left (-a e +b d \right ) \left (b x +a \right ) \ln \left (b x +a \right )}{b^{5} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((b*x+a)*(e*x+d)^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
\[ \frac {x e^{4}}{b^{4} \mathrm {sgn}\left (b x + a\right )} + \frac {4 \, {\left (b d e^{3} - a e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {b^{4} d^{4} + 2 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 22 \, a^{3} b d e^{3} + 13 \, a^{4} e^{4} + 18 \, {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 6 \, {\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} - 9 \, a^{2} b^{2} d e^{3} + 5 \, a^{3} b e^{4}\right )} x}{3 \, {\left (b x + a\right )}^{3} b^{5} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (b x + a\right )} {\left (e x + d\right )}^{4}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}\,{d x} \]________________________________________________________________________________________