\[ \int \frac {(a+b x) (d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (e x +d \right )^{3}}{3 b \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {3}{2}}}-\frac {2 e^{2} \left (-a e +b d \right )}{b^{4} \sqrt {\left (b x +a \right )^{2}}}-\frac {e \left (-a e +b d \right )^{2}}{2 b^{4} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {e^{3} \left (b x +a \right ) \ln \left (b x +a \right )}{b^{4} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((b*x+a)*(e*x+d)^3/(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
\[ \frac {e^{3} \log \left ({\left | b x + a \right |}\right )}{b^{4} \mathrm {sgn}\left (b x + a\right )} - \frac {18 \, {\left (b^{2} d e^{2} - a b e^{3}\right )} x^{2} + 9 \, {\left (b^{2} d^{2} e + 2 \, a b d e^{2} - 3 \, a^{2} e^{3}\right )} x + \frac {2 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3}}{b}}{6 \, {\left (b x + a\right )}^{3} b^{3} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (b x + a\right )} {\left (e x + d\right )}^{3}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}\,{d x} \]________________________________________________________________________________________