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