\[ \int \frac {(d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-a e +b d \right )^{2}}{4 b^{3} \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}}}-\frac {2 e \left (-a e +b d \right )}{3 b^{3} \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}}}-\frac {e^{2}}{2 b^{3} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((e*x+d)^2/(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
\[ -\frac {6 \, b^{2} x^{2} e^{2} + 8 \, b^{2} d x e + 3 \, b^{2} d^{2} + 4 \, a b x e^{2} + 2 \, a b d e + a^{2} e^{2}}{12 \, {\left (b x + a\right )}^{4} b^{3} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________