\[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx \]
Optimal antiderivative \[ \frac {c \left (e x +d \right ) \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}{4 e} \]
command
integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{4} \, {\left (c^{2} x^{4} e^{3} \mathrm {sgn}\left (x e + d\right ) + 4 \, c^{2} d x^{3} e^{2} \mathrm {sgn}\left (x e + d\right ) + 6 \, c^{2} d^{2} x^{2} e \mathrm {sgn}\left (x e + d\right ) + 4 \, c^{2} d^{3} x \mathrm {sgn}\left (x e + d\right )\right )} \sqrt {c} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________