\[ \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {1}{2 c e \left (e x +d \right ) \sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}} \]
command
integrate(1/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {e^{\left (-1\right )}}{2 \, {\left (x e + d\right )}^{2} c^{\frac {3}{2}} \mathrm {sgn}\left (x e + d\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________