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