\[ \int \frac {(d+e x)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (e x +d \right )^{\frac {3}{2}}}{c e \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}+\frac {8 d \sqrt {e x +d}}{c e \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}} \]
command
integrate((e*x+d)^(5/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {4 \, \sqrt {2} d e^{\left (-1\right )}}{\sqrt {c d} c} + \frac {2 \, {\left (\frac {2 \, d e^{\left (-1\right )}}{\sqrt {-{\left (x e + d\right )} c + 2 \, c d}} + \frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d} e^{\left (-1\right )}}{c}\right )}}{c} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________