\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x (d+e x)} \, dx \]
Optimal antiderivative \[ \frac {\left (-3 e x +4 d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{12}-\frac {3 d^{4} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8}-d^{4} \arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )+\frac {d^{2} \left (-3 e x +8 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{8} \]
command
integrate((-e^2*x^2+d^2)^(5/2)/x/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {3}{8} \, d^{4} \arcsin \left (\frac {x e}{d}\right ) \mathrm {sgn}\left (d\right ) - d^{4} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right ) + \frac {1}{24} \, {\left (32 \, d^{3} - {\left (15 \, d^{2} e - 2 \, {\left (3 \, x e^{3} - 4 \, d e^{2}\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________