\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^2 (d+e x)} \, dx \]
Optimal antiderivative \[ -\frac {\left (e x +3 d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 x}-\frac {3 d^{3} e \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2}+d^{3} e \arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )-\frac {d e \left (3 e x +2 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{2} \]
command
integrate((-e^2*x^2+d^2)^(5/2)/x^2/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {3}{2} \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e \mathrm {sgn}\left (d\right ) + d^{3} e \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right ) + \frac {d^{3} x e^{3}}{2 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{3} e^{\left (-1\right )}}{2 \, x} - \frac {1}{6} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (8 \, d^{2} e - {\left (2 \, x e^{3} - 3 \, d e^{2}\right )} x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________