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