\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^5 (d+e x)} \, dx \]
Optimal antiderivative \[ -\frac {\left (-4 e x +3 d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{12 x^{4}}-e^{4} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )-\frac {3 e^{4} \arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{8}+\frac {e^{2} \left (-8 e x +3 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{8 x^{2}} \]
command
integrate((-e^2*x^2+d^2)^(5/2)/x^5/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\arcsin \left (\frac {x e}{d}\right ) e^{4} \mathrm {sgn}\left (d\right ) - \frac {x^{4} {\left (\frac {8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{2}}{x} - \frac {120 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-2\right )}}{x^{3}} + \frac {24 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2}}{x^{2}} - 3 \, e^{4}\right )} e^{8}}{192 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4}} - \frac {3}{8} \, e^{4} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right ) - \frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{2}}{8 \, x} + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-2\right )}}{24 \, x^{3}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-4\right )}}{64 \, x^{4}} + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2}}{8 \, x^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________