\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^4 (d+e x)} \, dx \]
Optimal antiderivative \[ -\frac {\left (-3 e x +2 d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{6 x^{3}}+d \,e^{3} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )-\frac {3 d \,e^{3} \arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{2}+\frac {e^{2} \left (3 e x +2 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{2 x} \]
command
integrate((-e^2*x^2+d^2)^(5/2)/x^4/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ d \arcsin \left (\frac {x e}{d}\right ) e^{3} \mathrm {sgn}\left (d\right ) - \frac {3}{2} \, d e^{3} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right ) + \frac {{\left (d e^{3} - \frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d e}{x} - \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d e^{\left (-1\right )}}{x^{2}}\right )} x^{3} e^{6}}{24 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3}} + \frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d e}{8 \, x} + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d e^{\left (-1\right )}}{8 \, x^{2}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d e^{\left (-3\right )}}{24 \, x^{3}} + \sqrt {-x^{2} e^{2} + d^{2}} e^{3} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________