\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x (d+e x)^4} \, dx \]
Optimal antiderivative \[ 4 d \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )-d \arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )+\frac {8 d \left (-e x +d \right )}{\sqrt {-e^{2} x^{2}+d^{2}}}+\sqrt {-e^{2} x^{2}+d^{2}} \]
command
integrate((-e^2*x^2+d^2)^(5/2)/x/(e*x+d)^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ 4 \, d \arcsin \left (\frac {x e}{d}\right ) \mathrm {sgn}\left (d\right ) - d \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right ) + \sqrt {-x^{2} e^{2} + d^{2}} - \frac {16 \, d}{\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________