\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx \]
Optimal antiderivative \[ -\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{2 e \left (e x +d \right )}-\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{e \left (e x +d \right )^{3}}-\frac {15 d^{2} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e}-\frac {15 d \sqrt {-e^{2} x^{2}+d^{2}}}{2 e} \]
command
integrate((-e^2*x^2+d^2)^(5/2)/(e*x+d)^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {15}{2} \, d^{2} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\left (d\right ) + \frac {16 \, d^{2} e^{\left (-1\right )}}{\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1} - \frac {1}{2} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (8 \, d e^{\left (-1\right )} - x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________