\[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx \]
Optimal antiderivative \[ \frac {20 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 e^{2}}+\frac {8 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{e^{2} \left (e x +d \right )^{2}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{e^{2} \left (e x +d \right )^{4}}+\frac {10 d^{3} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2}}+\frac {10 d x \sqrt {-e^{2} x^{2}+d^{2}}}{e} \]
command
integrate(x*(-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
\[ 10 \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-2\right )} \mathrm {sgn}\left (d\right ) - \frac {16 \, d^{3} e^{\left (-2\right )}}{\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1} + \frac {1}{3} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (23 \, d^{2} e^{\left (-2\right )} - {\left (6 \, d e^{\left (-1\right )} - x\right )} x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________