\[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx \]
Optimal antiderivative \[ -\frac {d^{2} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{24 e}-\frac {\left (-5 e x +6 d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{30 e^{2}}-\frac {d^{6} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{16 e^{2}}-\frac {d^{4} x \sqrt {-e^{2} x^{2}+d^{2}}}{16 e} \]
command
integrate(x*(-e^2*x^2+d^2)^(5/2)/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{16} \, d^{6} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-2\right )} \mathrm {sgn}\left (d\right ) - \frac {1}{240} \, {\left (48 \, d^{5} e^{\left (-2\right )} - {\left (15 \, d^{4} e^{\left (-1\right )} + 2 \, {\left (48 \, d^{3} - {\left (35 \, d^{2} e - 4 \, {\left (5 \, x e^{3} - 6 \, d e^{2}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________