\[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx \]
Optimal antiderivative \[ \frac {d \left (-3 e x +4 d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{12 e^{3}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5 e^{3}}+\frac {d^{5} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 e^{3}}+\frac {d^{3} x \sqrt {-e^{2} x^{2}+d^{2}}}{8 e^{2}} \]
command
integrate(x^2*(-e^2*x^2+d^2)^(3/2)/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{8} \, d^{5} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} \mathrm {sgn}\left (d\right ) + \frac {1}{120} \, {\left (16 \, d^{4} e^{\left (-3\right )} - {\left (15 \, d^{3} e^{\left (-2\right )} - 2 \, {\left (4 \, d^{2} e^{\left (-1\right )} - 3 \, {\left (4 \, x e - 5 \, d\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________