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