\[ \int \frac {x^4}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {3 d^{3} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{5}}+\frac {x^{3} \left (-e x +d \right )}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {4 x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{3}}-\frac {d \left (-9 e x +16 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{6 e^{5}} \]
command
integrate(x^4/(e*x+d)/(-e^2*x^2+d^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {3}{2} \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} \mathrm {sgn}\left (d\right ) + \frac {2 \, d^{3} e^{\left (-5\right )}}{\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1} - \frac {1}{6} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (10 \, d^{2} e^{\left (-5\right )} + {\left (2 \, x e^{\left (-3\right )} - 3 \, d e^{\left (-4\right )}\right )} x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________