\[ \int \frac {x^3}{(d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx \]
Optimal antiderivative \[ \frac {d^{2} \left (-e x +d \right )^{3}}{5 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {13 d \left (-e x +d \right )^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{4}}+\frac {-\frac {32 e x}{15}+\frac {32 d}{15}}{e^{4} \sqrt {-e^{2} x^{2}+d^{2}}} \]
command
integrate(x^3/(e*x+d)^3/(-e^2*x^2+d^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \arcsin \left (\frac {x e}{d}\right ) e^{\left (-4\right )} \mathrm {sgn}\left (d\right ) - \frac {2 \, {\left (\frac {95 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + \frac {145 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-4\right )}}{x^{2}} + \frac {75 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-6\right )}}{x^{3}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-8\right )}}{x^{4}} + 22\right )} e^{\left (-4\right )}}{15 \, {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________