\[ \int \frac {x^5}{(d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx \]
Optimal antiderivative \[ \frac {d^{4} \left (-e x +d \right )^{3}}{5 e^{6} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {23 d^{3} \left (-e x +d \right )^{2}}{15 e^{6} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {13 d^{2} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{6}}+\frac {127 d^{2} \left (-e x +d \right )}{15 e^{6} \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {3 d \sqrt {-e^{2} x^{2}+d^{2}}}{e^{6}}-\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{5}} \]
command
integrate(x^5/(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
\[ \frac {13}{2} \, d^{2} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-6\right )} \mathrm {sgn}\left (d\right ) - \frac {1}{2} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (x e^{\left (-5\right )} - 6 \, d e^{\left (-6\right )}\right )} - \frac {2 \, {\left (\frac {445 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2} e^{\left (-2\right )}}{x} + \frac {665 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} e^{\left (-4\right )}}{x^{2}} + \frac {405 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{2} e^{\left (-6\right )}}{x^{3}} + \frac {90 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{2} e^{\left (-8\right )}}{x^{4}} + 107 \, d^{2}\right )} e^{\left (-6\right )}}{15 \, {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________