\[ \int \frac {x^5 \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx \]
Optimal antiderivative \[ \frac {d^{4} \left (-e x +d \right )^{4}}{5 e^{6} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {8 d^{3} \left (-e x +d \right )^{3}}{5 e^{6} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {18 d^{3} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{6}}+\frac {10 d^{2} \left (-e x +d \right )^{2}}{e^{6} \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {59 d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{6}}-\frac {2 d x \sqrt {-e^{2} x^{2}+d^{2}}}{e^{5}}+\frac {x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{4}} \]
command
integrate(x^5*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ 18 \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-6\right )} \mathrm {sgn}\left (d\right ) + \frac {1}{3} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (29 \, d^{2} e^{\left (-6\right )} + {\left (x e^{\left (-4\right )} - 6 \, d e^{\left (-5\right )}\right )} x\right )} - \frac {2 \, {\left (\frac {385 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{3} e^{\left (-2\right )}}{x} + \frac {575 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{3} e^{\left (-4\right )}}{x^{2}} + \frac {355 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{3} e^{\left (-6\right )}}{x^{3}} + \frac {80 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{3} e^{\left (-8\right )}}{x^{4}} + 93 \, d^{3}\right )} e^{\left (-6\right )}}{5 \, {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________