\[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx \]
Optimal antiderivative \[ \frac {65 d^{7} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{4 e^{6}}+\frac {d^{4} \left (-e x +d \right )^{4}}{e^{6} \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {515 d^{6} \sqrt {-e^{2} x^{2}+d^{2}}}{21 e^{6}}-\frac {49 d^{5} x \sqrt {-e^{2} x^{2}+d^{2}}}{4 e^{5}}+\frac {121 d^{4} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{21 e^{4}}-\frac {17 d^{3} x^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{6 e^{3}}+\frac {11 d^{2} x^{4} \sqrt {-e^{2} x^{2}+d^{2}}}{7 e^{2}}-\frac {2 d \,x^{5} \sqrt {-e^{2} x^{2}+d^{2}}}{3 e}+\frac {x^{6} \sqrt {-e^{2} x^{2}+d^{2}}}{7} \]
command
integrate(x^5*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {65}{4} \, d^{7} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-6\right )} \mathrm {sgn}\left (d\right ) - \frac {16 \, d^{7} e^{\left (-6\right )}}{\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1} + \frac {1}{84} \, {\left (1472 \, d^{6} e^{\left (-6\right )} - {\left (693 \, d^{5} e^{\left (-5\right )} - 2 \, {\left (200 \, d^{4} e^{\left (-4\right )} - {\left (119 \, d^{3} e^{\left (-3\right )} - 2 \, {\left (33 \, d^{2} e^{\left (-2\right )} - {\left (14 \, d e^{\left (-1\right )} - 3 \, x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________