\[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx \]
Optimal antiderivative \[ -\frac {239 d^{6} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{16 e^{5}}-\frac {d^{3} \left (-e x +d \right )^{4}}{e^{5} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {337 d^{5} \sqrt {-e^{2} x^{2}+d^{2}}}{15 e^{5}}+\frac {175 d^{4} x \sqrt {-e^{2} x^{2}+d^{2}}}{16 e^{4}}-\frac {71 d^{3} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{15 e^{3}}+\frac {47 d^{2} x^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{24 e^{2}}-\frac {4 d \,x^{4} \sqrt {-e^{2} x^{2}+d^{2}}}{5 e}+\frac {x^{5} \sqrt {-e^{2} x^{2}+d^{2}}}{6} \]
command
integrate(x^4*(-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
\[ -\frac {239}{16} \, d^{6} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} \mathrm {sgn}\left (d\right ) + \frac {16 \, d^{6} e^{\left (-5\right )}}{\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1} - \frac {1}{240} \, {\left (3712 \, d^{5} e^{\left (-5\right )} - {\left (1665 \, d^{4} e^{\left (-4\right )} - 2 \, {\left (448 \, d^{3} e^{\left (-3\right )} - {\left (235 \, d^{2} e^{\left (-2\right )} - 4 \, {\left (24 \, d e^{\left (-1\right )} - 5 \, x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________