\[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx \]
Optimal antiderivative \[ \frac {27 d^{5} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{4}}+\frac {d^{2} \left (-e x +d \right )^{4}}{e^{4} \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {101 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}{5 e^{4}}-\frac {19 d^{3} x \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{3}}+\frac {18 d^{2} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{5 e^{2}}-\frac {d \,x^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{e}+\frac {x^{4} \sqrt {-e^{2} x^{2}+d^{2}}}{5} \]
command
integrate(x^3*(-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
\[ \frac {27}{2} \, d^{5} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-4\right )} \mathrm {sgn}\left (d\right ) - \frac {16 \, d^{5} e^{\left (-4\right )}}{\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1} + \frac {1}{10} \, {\left (132 \, d^{4} e^{\left (-4\right )} - {\left (55 \, d^{3} e^{\left (-3\right )} - 2 \, {\left (13 \, d^{2} e^{\left (-2\right )} - {\left (5 \, d e^{\left (-1\right )} - x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________