\[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx \]
Optimal antiderivative \[ -\frac {95 d^{4} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 e^{3}}-\frac {d \left (-e x +d \right )^{4}}{e^{3} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {95 d^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{8 e^{3}}-\frac {95 d^{2} \left (-e x +d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{24 e^{3}}-\frac {19 d \left (-e x +d \right )^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{12 e^{3}}-\frac {\left (-e x +d \right )^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{4 e^{3}} \]
command
integrate(x^2*(-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
\[ -\frac {95}{8} \, d^{4} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} \mathrm {sgn}\left (d\right ) + \frac {16 \, d^{4} e^{\left (-3\right )}}{\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1} - \frac {1}{24} \, {\left (256 \, d^{3} e^{\left (-3\right )} - {\left (93 \, d^{2} e^{\left (-2\right )} - 2 \, {\left (16 \, d e^{\left (-1\right )} - 3 \, x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________