\[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx \]
Optimal antiderivative \[ \frac {d^{5} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{64 e^{4}}+\frac {4 d^{2} x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{63 e^{3}}-\frac {d \,x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{8 e^{2}}+\frac {x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{9 e}+\frac {d^{3} \left (-315 e x +128 d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5040 e^{5}}+\frac {3 d^{9} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{128 e^{5}}+\frac {3 d^{7} x \sqrt {-e^{2} x^{2}+d^{2}}}{128 e^{4}} \]
command
integrate(x^4*(-e^2*x^2+d^2)^(5/2)/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {3}{128} \, d^{9} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} \mathrm {sgn}\left (d\right ) + \frac {1}{40320} \, {\left (1024 \, d^{8} e^{\left (-5\right )} - {\left (945 \, d^{7} e^{\left (-4\right )} - 2 \, {\left (256 \, d^{6} e^{\left (-3\right )} - {\left (315 \, d^{5} e^{\left (-2\right )} - 4 \, {\left (48 \, d^{4} e^{\left (-1\right )} + 5 \, {\left (189 \, d^{3} - 2 \, {\left (80 \, d^{2} e - 7 \, {\left (8 \, x e^{3} - 9 \, d e^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________