\[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{d+e x} \, dx \]
Optimal antiderivative \[ \frac {5 d^{3} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{24}+\frac {d x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e}+\frac {5 d^{7} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{16 e}+\frac {5 d^{5} x \sqrt {-e^{2} x^{2}+d^{2}}}{16} \]
command
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {5}{16} \, d^{7} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\left (d\right ) + \frac {1}{336} \, {\left (48 \, d^{6} e^{\left (-1\right )} + {\left (231 \, d^{5} - 2 \, {\left (72 \, d^{4} e + {\left (91 \, d^{3} e^{2} - 4 \, {\left (18 \, d^{2} e^{3} - {\left (6 \, x e^{5} - 7 \, d e^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________