\[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^2} \, dx \]
Optimal antiderivative \[ \frac {7 d^{2} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{24}+\frac {7 d \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{30 e}+\frac {\left (-e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6 e}+\frac {7 d^{6} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{16 e}+\frac {7 d^{4} x \sqrt {-e^{2} x^{2}+d^{2}}}{16} \]
command
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (6720 \, d^{7} \arctan \left (\sqrt {\frac {2 \, d}{x e + d} - 1}\right ) e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + \frac {{\left (105 \, d^{7} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {11}{2}} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 595 \, d^{7} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {9}{2}} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 1686 \, d^{7} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 1386 \, d^{7} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 595 \, d^{7} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 105 \, d^{7} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} {\left (x e + d\right )}^{6}}{d^{6}}\right )} e^{\left (-8\right )}}{7680 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________