\[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {2 x}{15 d^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {1}{9 d e \left (e x +d \right )^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {1}{9 d^{2} e \left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {8 x}{45 d^{6} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {16 x}{45 d^{8} \sqrt {-e^{2} x^{2}+d^{2}}} \]
command
integrate(1/(e*x+d)^2/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{5760} \, {\left ({\left (\frac {3 \, {\left (315 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{2} + \frac {70 \, d}{x e + d} - 32\right )} e^{\left (-7\right )}}{d^{8} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {{\left (5 \, d^{64} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {9}{2}} e^{56} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{8} + 45 \, d^{64} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} e^{56} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{8} + 189 \, d^{64} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{56} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{8} + 525 \, d^{64} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{56} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{8} + 1575 \, d^{64} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{56} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{8}\right )} e^{\left (-63\right )}}{d^{72} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{9}}\right )} e^{7} + \frac {2048 i \, \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{8}}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________