\[ \int \frac {1}{(d+e x)^5 \sqrt {d^2-e^2 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{9 d e \left (e x +d \right )^{5}}-\frac {4 \sqrt {-e^{2} x^{2}+d^{2}}}{63 d^{2} e \left (e x +d \right )^{4}}-\frac {4 \sqrt {-e^{2} x^{2}+d^{2}}}{105 d^{3} e \left (e x +d \right )^{3}}-\frac {8 \sqrt {-e^{2} x^{2}+d^{2}}}{315 d^{4} e \left (e x +d \right )^{2}}-\frac {8 \sqrt {-e^{2} x^{2}+d^{2}}}{315 d^{5} e \left (e x +d \right )} \]
command
integrate(1/(e*x+d)^5/(-e^2*x^2+d^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{5040} \, {\left (-\frac {128 i \, \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{5}} + \frac {35 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {9}{2}} + 180 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} + 378 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} + 420 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} + 315 \, \sqrt {\frac {2 \, d}{x e + d} - 1}}{d^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________