\[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^5} \, dx \]
Optimal antiderivative \[ -\frac {35 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 e}-\frac {14 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{e \left (e x +d \right )^{2}}-\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{e \left (e x +d \right )^{4}}-\frac {35 d^{3} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e}-\frac {35 d x \sqrt {-e^{2} x^{2}+d^{2}}}{2} \]
command
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^5,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (840 \, d^{4} \arctan \left (\sqrt {\frac {2 \, d}{x e + d} - 1}\right ) e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 384 \, d^{4} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - \frac {{\left (87 \, d^{4} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 136 \, d^{4} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 57 \, d^{4} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} {\left (x e + d\right )}^{3}}{d^{3}}\right )} e^{\left (-5\right )}}{24 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________