\[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^4} \, dx \]
Optimal antiderivative \[ \frac {35 d \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{12 e}+\frac {7 \left (-e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4 e}+\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{e \left (e x +d \right )^{3}}+\frac {35 d^{4} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 e}+\frac {35 d^{2} x \sqrt {-e^{2} x^{2}+d^{2}}}{8} \]
command
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {35}{8} \, d^{4} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\left (d\right ) + \frac {1}{24} \, {\left (160 \, d^{3} e^{\left (-1\right )} - {\left (81 \, d^{2} + 2 \, {\left (3 \, x e^{2} - 16 \, d e\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________