\[ \int \frac {x^5 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {x^{4} \left (e x +d \right )}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {x^{2} \left (5 e x +4 d \right )}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\arctan \! \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{6}}+\frac {15 e x +8 d}{15 e^{6} \sqrt {-e^{2} x^{2}+d^{2}}} \]
command
integrate(x^5*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ -\arcsin \left (\frac {x e}{d}\right ) e^{\left (-6\right )} \mathrm {sgn}\left (d\right ) - \frac {{\left (8 \, d^{5} e^{\left (-6\right )} + {\left (15 \, d^{4} e^{\left (-5\right )} - {\left (20 \, d^{3} e^{\left (-4\right )} + {\left (35 \, d^{2} e^{\left (-3\right )} - {\left (23 \, x e^{\left (-1\right )} + 15 \, d e^{\left (-2\right )}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]