\[ \int \frac {x^5 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {d^{4} \left (e x +d \right )^{2}}{5 e^{6} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {22 d^{3} \left (e x +d \right )}{15 e^{6} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {2 d \arctan \! \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{6}}+\frac {2 d \left (23 e x +30 d \right )}{15 e^{6} \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{e^{6}} \]
command
integrate(x^5*(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
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ -2 \, d \arcsin \left (\frac {x e}{d}\right ) e^{\left (-6\right )} \mathrm {sgn}\left (d\right ) - \frac {{\left (56 \, d^{6} e^{\left (-6\right )} + {\left (30 \, d^{5} e^{\left (-5\right )} - {\left (140 \, d^{4} e^{\left (-4\right )} + {\left (70 \, d^{3} e^{\left (-3\right )} - {\left (105 \, d^{2} e^{\left (-2\right )} + {\left (46 \, d e^{\left (-1\right )} - 15 \, x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]