\[ \int \frac {x^3 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {d^{2} \left (e x +d \right )^{2}}{5 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d \left (e x +d \right )}{5 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 e x +5 d}{5 d \,e^{4} \sqrt {-e^{2} x^{2}+d^{2}}} \]
command
integrate(x^3*(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
\[ -\frac {{\left (2 \, d^{4} e^{\left (-4\right )} + {\left (x^{2} {\left (\frac {2 \, x e}{d} + 5\right )} - 5 \, d^{2} e^{\left (-2\right )}\right )} x^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{5 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]