\[ \int \frac {x (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (e x +d \right )^{2}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 \left (e x +d \right )}{15 d \,e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {4 x}{15 d^{3} e \sqrt {-e^{2} x^{2}+d^{2}}} \]
command
integrate(x*(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
\[ \frac {{\left ({\left (2 \, x {\left (\frac {2 \, x^{2} e^{3}}{d^{3}} - \frac {5 \, e}{d}\right )} - 5\right )} x^{2} - d^{2} e^{\left (-2\right )}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]