\[ \int \frac {x^2 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {d \left (e x +d \right )^{2}}{5 e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {7 \left (e x +d \right )}{15 e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {x}{15 d^{2} e^{2} \sqrt {-e^{2} x^{2}+d^{2}}} \]
command
integrate(x^2*(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
\[ \frac {{\left (4 \, d^{3} e^{\left (-3\right )} - {\left (x {\left (\frac {x^{2} e^{2}}{d^{2}} + 5\right )} + 10 \, d e^{\left (-1\right )}\right )} x^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]