\[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx \]
Optimal antiderivative \[ \frac {x^{2} \left (e x +d \right )}{7 d e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}-\frac {2 \left (-2 e x +d \right )}{35 d \,e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 x}{105 d^{3} e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {8 x}{105 d^{5} e^{2} \sqrt {-e^{2} x^{2}+d^{2}}} \]
command
integrate(x^2*(e*x+d)/(-e^2*x^2+d^2)^(9/2),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
\[ \frac {{\left ({\left ({\left (4 \, x^{2} {\left (\frac {2 \, x^{2} e^{4}}{d^{5}} - \frac {7 \, e^{2}}{d^{3}}\right )} + \frac {35}{d}\right )} x + 21 \, e^{\left (-1\right )}\right )} x^{2} - 6 \, d^{2} e^{\left (-3\right )}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{105 \, {\left (x^{2} e^{2} - d^{2}\right )}^{4}} \]