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