\[ \int \frac {x^4 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {d^{3} \left (e x +d \right )^{2}}{5 e^{5} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {17 d^{2} \left (e x +d \right )}{15 e^{5} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\arctan \! \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{5}}+\frac {\frac {26 e x}{15}+2 d}{e^{5} \sqrt {-e^{2} x^{2}+d^{2}}} \]
command
integrate(x^4*(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
\[ -\arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} \mathrm {sgn}\left (d\right ) - \frac {{\left (16 \, d^{5} e^{\left (-5\right )} + {\left (15 \, d^{4} e^{\left (-4\right )} - {\left (40 \, d^{3} e^{\left (-3\right )} + {\left (35 \, d^{2} e^{\left (-2\right )} - 2 \, {\left (15 \, d e^{\left (-1\right )} + 13 \, x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]