\[ \int \frac {\left (A+B x+C x^2\right ) \sqrt {d^2-e^2 x^2}}{d+e x} \, dx \]
Optimal antiderivative \[ -\frac {C \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 e^{3}}+\frac {\left (-B e +C d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{2 e^{3} \left (e x +d \right )}+\frac {d \left (C \,d^{2}-e \left (-2 A e +B d \right )\right ) \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{3}}+\frac {\left (C \,d^{2}-e \left (-2 A e +B d \right )\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{3}} \]
command
integrate((C*x^2+B*x+A)*(-e^2*x^2+d^2)^(1/2)/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{2} \, {\left (C d^{3} - B d^{2} e + 2 \, A d e^{2}\right )} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} \mathrm {sgn}\left (d\right ) + \frac {1}{6} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left ({\left (2 \, C x e^{\left (-1\right )} - 3 \, {\left (C d e^{3} - B e^{4}\right )} e^{\left (-5\right )}\right )} x + 2 \, {\left (2 \, C d^{2} e^{2} - 3 \, B d e^{3} + 3 \, A e^{4}\right )} e^{\left (-5\right )}\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________