\[ \int \frac {\left (A+B x+C x^2\right ) \sqrt {d^2-e^2 x^2}}{(d+e x)^3} \, dx \]
Optimal antiderivative \[ -\frac {\left (A \,e^{2}-B d e +C \,d^{2}\right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 d \,e^{3} \left (e x +d \right )^{3}}-\frac {C \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{e^{3} \left (e x +d \right )^{2}}+\frac {\left (-B e +3 C d \right ) \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{3}}+\frac {2 \left (-B e +3 C d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{e^{3} \left (e x +d \right )} \]
command
integrate((C*x^2+B*x+A)*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ {\left (3 \, C d - B e\right )} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} \mathrm {sgn}\left (d\right ) + \sqrt {-x^{2} e^{2} + d^{2}} C e^{\left (-3\right )} - \frac {2 \, {\left (\frac {24 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} C d^{2} e^{\left (-2\right )}}{x} + \frac {9 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} C d^{2} e^{\left (-4\right )}}{x^{2}} + 11 \, C d^{2} - 5 \, B d e - \frac {12 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} B d e^{\left (-1\right )}}{x} - \frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} B d e^{\left (-3\right )}}{x^{2}} - A e^{2} - \frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} A e^{\left (-2\right )}}{x^{2}}\right )} e^{\left (-3\right )}}{3 \, d {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________