\[ \int \frac {\left (A+B x+C x^2\right ) \sqrt {d^2-e^2 x^2}}{(d+e x)^2} \, dx \]
Optimal antiderivative \[ -\frac {\left (A \,e^{2}-B d e +C \,d^{2}\right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{d \,e^{3} \left (e x +d \right )^{2}}-\frac {C \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{2 e^{3} \left (e x +d \right )}-\frac {\left (5 C \,d^{2}-2 e \left (-A e +2 B d \right )\right ) \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{3}}-\frac {\left (5 C \,d^{2}-2 e \left (-A e +2 B d \right )\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{2 d \,e^{3}} \]
command
integrate((C*x^2+B*x+A)*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (8 \, C d^{3} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{3} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 8 \, B d^{2} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 8 \, A d \sqrt {\frac {2 \, d}{x e + d} - 1} e^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 4 \, {\left (5 \, C d^{3} e^{3} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 4 \, B d^{2} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 2 \, A d e^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} \arctan \left (\sqrt {\frac {2 \, d}{x e + d} - 1}\right ) + \frac {{\left (5 \, C d^{3} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{3} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 2 \, B d^{2} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3 \, C d^{3} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{3} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 2 \, B d^{2} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} {\left (x e + d\right )}^{2}}{d^{2}}\right )} e^{\left (-6\right )}}{4 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________