Nonlinear first order differential equation general solution

Given the equation:

$$\dfrac{d}{dx}y(x)+y(x)^2=ax^2+bx+c$$ one solution is: $$y(x)=\dfrac{1}{c_1+x}$$ Obviously, this is a trivial solution and this is what you can get from Wolfram Alpha. What is the most general solution of this equation? Thanks.

A way to write the general solution to the given first order inhomogeneous ordinary differential equation of Riccati type is in terms of parabolic cylinder functions, $D_\nu(x)$ is rational with numerator a sum of four weighted $D$ functions and denominator a sum of two weighted $D$ functions:

$$ y(x) = \\ \frac{-4 a^{3/4} C D_{\frac{b^2+4 a^{3/2}-4 a c}{8 a^{3/2}}}\left(\frac{b+2 a x}{\sqrt{2} a^{3/4}}\right)-\sqrt{2} (2 a x+b) D_{\frac{-b^2-4 a^{3/2}+4 a c}{8 a^{3/2}}}\left(\frac{\mathrm{i} (b+2 a x)}{\sqrt{2} a^{3/4}}\right)+\sqrt{2} C (2 a x+b) D_{\frac{b^2-4 a \left(c+\sqrt{a}\right)}{8 a^{3/2}}}\left(\frac{b+2 a x}{\sqrt{2} a^{3/4}}\right)-4 \mathrm{i} a^{3/4} D_{\frac{4 a \left(c+\sqrt{a}\right)-b^2}{8 a^{3/2}}}\left(\frac{\mathrm{i} (b+2 a x)}{\sqrt{2} a^{3/4}}\right)}{2 \sqrt{2a} \left(C D_{\frac{b^2-4 a \left(c+\sqrt{a}\right)}{8 a^{3/2}}}\left(\frac{b+2 a x}{\sqrt{2} a^{3/4}}\right)+D_{\frac{-b^2-4 a^{3/2}+4 a c}{8 a^{3/2}}}\left(\frac{\mathrm{i} (b+2 a x)}{\sqrt{2} a^{3/4}}\right)\right)} $$

Some of the complications in this expression are carefully balanced cancellations that occur when the roots of the right-hand side of the original equation are variously positive, zero, negative, or complex. If you have constraints on those coefficients, the complications might be reduced.