\[ \boxed { 2\, \left ( y \left ( x \right ) \right ) ^{3}{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) + \left ( y \left ( x \right ) \right ) ^{2} \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}-a{x}^{2}-bx-c=0} \]
Mathematica: cpu = 0.357545 (sec), leaf count = 41 \[ \text {DSolve}\left [-a x^2-b x-c+2 y(x)^3 y''(x)+y(x)^2 y'(x)^2=0,y(x),x\right ] \]
Maple: cpu = 1.903 (sec), leaf count = 0 \[ \text {could not solve} \]