\[ \boxed { 3\, \left ( {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) \right ) y \left ( x \right ) -2\, \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}-a{x}^{2}-bx-c=0} \]
Mathematica: cpu = 0.040505 (sec), leaf count = 36 \[ \text {DSolve}\left [-a x^2-b x-c+3 y(x) y''(x)-2 y'(x)^2=0,y(x),x\right ] \]
Maple: cpu = 1.903 (sec), leaf count = 207 \[ \left \{ y \left ( x \right ) ={\it RootOf} \left ( -2\,\arctan \left ( { \frac {2\,ax+b}{\sqrt {4\,ac-{b}^{2}}}} \right ) b-2\,\int ^{{\it \_Z}} \!{\frac {b}{\sqrt {4\,{{\it \_f}}^{4/3}{\it \_C1}\,{b}^{2}-36\,c{{ \it \_f}}^{2}a+9\,{b}^{2}{{\it \_f}}^{2}-2}}}{d{\it \_f}}\sqrt {4\,ac- {b}^{2}}+{\it \_C2}\,\sqrt {4\,ac-{b}^{2}} \right ) \left ( a{x}^{2}+bx +c \right ) ^{{\frac {3}{2}}},y \left ( x \right ) ={\it RootOf} \left ( - 2\,\arctan \left ( {\frac {2\,ax+b}{\sqrt {4\,ac-{b}^{2}}}} \right ) b+2 \,\int ^{{\it \_Z}}\!{\frac {b}{\sqrt {4\,{{\it \_f}}^{4/3}{\it \_C1} \,{b}^{2}-36\,c{{\it \_f}}^{2}a+9\,{b}^{2}{{\it \_f}}^{2}-2}}}{d{\it \_f}}\sqrt {4\,ac-{b}^{2}}+{\it \_C2}\,\sqrt {4\,ac-{b}^{2}} \right ) \left ( a{x}^{2}+bx+c \right ) ^{{\frac {3}{2}}} \right \} \]