\[ -a x^2-b x-c+3 y(x) y''(x)-2 y'(x)^2=0 \] ✓ Mathematica : cpu = 0.124793 (sec), leaf count = 118
DSolve[-c - b*x - a*x^2 - 2*Derivative[1][y][x]^2 + 3*y[x]*Derivative[2][y][x] == 0,y[x],x]
\[\text {Solve}\left [\int \frac {y(x)^{2/3}}{\left (a x^2+b x+c\right ) \sqrt {-\frac {2 \left (a x^2+b x+c\right )^3}{y(x)^2}+\frac {c_1 \left (a x^2+b x+c\right )}{y(x)^{2/3}}+9 \left (b^2-4 a c\right )}}d\frac {a x^2+b x+c}{y(x)^{2/3}}=-\int \frac {1}{3 \left (a x^2+b x+c\right )}dx+c_2,y(x)\right ]\] ✓ Maple : cpu = 1.24 (sec), leaf count = 207
dsolve(3*diff(diff(y(x),x),x)*y(x)-2*diff(y(x),x)^2-a*x^2-b*x-c=0,y(x))
\[y \left (x \right ) = \RootOf \left (-2 \left (\int _{}^{\textit {\_Z}}\frac {b}{\sqrt {4 \textit {\_f}^{\frac {4}{3}} c_{1} b^{2}-36 c \,\textit {\_f}^{2} a +9 b^{2} \textit {\_f}^{2}-2}}d \textit {\_f} \right ) \sqrt {4 c a -b^{2}}+c_{2} \sqrt {4 c a -b^{2}}-2 b \arctan \left (\frac {2 a x +b}{\sqrt {4 c a -b^{2}}}\right )\right ) \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}}\]