\[ h(y(x))+2 (1-y(x)) y(x) y''(x)-\left ((1-3 y(x)) y'(x)^2\right )=0 \] ✓ Mathematica : cpu = 0.595617 (sec), leaf count = 170
DSolve[h[y[x]] - (1 - 3*y[x])*Derivative[1][y][x]^2 + 2*(1 - y[x])*y[x]*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{(K[2]-1) \sqrt {K[2]} \sqrt {c_1+2 \int _1^{K[2]}\frac {e^{-2 \left (\log (1-K[1])+\frac {1}{2} \log (K[1])\right )} h(K[1])}{2 (K[1]-1) K[1]}dK[1]}}dK[2]\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[3]-1) \sqrt {K[3]} \sqrt {c_1+2 \int _1^{K[3]}\frac {e^{-2 \left (\log (1-K[1])+\frac {1}{2} \log (K[1])\right )} h(K[1])}{2 (K[1]-1) K[1]}dK[1]}}dK[3]\& \right ][x+c_2]\right \}\right \}\] ✓ Maple : cpu = 0.244 (sec), leaf count = 80
dsolve(2*y(x)*(1-y(x))*diff(diff(y(x),x),x)-(1-3*y(x))*diff(y(x),x)^2+h(y(x))=0,y(x))
\[\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_b} \left (c_{1}+\int \frac {h \left (\textit {\_b} \right )}{\left (\textit {\_b} -1\right )^{3} \textit {\_b}^{2}}d \textit {\_b} \right )}\, \left (\textit {\_b} -1\right )}d \textit {\_b} -x -c_{2} = 0\]