\[ -h(y(x))+3 (1-y(x)) y(x) y''(x)-2 (1-2 y(x)) y'(x)^2=0 \] ✓ Mathematica : cpu = 0.592316 (sec), leaf count = 186
DSolve[-h[y[x]] - 2*(1 - 2*y[x])*Derivative[1][y][x]^2 + 3*(1 - y[x])*y[x]*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{(1-K[2])^{2/3} K[2]^{2/3} \sqrt {c_1+2 \int _1^{K[2]}-\frac {\exp \left (-2 \left (\frac {2}{3} \log (1-K[1])+\frac {2}{3} \log (K[1])\right )\right ) h(K[1])}{3 (K[1]-1) K[1]}dK[1]}}dK[2]\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(1-K[3])^{2/3} K[3]^{2/3} \sqrt {c_1+2 \int _1^{K[3]}-\frac {\exp \left (-2 \left (\frac {2}{3} \log (1-K[1])+\frac {2}{3} \log (K[1])\right )\right ) h(K[1])}{3 (K[1]-1) K[1]}dK[1]}}dK[3]\& \right ][x+c_2]\right \}\right \}\] ✓ Maple : cpu = 0.253 (sec), leaf count = 119
dsolve(3*y(x)*(1-y(x))*diff(diff(y(x),x),x)-2*(1-2*y(x))*diff(y(x),x)^2-h(y(x))=0,y(x))
\[\int _{}^{y \left (x \right )}-\frac {\sqrt {9}}{3 \sqrt {\left (\textit {\_b} -1\right ) \left (c_{1}-\frac {2 \left (\int \frac {h \left (\textit {\_b} \right )}{\left (\textit {\_b}^{2}-\textit {\_b} \right )^{\frac {4}{3}} \textit {\_b} \left (\textit {\_b} -1\right )}d \textit {\_b} \right )}{3}\right ) \textit {\_b} \left (\textit {\_b} \left (\textit {\_b} -1\right )\right )^{\frac {1}{3}}}}d \textit {\_b} -x -c_{2} = 0\]