\[ -h(y(x))+(1-y(x)) y''(x)-3 (1-2 y(x)) y'(x)^2=0 \] ✓ Mathematica : cpu = 0.681475 (sec), leaf count = 168
DSolve[-h[y[x]] - 3*(1 - 2*y[x])*Derivative[1][y][x]^2 + (1 - y[x])*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{\frac {1}{2} (12-12 K[2])}}{(K[2]-1)^3 \sqrt {c_1+2 \int _1^{K[2]}-\frac {\exp (-2 (6 (K[1]-1)+3 \log (K[1]-1))) h(K[1])}{K[1]-1}dK[1]}}dK[2]\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{\frac {1}{2} (12-12 K[3])}}{(K[3]-1)^3 \sqrt {c_1+2 \int _1^{K[3]}-\frac {\exp (-2 (6 (K[1]-1)+3 \log (K[1]-1))) h(K[1])}{K[1]-1}dK[1]}}dK[3]\& \right ][x+c_2]\right \}\right \}\] ✓ Maple : cpu = 0.289 (sec), leaf count = 90
dsolve((1-y(x))*diff(diff(y(x),x),x)-3*(1-2*y(x))*diff(y(x),x)^2-h(y(x))=0,y(x))
\[\int _{}^{y \left (x \right )}\frac {{\mathrm e}^{-6 \textit {\_b}}}{\sqrt {-2 \left (\int \frac {{\mathrm e}^{-12 \textit {\_b}} h \left (\textit {\_b} \right )}{\left (\textit {\_b} -1\right )^{7}}d \textit {\_b} \right )+c_{1}}\, \left (\textit {\_b} -1\right )^{3}}d \textit {\_b} -x -c_{2} = 0\]