\[ \boxed { \left ( 1-y \left ( x \right ) \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) -3\, \left ( 1-2\,y \left ( x \right ) \right ) \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}-h \left ( y \left ( x \right ) \right ) =0} \]
Mathematica: cpu = 20.225068 (sec), leaf count = 164 \[ \left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}} -\frac {e^{\frac {1}{2} (12-12 K[2])}}{(K[2]-1)^3 \sqrt {2 \int _1^{K[2]} -\frac {h(K[1]) \exp (-2 (6 (K[1]-1)+3 \log (K[1]-1)))}{K[1]-1} \, dK[1]+c_1}} \, dK[2]\& \right ]\left [c_2+x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}} \frac {e^{\frac {1}{2} (12-12 K[3])}}{(K[3]-1)^3 \sqrt {2 \int _1^{K[3]} -\frac {h(K[1]) \exp (-2 (6 (K[1]-1)+3 \log (K[1]-1)))}{K[1]-1} \, dK[1]+c_1}} \, dK[3]\& \right ]\left [c_2+x\right ]\right \}\right \} \]
Maple: cpu = 0.249 (sec), leaf count = 90 \[ \left \{ \int ^{y \left ( x \right ) }\!{\frac {1}{ \left ( {\it \_b}-1 \right ) ^{3} \left ( {{\rm e}^{{\it \_b}}} \right ) ^{6}}{\frac {1}{ \sqrt {-2\,\int \!{\frac {h \left ( {\it \_b} \right ) }{ \left ( { {\rm e}^{{\it \_b}}} \right ) ^{12} \left ( {\it \_b}-1 \right ) ^{7}}} \,{\rm d}{\it \_b}+{\it \_C1}}}}}{d{\it \_b}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!-{\frac {1}{ \left ( {\it \_b}-1 \right ) ^{3} \left ( {{\rm e}^{{\it \_b}}} \right ) ^{6}}{\frac {1}{\sqrt {-2\,\int \!{\frac {h \left ( {\it \_b} \right ) }{ \left ( {{\rm e}^{{\it \_b}}} \right ) ^{12} \left ( {\it \_b}-1 \right ) ^{7}}}\,{\rm d}{\it \_b}+{ \it \_C1}}}}}{d{\it \_b}}-x-{\it \_C2}=0 \right \} \]