\[ \boxed { 2\,y \left ( x \right ) \left ( 1-y \left ( x \right ) \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) - \left ( 1-3\,y \left ( x \right ) \right ) \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+h \left ( y \left ( x \right ) \right ) =0} \]
Mathematica: cpu = 36.216099 (sec), leaf count = 166 \[ \left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}} -\frac {1}{(K[2]-1) \sqrt {K[2]} \sqrt {2 \int _1^{K[2]} \frac {h(K[1]) e^{-2 \left (\log (1-K[1])+\frac {1}{2} \log (K[1])\right )}}{2 (K[1]-1) K[1]} \, dK[1]+c_1}} \, dK[2]\& \right ]\left [c_2+x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}} \frac {1}{(K[3]-1) \sqrt {K[3]} \sqrt {2 \int _1^{K[3]} \frac {h(K[1]) e^{-2 \left (\log (1-K[1])+\frac {1}{2} \log (K[1])\right )}}{2 (K[1]-1) K[1]} \, dK[1]+c_1}} \, dK[3]\& \right ]\left [c_2+x\right ]\right \}\right \} \]
Maple: cpu = 0.187 (sec), leaf count = 84 \[ \left \{ \int ^{y \left ( x \right ) }\!{\frac {1}{{\it \_b}-1}{\frac {1 }{\sqrt {\int \!{\frac {h \left ( {\it \_b} \right ) }{{{\it \_b}}^{2} \left ( {\it \_b}-1 \right ) ^{3}}}\,{\rm d}{\it \_b}{\it \_b}+{\it \_C1}\,{\it \_b}}}}}{d{\it \_b}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!-{\frac {1}{{\it \_b}-1}{\frac {1}{\sqrt {\int \!{\frac {h \left ( {\it \_b} \right ) }{{{\it \_b}}^{2} \left ( {\it \_b}-1 \right ) ^{3}}}\,{\rm d}{\it \_b}{\it \_b}+{\it \_C1}\,{\it \_b}}}}}{d {\it \_b}}-x-{\it \_C2}=0 \right \} \]