\[ \boxed { 3\,y \left ( x \right ) \left ( 1-y \left ( x \right ) \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) -2\, \left ( 1-2\,y \left ( x \right ) \right ) \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}-h \left ( y \left ( x \right ) \right ) =0} \]
Mathematica: cpu = 21.362213 (sec), leaf count = 182 \[ \left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}} -\frac {1}{(1-K[2])^{2/3} K[2]^{2/3} \sqrt {2 \int _1^{K[2]} -\frac {h(K[1]) \exp \left (-2 \left (\frac {2}{3} \log (1-K[1])+\frac {2}{3} \log (K[1])\right )\right )}{3 (K[1]-1) K[1]} \, dK[1]+c_1}} \, dK[2]\& \right ]\left [c_2+x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}} \frac {1}{(1-K[3])^{2/3} K[3]^{2/3} \sqrt {2 \int _1^{K[3]} -\frac {h(K[1]) \exp \left (-2 \left (\frac {2}{3} \log (1-K[1])+\frac {2}{3} \log (K[1])\right )\right )}{3 (K[1]-1) K[1]} \, dK[1]+c_1}} \, dK[3]\& \right ]\left [c_2+x\right ]\right \}\right \} \]
Maple: cpu = 0.171 (sec), leaf count = 219 \[ \left \{ \int ^{y \left ( x \right ) }\!-3\,{\frac {1}{\sqrt {-6\,{{\it \_b}}^{2}\sqrt [3]{{\it \_b}\, \left ( {\it \_b}-1 \right ) }\int \!{ \frac {h \left ( {\it \_b} \right ) }{ \left ( {{\it \_b}}^{2}-{\it \_b} \right ) ^{4/3}{\it \_b}\, \left ( {\it \_b}-1 \right ) }}\,{\rm d}{\it \_b}+9\,{{\it \_b}}^{2}\sqrt [3]{{\it \_b}\, \left ( {\it \_b}-1 \right ) }{\it \_C1}+6\,{\it \_b}\,\sqrt [3]{{\it \_b}\, \left ( {\it \_b}-1 \right ) }\int \!{\frac {h \left ( {\it \_b} \right ) }{ \left ( {{ \it \_b}}^{2}-{\it \_b} \right ) ^{4/3}{\it \_b}\, \left ( {\it \_b}-1 \right ) }}\,{\rm d}{\it \_b}-9\,{\it \_b}\,\sqrt [3]{{\it \_b}\, \left ( {\it \_b}-1 \right ) }{\it \_C1}}}}{d{\it \_b}}-x-{\it \_C2}=0, \int ^{y \left ( x \right ) }\!3\,{\frac {1}{\sqrt {-6\,{{\it \_b}}^{2} \sqrt [3]{{\it \_b}\, \left ( {\it \_b}-1 \right ) }\int \!{\frac {h \left ( {\it \_b} \right ) }{ \left ( {{\it \_b}}^{2}-{\it \_b} \right ) ^{4/3}{\it \_b}\, \left ( {\it \_b}-1 \right ) }}\,{\rm d}{\it \_b}+9\,{ {\it \_b}}^{2}\sqrt [3]{{\it \_b}\, \left ( {\it \_b}-1 \right ) }{\it \_C1}+6\,{\it \_b}\,\sqrt [3]{{\it \_b}\, \left ( {\it \_b}-1 \right ) } \int \!{\frac {h \left ( {\it \_b} \right ) }{ \left ( {{\it \_b}}^{2}-{ \it \_b} \right ) ^{4/3}{\it \_b}\, \left ( {\it \_b}-1 \right ) }} \,{\rm d}{\it \_b}-9\,{\it \_b}\,\sqrt [3]{{\it \_b}\, \left ( {\it \_b }-1 \right ) }{\it \_C1}}}}{d{\it \_b}}-x-{\it \_C2}=0 \right \} \]