\[ \boxed { ay \left ( x \right ) \left ( -1+y \left ( x \right ) \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) + \left ( by \left ( x \right ) +c \right ) \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+h \left ( y \left ( x \right ) \right ) =0} \]
Mathematica: cpu = 25.014176 (sec), leaf count = 222 \[ \left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}} -\frac {K[2]^{-\frac {c}{a}} (1-K[2])^{\frac {1}{2} \left (\frac {2 b}{a}+\frac {2 c}{a}\right )}}{\sqrt {2 \int _1^{K[2]} -\frac {h(K[1]) \exp \left (-\frac {2 (c \log (K[1])-(b+c) \log (1-K[1]))}{a}\right )}{a (K[1]-1) K[1]} \, dK[1]+c_1}} \, dK[2]\& \right ]\left [c_2+x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}} \frac {K[3]^{-\frac {c}{a}} (1-K[3])^{\frac {1}{2} \left (\frac {2 b}{a}+\frac {2 c}{a}\right )}}{\sqrt {2 \int _1^{K[3]} -\frac {h(K[1]) \exp \left (-\frac {2 (c \log (K[1])-(b+c) \log (1-K[1]))}{a}\right )}{a (K[1]-1) K[1]} \, dK[1]+c_1}} \, dK[3]\& \right ]\left [c_2+x\right ]\right \}\right \} \]
Maple: cpu = 0.562 (sec), leaf count = 192 \[ \left \{ \int ^{y \left ( x \right ) }\!{a{\frac {1}{\sqrt {-a \left ( -{ \it \_C1}\,a+2\,\int \!{\frac {h \left ( {\it \_b} \right ) }{{\it \_b} \, \left ( {\it \_b}-1 \right ) } \left ( \left ( {\it \_b}-1 \right ) ^{{ \frac {b}{a}}} \right ) ^{2} \left ( \left ( {\it \_b}-1 \right ) ^{{ \frac {c}{a}}} \right ) ^{2} \left ( {{\it \_b}}^{{\frac {c}{a}}} \right ) ^{-2}}\,{\rm d}{\it \_b} \right ) }}} \left ( {{\it \_b}}^{{ \frac {c}{a}}} \right ) ^{-1} \left ( \left ( {\it \_b}-1 \right ) ^{-{ \frac {c+b}{a}}} \right ) ^{-1}}{d{\it \_b}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!-{a{\frac {1}{\sqrt {-a \left ( -{\it \_C1}\,a+2\, \int \!{\frac {h \left ( {\it \_b} \right ) }{{\it \_b}\, \left ( {\it \_b}-1 \right ) } \left ( \left ( {\it \_b}-1 \right ) ^{{\frac {b}{a}}} \right ) ^{2} \left ( \left ( {\it \_b}-1 \right ) ^{{\frac {c}{a}}} \right ) ^{2} \left ( {{\it \_b}}^{{\frac {c}{a}}} \right ) ^{-2}} \,{\rm d}{\it \_b} \right ) }}} \left ( {{\it \_b}}^{{\frac {c}{a}}} \right ) ^{-1} \left ( \left ( {\it \_b}-1 \right ) ^{-{\frac {c+b}{a}}} \right ) ^{-1}}{d{\it \_b}}-x-{\it \_C2}=0 \right \} \]