\[ a (y(x)-1) y(x) y''(x)+y'(x)^2 (b y(x)+c)+h(y(x))=0 \] ✓ Mathematica : cpu = 0.992295 (sec), leaf count = 226
DSolve[h[y[x]] + (c + b*y[x])*Derivative[1][y][x]^2 + a*(-1 + y[x])*y[x]*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {(1-K[2])^{\frac {1}{2} \left (\frac {2 b}{a}+\frac {2 c}{a}\right )} K[2]^{-\frac {c}{a}}}{\sqrt {c_1+2 \int _1^{K[2]}-\frac {\exp \left (-\frac {2 (c \log (K[1])-(b+c) \log (1-K[1]))}{a}\right ) h(K[1])}{a (K[1]-1) K[1]}dK[1]}}dK[2]\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {(1-K[3])^{\frac {1}{2} \left (\frac {2 b}{a}+\frac {2 c}{a}\right )} K[3]^{-\frac {c}{a}}}{\sqrt {c_1+2 \int _1^{K[3]}-\frac {\exp \left (-\frac {2 (c \log (K[1])-(b+c) \log (1-K[1]))}{a}\right ) h(K[1])}{a (K[1]-1) K[1]}dK[1]}}dK[3]\& \right ][x+c_2]\right \}\right \}\] ✓ Maple : cpu = 0.415 (sec), leaf count = 194
dsolve(a*y(x)*(-1+y(x))*diff(diff(y(x),x),x)+(b*y(x)+c)*diff(y(x),x)^2+h(y(x))=0,y(x))
\[\int _{}^{y \left (x \right )}\frac {a \,\textit {\_b}^{-\frac {c}{a}} \left (\textit {\_b} -1\right )^{-\frac {-b -c}{a}}}{\sqrt {a \left (c_{1} a -2 \left (\int \frac {\textit {\_b}^{-\frac {2 c}{a}} \left (\textit {\_b} -1\right )^{\frac {2 b}{a}} \left (\textit {\_b} -1\right )^{\frac {2 c}{a}} h \left (\textit {\_b} \right )}{\textit {\_b} \left (\textit {\_b} -1\right )}d \textit {\_b} \right )\right )}}d \textit {\_b} -x -c_{2} = 0\]