\[ a y'(x)^2+b y(x)^2 y'(x)+c y(x)^4+y(x) y''(x)=0 \] ✓ Mathematica : cpu = 74.0185 (sec), leaf count = 105
DSolve[c*y[x]^4 + b*y[x]^2*Derivative[1][y][x] + a*Derivative[1][y][x]^2 + y[x]*Derivative[2][y][x] == 0,y[x],x]
\[\text {Solve}\left [\int _1^{y(x)}\frac {1}{K[2]^2 \text {InverseFunction}\left [\frac {\log (c+\text {$\#$1} (b+(a+2) \text {$\#$1}))-\frac {2 b \arctan \left (\frac {b+2 (a+2) \text {$\#$1}}{\sqrt {4 (a+2) c-b^2}}\right )}{\sqrt {4 (a+2) c-b^2}}}{2 (a+2)}\& \right ][c_1-\log (K[2])]}dK[2]=x-c_2,y(x)\right ]\] ✓ Maple : cpu = 0.572 (sec), leaf count = 155
dsolve(diff(diff(y(x),x),x)*y(x)+a*diff(y(x),x)^2+b*y(x)^2*diff(y(x),x)+c*y(x)^4=0,y(x))
\[\int _{}^{y \left (x \right )}\frac {2 a +4}{\tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} \,\textit {\_a}^{2} b -2 a \ln \left (\textit {\_a} \right ) \sqrt {\textit {\_a}^{4} \left (4 a c -b^{2}+8 c \right )}-\ln \left (\frac {\textit {\_a}^{4} \left (\tan \left (\textit {\_Z} \right )^{2}+1\right ) \left (4 a c -b^{2}+8 c \right )}{4 a +8}\right ) \sqrt {\textit {\_a}^{4} \left (4 a c -b^{2}+8 c \right )}+c_{1} \sqrt {\textit {\_a}^{4} \left (4 a c -b^{2}+8 c \right )}\right )\right ) \sqrt {\textit {\_a}^{4} \left (4 \left (a +2\right ) c -b^{2}\right )}-b \,\textit {\_a}^{2}}d \textit {\_a} -x -c_{2} = 0\]