\[ a y(x) y''(x)+b y'(x)^2+\text {c0}+\text {c1} y(x)+\text {c2} y(x)^2+\text {c3} y(x)^3+\text {c4} y(x)^4=0 \] ✗ Mathematica : cpu = 103.961 (sec), leaf count = 0 , could not solve
DSolve[c0 + c1*y[x] + c2*y[x]^2 + c3*y[x]^3 + c4*y[x]^4 + b*Derivative[1][y][x]^2 + a*y[x]*Derivative[2][y][x] == 0, y[x], x]
✓ Maple : cpu = 0.581 (sec), leaf count = 1028
\[ \left \{ \int ^{y \left ( x \right ) }\!{b \left ( 6\,{a}^{4}+25\,{a}^{3}b+35\,{a}^{2}{b}^{2}+20\,a{b}^{3}+4\,{b}^{4} \right ) {{\it \_a}}^{2\,{\frac {b}{a}}}{\frac {1}{\sqrt {-{{\it \_a}}^{2\,{\frac {b}{a}}}b \left ( 6\,{a}^{4}+25\,{a}^{3}b+35\,{a}^{2}{b}^{2}+20\,a{b}^{3}+4\,{b}^{4} \right ) \left ( 25\,{{\it \_a}}^{2\,{\frac {b}{a}}}{a}^{3}b{\it c0}+35\,{{\it \_a}}^{2\,{\frac {b}{a}}}{a}^{2}{b}^{2}{\it c0}+20\,{{\it \_a}}^{2\,{\frac {b}{a}}}a{b}^{3}{\it c0}+3\,{{\it \_a}}^{2\,{\frac {2\,a+b}{a}}}{a}^{3}b{\it c4}+11\,{{\it \_a}}^{2\,{\frac {2\,a+b}{a}}}{a}^{2}{b}^{2}{\it c4}+6\,{{\it \_a}}^{2\,{\frac {b}{a}}}{a}^{4}{\it c0}+4\,{{\it \_a}}^{2\,{\frac {b}{a}}}{b}^{4}{\it c0}+4\,{{\it \_a}}^{{\frac {a+2\,b}{a}}}{b}^{4}{\it c1}+4\,{{\it \_a}}^{2\,{\frac {a+b}{a}}}{b}^{4}{\it c2}+4\,{{\it \_a}}^{{\frac {3\,a+2\,b}{a}}}{b}^{4}{\it c3}+4\,{{\it \_a}}^{2\,{\frac {2\,a+b}{a}}}{b}^{4}{\it c4}-6\,{\it \_C1}\,{a}^{4}b-25\,{\it \_C1}\,{a}^{3}{b}^{2}-35\,{\it \_C1}\,{a}^{2}{b}^{3}-20\,{\it \_C1}\,a{b}^{4}+12\,{{\it \_a}}^{2\,{\frac {2\,a+b}{a}}}a{b}^{3}{\it c4}+4\,{{\it \_a}}^{{\frac {3\,a+2\,b}{a}}}{a}^{3}b{\it c3}+14\,{{\it \_a}}^{{\frac {3\,a+2\,b}{a}}}{a}^{2}{b}^{2}{\it c3}+14\,{{\it \_a}}^{{\frac {3\,a+2\,b}{a}}}a{b}^{3}{\it c3}+6\,{{\it \_a}}^{2\,{\frac {a+b}{a}}}{a}^{3}b{\it c2}+19\,{{\it \_a}}^{2\,{\frac {a+b}{a}}}{a}^{2}{b}^{2}{\it c2}+16\,{{\it \_a}}^{2\,{\frac {a+b}{a}}}a{b}^{3}{\it c2}+12\,{{\it \_a}}^{{\frac {a+2\,b}{a}}}{a}^{3}b{\it c1}+26\,{{\it \_a}}^{{\frac {a+2\,b}{a}}}{a}^{2}{b}^{2}{\it c1}+18\,{{\it \_a}}^{{\frac {a+2\,b}{a}}}a{b}^{3}{\it c1}-4\,{\it \_C1}\,{b}^{5} \right ) }}}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!-{b \left ( 6\,{a}^{4}+25\,{a}^{3}b+35\,{a}^{2}{b}^{2}+20\,a{b}^{3}+4\,{b}^{4} \right ) {{\it \_a}}^{2\,{\frac {b}{a}}}{\frac {1}{\sqrt {-{{\it \_a}}^{2\,{\frac {b}{a}}}b \left ( 6\,{a}^{4}+25\,{a}^{3}b+35\,{a}^{2}{b}^{2}+20\,a{b}^{3}+4\,{b}^{4} \right ) \left ( 25\,{{\it \_a}}^{2\,{\frac {b}{a}}}{a}^{3}b{\it c0}+35\,{{\it \_a}}^{2\,{\frac {b}{a}}}{a}^{2}{b}^{2}{\it c0}+20\,{{\it \_a}}^{2\,{\frac {b}{a}}}a{b}^{3}{\it c0}+3\,{{\it \_a}}^{2\,{\frac {2\,a+b}{a}}}{a}^{3}b{\it c4}+11\,{{\it \_a}}^{2\,{\frac {2\,a+b}{a}}}{a}^{2}{b}^{2}{\it c4}+6\,{{\it \_a}}^{2\,{\frac {b}{a}}}{a}^{4}{\it c0}+4\,{{\it \_a}}^{2\,{\frac {b}{a}}}{b}^{4}{\it c0}+4\,{{\it \_a}}^{{\frac {a+2\,b}{a}}}{b}^{4}{\it c1}+4\,{{\it \_a}}^{2\,{\frac {a+b}{a}}}{b}^{4}{\it c2}+4\,{{\it \_a}}^{{\frac {3\,a+2\,b}{a}}}{b}^{4}{\it c3}+4\,{{\it \_a}}^{2\,{\frac {2\,a+b}{a}}}{b}^{4}{\it c4}-6\,{\it \_C1}\,{a}^{4}b-25\,{\it \_C1}\,{a}^{3}{b}^{2}-35\,{\it \_C1}\,{a}^{2}{b}^{3}-20\,{\it \_C1}\,a{b}^{4}+12\,{{\it \_a}}^{2\,{\frac {2\,a+b}{a}}}a{b}^{3}{\it c4}+4\,{{\it \_a}}^{{\frac {3\,a+2\,b}{a}}}{a}^{3}b{\it c3}+14\,{{\it \_a}}^{{\frac {3\,a+2\,b}{a}}}{a}^{2}{b}^{2}{\it c3}+14\,{{\it \_a}}^{{\frac {3\,a+2\,b}{a}}}a{b}^{3}{\it c3}+6\,{{\it \_a}}^{2\,{\frac {a+b}{a}}}{a}^{3}b{\it c2}+19\,{{\it \_a}}^{2\,{\frac {a+b}{a}}}{a}^{2}{b}^{2}{\it c2}+16\,{{\it \_a}}^{2\,{\frac {a+b}{a}}}a{b}^{3}{\it c2}+12\,{{\it \_a}}^{{\frac {a+2\,b}{a}}}{a}^{3}b{\it c1}+26\,{{\it \_a}}^{{\frac {a+2\,b}{a}}}{a}^{2}{b}^{2}{\it c1}+18\,{{\it \_a}}^{{\frac {a+2\,b}{a}}}a{b}^{3}{\it c1}-4\,{\it \_C1}\,{b}^{5} \right ) }}}}{d{\it \_a}}-x-{\it \_C2}=0 \right \} \]