\[ \boxed { ay \left ( x \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +b \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+{\it c4}\, \left ( y \left ( x \right ) \right ) ^{4}+{\it c3}\, \left ( y \left ( x \right ) \right ) ^{3}+{\it c2}\, \left ( y \left ( x \right ) \right ) ^{2}+{\it c1}\,y \left ( x \right ) +{\it c0}=0} \]
Mathematica: cpu = 104.001206 (sec), leaf count = 46 \[ \text {DSolve}\left [a y(x) y''(x)+b y'(x)^2+\text {c0}+\text {c1} y(x)+\text {c2} y(x)^2+\text {c3} y(x)^3+\text {c4} y(x)^4=0,y(x),x\right ] \]
Maple: cpu = 1.810 (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 ( 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 \_a}}^{2\,{\frac {b}{a}}}{a}^{4}{ \it c0}+4\,{{\it \_a}}^{2\,{\frac {b}{a}}}{b}^{4}{\it c0}-4\,{\it \_C1 }\,{b}^{5}-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}+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}+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}+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} \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 ( 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 \_a} }^{2\,{\frac {b}{a}}}{a}^{4}{\it c0}+4\,{{\it \_a}}^{2\,{\frac {b}{a}} }{b}^{4}{\it c0}-4\,{\it \_C1}\,{b}^{5}-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}+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}+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}+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} \right ) }}}}{d{\it \_a}}-x-{\it \_C2}=0 \right \} \]