\[ y(x)^2 \left (a y(x)^3+1\right )+2 y(x) y''(x)-6 y'(x)^2=0 \] ✓ Mathematica : cpu = 20.5057 (sec), leaf count = 2761
\[\left \{\text {Solve}\left [-\frac {4 \left (F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right )}}\right )|-\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}\right ) \text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]+\Pi \left (\frac {\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ] \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ] \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )};\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right )}}\right )|-\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right )\right ) y(x) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right )}} \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ] \text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ] \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ){}^2 \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right ){}^2}} \sqrt {4 c_1 y(x)^6+4 a y(x)^5+y(x)^2}}=x+c_2,y(x)\right ],\text {Solve}\left [\frac {4 \left (F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right )}}\right )|-\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}\right ) \text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]+\Pi \left (\frac {\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ] \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ] \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )};\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right )}}\right )|-\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right )\right ) y(x) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right )}} \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ] \text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ] \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ){}^2 \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right ){}^2}} \sqrt {4 c_1 y(x)^6+4 a y(x)^5+y(x)^2}}=x+c_2,y(x)\right ]\right \}\] ✓ Maple : cpu = 0.937 (sec), leaf count = 71
\[\int _{}^{y \relax (x )}-\frac {2}{\sqrt {4 \textit {\_a}^{4} c_{1}+4 a \,\textit {\_a}^{3}+1}\, \textit {\_a}}d \textit {\_a} -x -c_{2} = 0\]