\[ \boxed { 2\, \left ( {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) \right ) y \left ( x \right ) -6\, \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+ \left ( 1+a \left ( y \left ( x \right ) \right ) ^{3} \right ) \left ( y \left ( x \right ) \right ) ^{2}=0} \]
Mathematica: cpu = 18.186309 (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 = 1.638 (sec), leaf count = 71 \[ \left \{ \int ^{y \left ( x \right ) }\!-2\,{\frac {1}{\sqrt {4\,{\it \_C1}\,{{\it \_a}}^{4}+4\,a{{\it \_a}}^{3}+1}{\it \_a}}}{d{\it \_a}}-x -{\it \_C2}=0,\int ^{y \left ( x \right ) }\!2\,{\frac {1}{\sqrt {4\,{ \it \_C1}\,{{\it \_a}}^{4}+4\,a{{\it \_a}}^{3}+1}{\it \_a}}}{d{\it \_a }}-x-{\it \_C2}=0 \right \} \]