\[ \boxed { \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{3}+x \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}-y \left ( x \right ) =0} \]
Mathematica: cpu = 52.151622 (sec), leaf count = 1758 \[ \left \{\left \{y(x)\to \frac {1}{2} \left (\frac {4\ 2^{2/3} x^4}{3 \left (-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54\right ){}^{2/3}}+\frac {4 \sqrt [3]{2} x^3}{3 \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}+\frac {8\ 2^{2/3} x^3}{\left (-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54\right ){}^{2/3}}+\frac {6 \sqrt [3]{2} x^2}{\sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}+\frac {18\ 2^{2/3} x^2}{\left (-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54\right ){}^{2/3}}-\frac {x^2}{3}+\frac {1}{3} 2^{2/3} \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54} x-\frac {\sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54} x}{3 \sqrt [3]{2}}+\frac {9 \sqrt [3]{2} x}{\sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}+\frac {18\ 2^{2/3} x}{\left (-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54\right ){}^{2/3}}+x+2 c_1+\frac {\left (-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54\right ){}^{2/3}}{12\ 2^{2/3}}+\frac {\sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}{2 \sqrt [3]{2}}+\frac {9}{2^{2/3} \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}+\frac {27}{2 \sqrt [3]{2} \left (-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54\right ){}^{2/3}}+\frac {9}{4}\right )\right \},\left \{y(x)\to \frac {1}{2} \left (3 \left (\frac {1}{6} (3-2 x)-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}{12 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (-4 x^2-12 x-9\right )}{6\ 2^{2/3} \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}\right ){}^2+4 x \left (\frac {1}{6} (3-2 x)-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}{12 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (-4 x^2-12 x-9\right )}{6\ 2^{2/3} \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}\right )-2 x+2 c_1\right )\right \},\left \{y(x)\to \frac {1}{2} \left (3 \left (\frac {1}{6} (3-2 x)-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}{12 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (-4 x^2-12 x-9\right )}{6\ 2^{2/3} \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}\right ){}^2+4 x \left (\frac {1}{6} (3-2 x)-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}{12 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (-4 x^2-12 x-9\right )}{6\ 2^{2/3} \sqrt [3]{-16 x^3-72 x^2-108 x+216 c_1+\sqrt {4 \left (-4 x^2-12 x-9\right )^3+\left (-16 x^3-72 x^2-108 x+216 c_1+54\right ){}^2}+54}}\right )-2 x+2 c_1\right )\right \}\right \} \]
Maple: cpu = 0.031 (sec), leaf count = 1473 \[ \left \{ y \left ( x \right ) =0,y \left ( x \right ) = \left ( {\frac {1}{ 6}\sqrt [3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1}\,{x}^{3}-216\,{x}^{2}{\it \_C1}-24\,{x}^{3}+324\,{{ \it \_C1}}^{2}-324\,{\it \_C1}\,x-108\,{x}^{2}+162\,{\it \_C1}-162\,x} }}-6\,{\frac {-x/3-1/9\,{x}^{2}-1/4}{\sqrt [3]{-36\,{x}^{2}-54\,x+108 \,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1}\,{x}^{3}-216\,{x} ^{2}{\it \_C1}-24\,{x}^{3}+324\,{{\it \_C1}}^{2}-324\,{\it \_C1}\,x- 108\,{x}^{2}+162\,{\it \_C1}-162\,x}}}}-{\frac {x}{3}}+{\frac {1}{2}} \right ) ^{3}+x \left ( {\frac {1}{6}\sqrt [3]{-36\,{x}^{2}-54\,x+108\, {\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1}\,{x}^{3}-216\,{x}^{ 2}{\it \_C1}-24\,{x}^{3}+324\,{{\it \_C1}}^{2}-324\,{\it \_C1}\,x-108 \,{x}^{2}+162\,{\it \_C1}-162\,x}}}-6\,{\frac {-x/3-1/9\,{x}^{2}-1/4}{ \sqrt [3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {- 48\,{\it \_C1}\,{x}^{3}-216\,{x}^{2}{\it \_C1}-24\,{x}^{3}+324\,{{\it \_C1}}^{2}-324\,{\it \_C1}\,x-108\,{x}^{2}+162\,{\it \_C1}-162\,x}}}}- {\frac {x}{3}}+{\frac {1}{2}} \right ) ^{2},y \left ( x \right ) = \left ( -{\frac {1}{12}\sqrt [3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8 \,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1}\,{x}^{3}-216\,{x}^{2}{\it \_C1} -24\,{x}^{3}+324\,{{\it \_C1}}^{2}-324\,{\it \_C1}\,x-108\,{x}^{2}+162 \,{\it \_C1}-162\,x}}}+3\,{\frac {-x/3-1/9\,{x}^{2}-1/4}{\sqrt [3]{-36 \,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1 }\,{x}^{3}-216\,{x}^{2}{\it \_C1}-24\,{x}^{3}+324\,{{\it \_C1}}^{2}- 324\,{\it \_C1}\,x-108\,{x}^{2}+162\,{\it \_C1}-162\,x}}}}-{\frac {x}{ 3}}+{\frac {1}{2}}-{\frac {i}{2}}\sqrt {3} \left ( {\frac {1}{6}\sqrt [ 3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-48\,{ \it \_C1}\,{x}^{3}-216\,{x}^{2}{\it \_C1}-24\,{x}^{3}+324\,{{\it \_C1} }^{2}-324\,{\it \_C1}\,x-108\,{x}^{2}+162\,{\it \_C1}-162\,x}}}+6\,{ \frac {-x/3-1/9\,{x}^{2}-1/4}{\sqrt [3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1}\,{x}^{3}-216\,{x}^{2}{ \it \_C1}-24\,{x}^{3}+324\,{{\it \_C1}}^{2}-324\,{\it \_C1}\,x-108\,{x }^{2}+162\,{\it \_C1}-162\,x}}}} \right ) \right ) ^{3}+x \left ( -{ \frac {1}{12}\sqrt [3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+ 27+6\,\sqrt {-48\,{\it \_C1}\,{x}^{3}-216\,{x}^{2}{\it \_C1}-24\,{x}^{ 3}+324\,{{\it \_C1}}^{2}-324\,{\it \_C1}\,x-108\,{x}^{2}+162\,{\it \_C1}-162\,x}}}+3\,{\frac {-x/3-1/9\,{x}^{2}-1/4}{\sqrt [3]{-36\,{x}^{ 2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1}\,{x}^ {3}-216\,{x}^{2}{\it \_C1}-24\,{x}^{3}+324\,{{\it \_C1}}^{2}-324\,{ \it \_C1}\,x-108\,{x}^{2}+162\,{\it \_C1}-162\,x}}}}-{\frac {x}{3}}+{ \frac {1}{2}}-{\frac {i}{2}}\sqrt {3} \left ( {\frac {1}{6}\sqrt [3]{- 36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1}\,{x}^{3}-216\,{x}^{2}{\it \_C1}-24\,{x}^{3}+324\,{{\it \_C1}}^{2 }-324\,{\it \_C1}\,x-108\,{x}^{2}+162\,{\it \_C1}-162\,x}}}+6\,{\frac {-x/3-1/9\,{x}^{2}-1/4}{\sqrt [3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8 \,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1}\,{x}^{3}-216\,{x}^{2}{\it \_C1} -24\,{x}^{3}+324\,{{\it \_C1}}^{2}-324\,{\it \_C1}\,x-108\,{x}^{2}+162 \,{\it \_C1}-162\,x}}}} \right ) \right ) ^{2},y \left ( x \right ) = \left ( -{\frac {1}{12}\sqrt [3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8 \,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1}\,{x}^{3}-216\,{x}^{2}{\it \_C1} -24\,{x}^{3}+324\,{{\it \_C1}}^{2}-324\,{\it \_C1}\,x-108\,{x}^{2}+162 \,{\it \_C1}-162\,x}}}+3\,{\frac {-x/3-1/9\,{x}^{2}-1/4}{\sqrt [3]{-36 \,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1 }\,{x}^{3}-216\,{x}^{2}{\it \_C1}-24\,{x}^{3}+324\,{{\it \_C1}}^{2}- 324\,{\it \_C1}\,x-108\,{x}^{2}+162\,{\it \_C1}-162\,x}}}}-{\frac {x}{ 3}}+{\frac {1}{2}}+{\frac {i}{2}}\sqrt {3} \left ( {\frac {1}{6}\sqrt [ 3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-48\,{ \it \_C1}\,{x}^{3}-216\,{x}^{2}{\it \_C1}-24\,{x}^{3}+324\,{{\it \_C1} }^{2}-324\,{\it \_C1}\,x-108\,{x}^{2}+162\,{\it \_C1}-162\,x}}}+6\,{ \frac {-x/3-1/9\,{x}^{2}-1/4}{\sqrt [3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1}\,{x}^{3}-216\,{x}^{2}{ \it \_C1}-24\,{x}^{3}+324\,{{\it \_C1}}^{2}-324\,{\it \_C1}\,x-108\,{x }^{2}+162\,{\it \_C1}-162\,x}}}} \right ) \right ) ^{3}+x \left ( -{ \frac {1}{12}\sqrt [3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+ 27+6\,\sqrt {-48\,{\it \_C1}\,{x}^{3}-216\,{x}^{2}{\it \_C1}-24\,{x}^{ 3}+324\,{{\it \_C1}}^{2}-324\,{\it \_C1}\,x-108\,{x}^{2}+162\,{\it \_C1}-162\,x}}}+3\,{\frac {-x/3-1/9\,{x}^{2}-1/4}{\sqrt [3]{-36\,{x}^{ 2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1}\,{x}^ {3}-216\,{x}^{2}{\it \_C1}-24\,{x}^{3}+324\,{{\it \_C1}}^{2}-324\,{ \it \_C1}\,x-108\,{x}^{2}+162\,{\it \_C1}-162\,x}}}}-{\frac {x}{3}}+{ \frac {1}{2}}+{\frac {i}{2}}\sqrt {3} \left ( {\frac {1}{6}\sqrt [3]{- 36\,{x}^{2}-54\,x+108\,{\it \_C1}-8\,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1}\,{x}^{3}-216\,{x}^{2}{\it \_C1}-24\,{x}^{3}+324\,{{\it \_C1}}^{2 }-324\,{\it \_C1}\,x-108\,{x}^{2}+162\,{\it \_C1}-162\,x}}}+6\,{\frac {-x/3-1/9\,{x}^{2}-1/4}{\sqrt [3]{-36\,{x}^{2}-54\,x+108\,{\it \_C1}-8 \,{x}^{3}+27+6\,\sqrt {-48\,{\it \_C1}\,{x}^{3}-216\,{x}^{2}{\it \_C1} -24\,{x}^{3}+324\,{{\it \_C1}}^{2}-324\,{\it \_C1}\,x-108\,{x}^{2}+162 \,{\it \_C1}-162\,x}}}} \right ) \right ) ^{2} \right \} \]