\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {x}{-y \left ( x \right ) +{x}^{4}+2\,{x}^{2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( y \left ( x \right ) \right ) ^{4}}}=0} \]
Mathematica: cpu = 0.043006 (sec), leaf count = 510 \[ \left \{\left \{y(x)\to \frac {\sqrt [3]{144 c_1 x^2+\sqrt {4 \left (12 x^2-4 c_1^2\right ){}^3+\left (144 c_1 x^2+16 c_1^3-108\right ){}^2}+16 c_1^3-108}}{6 \sqrt [3]{2}}-\frac {12 x^2-4 c_1^2}{3\ 2^{2/3} \sqrt [3]{144 c_1 x^2+\sqrt {4 \left (12 x^2-4 c_1^2\right ){}^3+\left (144 c_1 x^2+16 c_1^3-108\right ){}^2}+16 c_1^3-108}}+\frac {c_1}{3}\right \},\left \{y(x)\to -\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{144 c_1 x^2+\sqrt {4 \left (12 x^2-4 c_1^2\right ){}^3+\left (144 c_1 x^2+16 c_1^3-108\right ){}^2}+16 c_1^3-108}}{12 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (12 x^2-4 c_1^2\right )}{6\ 2^{2/3} \sqrt [3]{144 c_1 x^2+\sqrt {4 \left (12 x^2-4 c_1^2\right ){}^3+\left (144 c_1 x^2+16 c_1^3-108\right ){}^2}+16 c_1^3-108}}+\frac {c_1}{3}\right \},\left \{y(x)\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{144 c_1 x^2+\sqrt {4 \left (12 x^2-4 c_1^2\right ){}^3+\left (144 c_1 x^2+16 c_1^3-108\right ){}^2}+16 c_1^3-108}}{12 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (12 x^2-4 c_1^2\right )}{6\ 2^{2/3} \sqrt [3]{144 c_1 x^2+\sqrt {4 \left (12 x^2-4 c_1^2\right ){}^3+\left (144 c_1 x^2+16 c_1^3-108\right ){}^2}+16 c_1^3-108}}+\frac {c_1}{3}\right \}\right \} \]
Maple: cpu = 0.156 (sec), leaf count = 621 \[ \left \{ y \left ( x \right ) ={\frac {1}{12} \left ( -2\,{\it \_C1}\, \sqrt [3]{-36\,{x}^{2}{\it \_C1}-54-{{\it \_C1}}^{3}+6\,\sqrt {48\,{x} ^{6}+24\,{x}^{4}{{\it \_C1}}^{2}+ \left ( 3\,{{\it \_C1}}^{4}+108\,{ \it \_C1} \right ) {x}^{2}+3\,{{\it \_C1}}^{3}+81}}+ \left ( i \left ( - 36\,{x}^{2}{\it \_C1}-54-{{\it \_C1}}^{3}+6\,\sqrt {48\,{x}^{6}+24\,{x }^{4}{{\it \_C1}}^{2}+ \left ( 3\,{{\it \_C1}}^{4}+108\,{\it \_C1} \right ) {x}^{2}+3\,{{\it \_C1}}^{3}+81} \right ) ^{{\frac {2}{3}}}+12 \,i{x}^{2}-i{{\it \_C1}}^{2} \right ) \sqrt {3}- \left ( -36\,{x}^{2}{ \it \_C1}-54-{{\it \_C1}}^{3}+6\,\sqrt {48\,{x}^{6}+24\,{x}^{4}{{\it \_C1}}^{2}+ \left ( 3\,{{\it \_C1}}^{4}+108\,{\it \_C1} \right ) {x}^{2} +3\,{{\it \_C1}}^{3}+81} \right ) ^{{\frac {2}{3}}}+12\,{x}^{2}-{{\it \_C1}}^{2} \right ) {\frac {1}{\sqrt [3]{-36\,{x}^{2}{\it \_C1}-54-{{ \it \_C1}}^{3}+6\,\sqrt {48\,{x}^{6}+24\,{x}^{4}{{\it \_C1}}^{2}+ \left ( 3\,{{\it \_C1}}^{4}+108\,{\it \_C1} \right ) {x}^{2}+3\,{{\it \_C1}}^{3}+81}}}}},y \left ( x \right ) =-{\frac {1}{12} \left ( 2\,{\it \_C1}\,\sqrt [3]{-36\,{x}^{2}{\it \_C1}-54-{{\it \_C1}}^{3}+6\,\sqrt { 48\,{x}^{6}+24\,{x}^{4}{{\it \_C1}}^{2}+ \left ( 3\,{{\it \_C1}}^{4}+ 108\,{\it \_C1} \right ) {x}^{2}+3\,{{\it \_C1}}^{3}+81}}+ \left ( i \left ( -36\,{x}^{2}{\it \_C1}-54-{{\it \_C1}}^{3}+6\,\sqrt {48\,{x}^{ 6}+24\,{x}^{4}{{\it \_C1}}^{2}+ \left ( 3\,{{\it \_C1}}^{4}+108\,{\it \_C1} \right ) {x}^{2}+3\,{{\it \_C1}}^{3}+81} \right ) ^{{\frac {2}{3}} }+12\,i{x}^{2}-i{{\it \_C1}}^{2} \right ) \sqrt {3}+ \left ( -36\,{x}^{2 }{\it \_C1}-54-{{\it \_C1}}^{3}+6\,\sqrt {48\,{x}^{6}+24\,{x}^{4}{{ \it \_C1}}^{2}+ \left ( 3\,{{\it \_C1}}^{4}+108\,{\it \_C1} \right ) {x} ^{2}+3\,{{\it \_C1}}^{3}+81} \right ) ^{{\frac {2}{3}}}-12\,{x}^{2}+{{ \it \_C1}}^{2} \right ) {\frac {1}{\sqrt [3]{-36\,{x}^{2}{\it \_C1}-54- {{\it \_C1}}^{3}+6\,\sqrt {48\,{x}^{6}+24\,{x}^{4}{{\it \_C1}}^{2}+ \left ( 3\,{{\it \_C1}}^{4}+108\,{\it \_C1} \right ) {x}^{2}+3\,{{\it \_C1}}^{3}+81}}}}},y \left ( x \right ) ={\frac {1}{6}\sqrt [3]{-36\,{x} ^{2}{\it \_C1}-54-{{\it \_C1}}^{3}+6\,\sqrt {3\,{x}^{2}{{\it \_C1}}^{4 }+24\,{x}^{4}{{\it \_C1}}^{2}+48\,{x}^{6}+3\,{{\it \_C1}}^{3}+108\,{x} ^{2}{\it \_C1}+81}}}+{\frac {{{\it \_C1}}^{2}-12\,{x}^{2}}{6}{\frac {1 }{\sqrt [3]{-36\,{x}^{2}{\it \_C1}-54-{{\it \_C1}}^{3}+6\,\sqrt {3\,{x }^{2}{{\it \_C1}}^{4}+24\,{x}^{4}{{\it \_C1}}^{2}+48\,{x}^{6}+3\,{{ \it \_C1}}^{3}+108\,{x}^{2}{\it \_C1}+81}}}}}-{\frac {{\it \_C1}}{6}} \right \} \]