\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {x \left ( {x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}+1 \right ) }{- \left ( y \left ( x \right ) \right ) ^{3}-{x}^{2}y \left ( x \right ) -y \left ( x \right ) + \left ( y \left ( x \right ) \right ) ^{6}+3\,{x}^{2} \left ( y \left ( x \right ) \right ) ^{4}+3\,{x}^{4} \left ( y \left ( x \right ) \right ) ^{2}+{x}^{6}}}=0} \]
Mathematica: cpu = 0.055007 (sec), leaf count = 326 \[ \left \{\left \{y(x)\to \text {Root}\left [4 \text {$\#$1}^5-4 \text {$\#$1}^4 c_1+8 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (2-8 c_1 x^2\right )+4 \text {$\#$1} x^4-4 c_1 x^4+2 x^2+1\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [4 \text {$\#$1}^5-4 \text {$\#$1}^4 c_1+8 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (2-8 c_1 x^2\right )+4 \text {$\#$1} x^4-4 c_1 x^4+2 x^2+1\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [4 \text {$\#$1}^5-4 \text {$\#$1}^4 c_1+8 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (2-8 c_1 x^2\right )+4 \text {$\#$1} x^4-4 c_1 x^4+2 x^2+1\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [4 \text {$\#$1}^5-4 \text {$\#$1}^4 c_1+8 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (2-8 c_1 x^2\right )+4 \text {$\#$1} x^4-4 c_1 x^4+2 x^2+1\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [4 \text {$\#$1}^5-4 \text {$\#$1}^4 c_1+8 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (2-8 c_1 x^2\right )+4 \text {$\#$1} x^4-4 c_1 x^4+2 x^2+1\& ,5\right ]\right \}\right \} \]
Maple: cpu = 0.249 (sec), leaf count = 33 \[ \left \{ -{\frac {1}{4\, \left ( \left ( y \left ( x \right ) \right ) ^{ 2}+{x}^{2} \right ) ^{2}}}-{\frac {1}{2\, \left ( y \left ( x \right ) \right ) ^{2}+2\,{x}^{2}}}-y \left ( x \right ) +{\it \_C1}=0 \right \} \]