\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {x-y \left ( x \right ) +\sqrt {y \left ( x \right ) }}{x-y \left ( x \right ) +\sqrt {y \left ( x \right ) }+1}}=0} \]
Mathematica: cpu = 0.094512 (sec), leaf count = 943 \[ \left \{\left \{y(x)\to \text {Root}\left [x^6-2 e^{3 c_1} x^3+e^{6 c_1}+\text {$\#$1}^6+(-6 x-6) \text {$\#$1}^5+\left (15 x^2+24 x+9\right ) \text {$\#$1}^4+\left (-20 x^3-36 x^2-18 x+2 e^{3 c_1}-4\right ) \text {$\#$1}^3+\left (15 x^4+24 x^3+9 x^2-6 e^{3 c_1} x-6 e^{3 c_1}\right ) \text {$\#$1}^2+\left (-6 x^5-6 x^4+6 e^{3 c_1} x^2+6 e^{3 c_1} x\right ) \text {$\#$1}\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [x^6-2 e^{3 c_1} x^3+e^{6 c_1}+\text {$\#$1}^6+(-6 x-6) \text {$\#$1}^5+\left (15 x^2+24 x+9\right ) \text {$\#$1}^4+\left (-20 x^3-36 x^2-18 x+2 e^{3 c_1}-4\right ) \text {$\#$1}^3+\left (15 x^4+24 x^3+9 x^2-6 e^{3 c_1} x-6 e^{3 c_1}\right ) \text {$\#$1}^2+\left (-6 x^5-6 x^4+6 e^{3 c_1} x^2+6 e^{3 c_1} x\right ) \text {$\#$1}\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [x^6-2 e^{3 c_1} x^3+e^{6 c_1}+\text {$\#$1}^6+(-6 x-6) \text {$\#$1}^5+\left (15 x^2+24 x+9\right ) \text {$\#$1}^4+\left (-20 x^3-36 x^2-18 x+2 e^{3 c_1}-4\right ) \text {$\#$1}^3+\left (15 x^4+24 x^3+9 x^2-6 e^{3 c_1} x-6 e^{3 c_1}\right ) \text {$\#$1}^2+\left (-6 x^5-6 x^4+6 e^{3 c_1} x^2+6 e^{3 c_1} x\right ) \text {$\#$1}\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [x^6-2 e^{3 c_1} x^3+e^{6 c_1}+\text {$\#$1}^6+(-6 x-6) \text {$\#$1}^5+\left (15 x^2+24 x+9\right ) \text {$\#$1}^4+\left (-20 x^3-36 x^2-18 x+2 e^{3 c_1}-4\right ) \text {$\#$1}^3+\left (15 x^4+24 x^3+9 x^2-6 e^{3 c_1} x-6 e^{3 c_1}\right ) \text {$\#$1}^2+\left (-6 x^5-6 x^4+6 e^{3 c_1} x^2+6 e^{3 c_1} x\right ) \text {$\#$1}\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [x^6-2 e^{3 c_1} x^3+e^{6 c_1}+\text {$\#$1}^6+(-6 x-6) \text {$\#$1}^5+\left (15 x^2+24 x+9\right ) \text {$\#$1}^4+\left (-20 x^3-36 x^2-18 x+2 e^{3 c_1}-4\right ) \text {$\#$1}^3+\left (15 x^4+24 x^3+9 x^2-6 e^{3 c_1} x-6 e^{3 c_1}\right ) \text {$\#$1}^2+\left (-6 x^5-6 x^4+6 e^{3 c_1} x^2+6 e^{3 c_1} x\right ) \text {$\#$1}\& ,5\right ]\right \},\left \{y(x)\to \text {Root}\left [x^6-2 e^{3 c_1} x^3+e^{6 c_1}+\text {$\#$1}^6+(-6 x-6) \text {$\#$1}^5+\left (15 x^2+24 x+9\right ) \text {$\#$1}^4+\left (-20 x^3-36 x^2-18 x+2 e^{3 c_1}-4\right ) \text {$\#$1}^3+\left (15 x^4+24 x^3+9 x^2-6 e^{3 c_1} x-6 e^{3 c_1}\right ) \text {$\#$1}^2+\left (-6 x^5-6 x^4+6 e^{3 c_1} x^2+6 e^{3 c_1} x\right ) \text {$\#$1}\& ,6\right ]\right \}\right \} \]
Maple: cpu = 0.078 (sec), leaf count = 60 \[ \left \{ \sqrt {y \left ( x \right ) -2\,\sqrt {y \left ( x \right ) }-x}y \left ( x \right ) +\sqrt {y \left ( x \right ) -2\,\sqrt {y \left ( x \right ) }-x}\sqrt {y \left ( x \right ) }-\sqrt {y \left ( x \right ) -2 \,\sqrt {y \left ( x \right ) }-x}x-{\it \_C1}=0 \right \} \]