\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) -{\frac { \left ( y \left ( x \right ) \right ) ^{2}+1}{ \left | y \left ( x \right ) +\sqrt {1+y \left ( x \right ) } \right | \left ( 1+x \right ) ^{3/2}}}=0} \]
Mathematica: cpu = 558.947977 (sec), leaf count = 47 \[ \left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}} \frac {\left | K[1]+\sqrt {K[1]+1}\right | }{K[1]^2+1} \, dK[1]\& \right ]\left [c_1-\frac {2}{\sqrt {x+1}}\right ]\right \}\right \} \]
Maple: cpu = 0.094 (sec), leaf count = 35 \[ \left \{ -2\,{\frac {1}{\sqrt {1+x}}}-\int ^{y \left ( x \right ) }\!{ \frac {1}{{{\it \_a}}^{2}+1} \left | {\it \_a}+\sqrt {{\it \_a}+1} \right | }{d{\it \_a}}+{\it \_C1}=0 \right \} \]
Sage: cpu = 0 (sec), leaf count = 0 \[ \text {Maxima was unable to solve this ODE} \]