\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) =1/2\,{\frac {1}{x \left ( 1+x \right ) } \left ( y \left ( x \right ) \sin \left ( 3/2\,{\frac {y \left ( x \right ) }{x}} \right ) \cos \left ( 1/2\,{\frac {y \left ( x \right ) }{x}} \right ) x+y \left ( x \right ) \sin \left ( 3/2\,{\frac {y \left ( x \right ) }{x}} \right ) \cos \left ( 1/2\,{\frac {y \left ( x \right ) }{x}} \right ) +y \left ( x \right ) \cos \left ( 1/2\,{\frac {y \left ( x \right ) }{x}} \right ) \sin \left ( 1/2\,{\frac {y \left ( x \right ) }{x}} \right ) x+y \left ( x \right ) \cos \left ( 1/2\,{\frac {y \left ( x \right ) }{x}} \right ) \sin \left ( 1/2\,{\frac {y \left ( x \right ) }{x}} \right ) -\sin \left ( {\frac {y \left ( x \right ) }{x}} \right ) y \left ( x \right ) x-y \left ( x \right ) \sin \left ( {\frac {y \left ( x \right ) }{x}} \right ) +2\,\sin \left ( {\frac {y \left ( x \right ) }{x}} \right ) \cos \left ( 1/2\,{\frac {y \left ( x \right ) }{x}} \right ) \sin \left ( 1/2\,{\frac {y \left ( x \right ) }{x}} \right ) x \right ) \left ( \cos \left ( {\frac {y \left ( x \right ) }{x}} \right ) \right ) ^{-1} \left ( \cos \left ( 1/2\,{\frac {y \left ( x \right ) }{x}} \right ) \right ) ^{-1} \left ( \sin \left ( 1/2\,{\frac {y \left ( x \right ) }{x}} \right ) \right ) ^{-1}}=0} \]
Mathematica: cpu = 0.058507 (sec), leaf count = 19 \[ \left \{\left \{y(x)\to x \sin ^{-1}\left (\frac {e^{c_1} x}{x+1}\right )\right \}\right \} \]
Maple: cpu = 0.125 (sec), leaf count = 27 \[ \left \{ y \left ( x \right ) ={\frac {x}{2}\arccos \left ( {\frac {{\it \_C1}\,{x}^{2}+{x}^{2}+2\,x+1}{ \left ( 1+x \right ) ^{2}}} \right ) } \right \} \]