\[ \boxed { \left ( f \left ( y \left ( x \right ) +x \right ) +1 \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +f \left ( y \left ( x \right ) +x \right ) =0} \]
Mathematica: cpu = 32.940183 (sec), leaf count = 49 \[ \text {Solve}\left [\int _1^{y(x)} \left (-\int _1^x f'(K[1]+K[2]) \, dK[1]+f(K[2]+x)+1\right ) \, dK[2]+\int _1^x f(K[1]+y(x)) \, dK[1]=c_1,y(x)\right ] \]
Maple: cpu = 0.031 (sec), leaf count = 22 \[ \left \{ y \left ( x \right ) =-x+{\it RootOf} \left ( -x+\int ^{{\it \_Z }}\!1+f \left ( {\it \_a} \right ) {d{\it \_a}}+{\it \_C1} \right ) \right \} \]