\[ (f(y(x)+x)+1) y'(x)+f(y(x)+x)=0 \] ✓ Mathematica : cpu = 55.4958 (sec), leaf count = 49
\[\text {Solve}\left [c_1=\int _1^{y(x)} \left (-\int _1^x f'(K[1]+K[2]) \, dK[1]+f(K[2]+x)+1\right ) \, dK[2]+\int _1^x f(K[1]+y(x)) \, dK[1],y(x)\right ]\]
✓ Maple : cpu = 0.039 (sec), leaf count = 22
\[ \left \{ y \left ( x \right ) =-x+{\it RootOf} \left ( -x+\int ^{{\it \_Z}}\!1+f \left ( {\it \_a} \right ) {d{\it \_a}}+{\it \_C1} \right ) \right \} \]