\[ (f(y(x)+x)+1) y'(x)+f(y(x)+x)=0 \] ✓ Mathematica : cpu = 0.127981 (sec), leaf count = 52
\[\text {Solve}\left [\int _1^{y(x)}\left (f(x+K[2])-\int _1^xf'(K[1]+K[2])dK[1]+1\right )dK[2]+\int _1^xf(K[1]+y(x))dK[1]=c_1,y(x)\right ]\] ✓ Maple : cpu = 0.036 (sec), leaf count = 22
\[\{y \left (x \right ) = -x +\RootOf \left (c_{1}-x +\int _{}^{\textit {\_Z}}\left (f \left (\textit {\_a} \right )+1\right )d \textit {\_a} \right )\}\]