Note that x+x2+1​1​⋅x−x2+1​x−x2+1​​=1x2+1​−x​. Hence
x+x2+1​−x+x2+1​1​=2x,
which is rational if and only if x is rational. To show that this is the only answer, show that each of the other sets is different from the set of rational numbers.