Clear fractions, then set it up as a polynomial for which you're looking for the roots. If it had an integer solution a, then even without solving that polynomial, you could could restrict what a has to be to just a few choices. None of those choices are solutions, and therefore, there are no integer solutions to that equation.

Ex: Does x^2 - 7x +3 = 0 have an integer solution? You can answer that w/o solving that quadratic. If it had an integer solution a, then it would've factored with (x-a) being a factor (and the other also being an integer - this is a theorem). Thus a would divide 3.

So if this has an integer solution, it must be in the set {-1, 1, -3, 3}. But none of those is a solution, which you can see by just checking them. Therefore, that polynomial doesn't have an integer root.