You can't compare AIC of IMAX(1,1) and ARX(1), because in the former there's a differencing and in the latter there is none.
You could go for $R^2$ like measure called FVU. Make sure you apply it on the dependent variable itself, not the differences. This is a "mechanical" approach. I'd hold off on this one though.
The real problem is that you're comparing very different specifications: non-stationary (unit root) and stationary. They're fundamentally different. For instance, if the constant is significant in IMAX(1,1) then your process will wander away from where it starts, while ARX(1) will be jumping around the unconditional mean, i.e. mean-reverting.
You have to make up your mind as to which approach to use using other means than in-sample fit statistics such as AIC. For instance, you may examine the autoregressive coefficient in ARX(1), whether it's close to 1 or not. If it's close then maybe ARX(1) is not a good fit at all. On the other hand, if the residuals from IMAX(1) have very strong negative autocorrelation at lag 1 this may indicate over-differencing etc.