I am trying to compute the following expression: $$ \mathbb{E} e^{\lambda \prod_{i=1}^n \left( (Z_i-\alpha_i)(Z'_i-\alpha_i) + \beta_i \right)} $$ where $\alpha_i,\beta_i \in \mathbb{R}$, and $Z_1,\dots,Z_n,Z'_1,\dots,Z'_n$ are i.i.d. Rademacher r.v.'s (uniform on $\{-1,1\}$). Given all the symmetry and independence, it looks like this should have a nicer expression or expansion, as a function of $\lambda,\alpha_i,\beta_i$, but I can't figure it out. Am I missing a simple step?
Note: I'd be happy with a nice-looking non-trivial upper bound as well.