Let $ G(x)$ be the following function $ $ G(x) = \sum_{i \in I} \alpha_i (t+x)^{\lambda_i} x^{1-\lambda_i} $ $ where we know that the coefficients sum up to one ($ \sum_{i \in I} \alpha_i = 1$ ) and that the powers appear symmetrically in the sum (i.e. $ \forall \lambda_i, i\in I$ $ \exists j \in I$ s.t. $ \lambda_j = 1-\lambda_i$ ). Then I am interested in the asymptotic behavior of $ $ \exp\left(\int^R_0 \frac{n+1}{G(x)}dx \right),$ $ where $ n$ is an integer.

More specifically, I want to show that this exponential goes asymptotically as $ R^{n+1}$ , i.e. $ $ \lim_{R \rightarrow \infty} \frac{\exp\left(\int^R_0 \frac{n+1}{G(x)}dx \right)}{R^{n+1}} = 1, $ $ or find conditions on $ \alpha_i, \lambda_i$ so that it’s true.