# Asymptotic behavior of an exponential of an integral

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.