How to calculate $\mathbb{E}\left[ \exp\left(C \sqrt{\sum_{i=1}^{n} X_i^2} \right) \right]$ with $X \sim \mathcal{N}(0, I_n)$?

Question

Let $X \sim \mathcal{N}(0, I_n)$ be a multivariate normal random vector and let $C > 0$ be some fixed constant. I wonder how can we calculate the following expectation:

\begin{align} &\mathbb{E}\left[ \exp\left(C \sqrt{\sum_{i=1}^{n} X_i^2} \right) \right] \\ =&\mathbb{E}\Big[ \exp\Big(C \sqrt{V}\Big)\Big] \\ =& \int _{\mathbb{R}^{n}} \frac{1}{(2\pi)^{n/2}} \exp\left[-\frac{C}{2} \Big(\sum_{i=1}^{n} x_i^2\Big)^{3/2}\right] dx_1 \cdots dx_n \end{align} where $V = X_1^2 + \cdots + X_n^2$ is a chi-square random variable with $n$ degrees of freedom.

I wonder is there a closed-form solution?