Mean of the Euclidean Norm of a Gaussian Mixture

Question

The single Gaussian case is answered here, but I would like to know what is $E[\mathbf{\|x\|}]$ when $\mathbf{x} \sim \sum_{i=1}^{N} w_i p_g(\mathbf{m}_i, \mathbf{P}_i)$, where $p_g(\cdot)$ is a single multivariate Gaussian probability density function, and $w_i$ are scalar weights which all sum to 1? I would greatly appreciate any insights. My motivation for this is to find a concentration inequality that relates to $P(\|\mathbf{x}\| \leq \rho) > 1 - \beta$ where $\beta$ is a small, positive number less than 1.