Distribution of the $l_2$-norm of gaussian vector

Question

Let $Y_k \sim N(\mu_k, \sigma_k^2)$. For $\sigma_k = \sigma$ the squared norm of $Y = (Y_1, \ldots, Y_n)$ follows the noncentral chi square distribution.

What is the distribution in the general case? The derivation of the pdf found on wikipedia doesn't generalize to our case.