Is there a closed form distribution for the transformation given by $Y = \sqrt{X_1^2 + X_2^2}$, when $X_1, X_2$ are jointly normal but dependent random variables with different variance?

OBS: I know, when $X_1$ and $X_2$ are independent and have the same variance, the resultant distribution is $Y \sim {\rm Rice}(\sqrt{\mu_1^2 + \mu_2^2},\sigma)$, but in this case $X_1$ and $X_2$ are linear transformations of the same original distributions, so, they will be dependent and have different variance.

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    $\begingroup$ There is even the case of $X_1, X_2$ dependent jointly normal random variables with identical variance. $\endgroup$ – Dilip Sarwate Sep 30 '14 at 19:53