Distribution arising from randomly distributed mean and variance

[omg!]

forgive for my ignorance, but i have a practical problem that i don't know how to approach:

$$X\sim\mathcal{N}(\mu,\sigma^2)$$
where $$\mu\sim\mathcal{N}(\mu_{\mu},\sigma_{\mu}^2)$$
and $$\sigma\sim\mathcal{N}(\mu_{\sigma},\sigma_{\sigma}^2)$$

what is the resulting distribution of $$X$$, in terms of $$\mu_{\mu},\sigma_{\mu},\mu_{\sigma},\sigma_{\sigma}$$?

 

[EnumaElish]

X's distribution is conditional on $\mu$ and $\sigma^2$. Use Bayes' theorem (continuous version of Eq. 7 in http://mathworld.wolfram.com/BayesTheorem.html where integral replaces summation) to derive the unconditional distribution of X.