# sum of exponentially distributed random numbers

I have a set of exponentially distributed random variables $X_i \sim \exp(\mu_i)$ with rates being also random with some distribution. Is there a way to find the distribution (or the CDF if it is easier) of $Y$ the sum of $X_i$,

$$Y=\sum\limits_{i=1}^N X_i,$$

other than the following:

$$f_Y(x) = \sum\limits_{i=1}^N \prod\limits_{j=i} \frac{\mu_i}{\mu_j-\mu_i} \mu_i e^{-\mu_i x}.$$

I found the above expression earlier, where $\mu_i$ is different for each $X_i$, but in my case, the rates are random with some distribution $f_\mu (\mu)$.

Is there any other expression for the sum?

Edit: My Question was marked as duplicate, but I never found where the duplicate original was. As far as I can see, the closest questions are either dealing with identical rates of the exponential, which is not my case or had the formula that I stated, which is not helpful for me.