How do I evaluate the following combination of random variables? Is it martingale? I'm about to analyse the following expression
$$Z_n:=\prod_{k=1}^n \left(\frac{\frac{Y_k}{\prod_{i=1}^k X_i}}{\sum_{j=1}^k \frac{Y_j}{\prod_{i=1}^j X_i}} \right),$$
where $Y_j$ for all $j\in \mathbb{N}$, are independent and $\Gamma(\beta,1)$-distributed random variables and $X_i$, for all $i \in \mathbb{N}$ independent identically BetaDistr.($\alpha,\beta$). I have presumtion that this expression is a martingale and tried to prove the martingale property. Let $\mathcal{F_n}=\{X_i| Y_i: i \le n\}$
$$\mathbb{E}[Z_{n+1}|\mathcal{F}_n]=Z_n\mathbb{E}\left[\frac{\frac{Y_{n+1}}{\prod_{i=1}^{n+1} X_i}}{\sum_{j=1}^{n+1} \frac{Y_j}{\prod_{i=1}^j X_i}} |\mathcal{F}_n \right]=Z_n \frac{1}{\prod_{i=1}^n X_i}\mathbb{E}\left[\frac{\frac{Y_{n+1}}{X_{n+1}}}{\sum_{j=1}^{n} \frac{Y_j}{\prod_{i=1}^j X_i}+\frac{Y_{n+1}}{\prod_{i=1}^{n+1} X_i} } |\mathcal{F}_n \right]=?$$
I could calculate only till this step. Does somebody see how to proceed further? Maybe somebody has already had experience with this combination of random variables and could give me a hint what are another options to evaluate this expression?

I assume that the family $\left(X_i,Y_j,i,j\geqslant 1\right)$ is independent.
There exists a formula for the conditional expectation of a function of two independent vectors with respect to the first vector. Let $f\colon \mathbb R^{k}\times\mathbb R^\ell\to\mathbb R$ be a measurable function and $U$ and $V$ two independent random vectors with values in $\mathbb R^{k}$ and $\mathbb R^\ell$ respectively. Then
$$
\mathbb E\left[f\left(U,V\right)\mid U\right]=g\left(U\right),
$$
where the function $g\colon\mathbb R^k\to\mathbb R$ is defined by $g\left(u\right)=\mathbb E\left[f\left(u,V\right) \right]$.
So it seems that a first step in the question is to compute 
$$
\mathbb E\left[
\frac{\frac{Y}{X}}{a+b\frac{Y}{X}}
\right]
$$
for fixed real numbers $a$ and $b$, where $Y$ has $\Gamma(\beta,1)$-distribution and $X$ is independent of $Y$ and has a $\beta$-distribution.
