Does there exist a nicer form for $\beta(x + a, y + b) / \beta(a, b)$? I have the expression
$$\displaystyle\frac{\beta(x + a, y + b)}{\beta(a, b)}$$
where $\beta(a_1,a_2) = \displaystyle\frac{\Gamma(a_1)\Gamma(a_2)}{\Gamma(a_1+a_2)}$.
I have a feeling this should have a closed-form which is intuitive and makes less heavy use of the Beta function. Can someone describe to me whether this is true?
Here, $x$ and $y$ are integers larger than $0.$

I think Michael Hardy gives your answer. I only want to notify that if $y=0$ then 
$x$th raw moment of Beta distribution $\mu_x=E(T^x)=\frac{\int_0^1 t^{x+a-1}(1-t)^{b-1}~dt}{\beta(a,b)}=\frac{\beta(a+x,b)}{\beta(a,b)}$ if $x>-a$
