Finding the conditional distribution

[Bazzinga]

Hey guys, I'm trying to find a conditional distribution based on the following information:

$Y|u Poisson(u \lambda)$, where $u~Gamma( \phi)$ and $Y~NegBinomial(\frac{\lambda \phi}{1+ \lambda \phi}, \phi^{-1})$

I want to find the conditional distribution $u|Y$

Here's what I've got so far:

$f(u|y)= \frac{f(u, y)}{p(y)} = \frac{p(y|u)}{p(y)} f(u)$
$=\frac{(u \lambda)^{y}e^{-u \lambda}}{y!} \frac{u^{ \alpha - 1}e^{-u/ \beta}}{ \Gamma ( \alpha) \beta^{ \alpha}} \frac{y! \Gamma ( \alpha)}{\Gamma (y+ \alpha)} ( \frac{1+\lambda \beta}{ \lambda \beta})^{y} (1+ \lambda \beta)^{ \alpha}$

where $\beta = \phi$ and $\alpha = \phi^{-1}$
(I'm using $\beta$ and $\alpha$ right now because that's what it does in the notes)

I'm not sure where to go from here. I can cancel some terms out, but then I'm not sure what distribution I'm supposed to end up with. Could anyone give me a push in the right direction?

 

[haruspex]

I can cancel some terms out

Then please do so and post what you get.