How to write this integral in terms of Gamma functions

Question

$\int_0^1 dx \ x^{\alpha n}(1-x)^{\beta n} = \frac{\Gamma(\alpha n +1)\Gamma(\beta n +1)}{\Gamma(\alpha n + \beta n + 2)}$ for Re($\alpha$n) > -$1$ and Re($\beta$n) > -$1$.

I want to manipulate the left side in order to arrive at the result on the right but I fail to make it work. Reparametrizations to write an integral in form of a Gamma function are clear to me but how do I end up with three different ones? Thanks in advance for any help!

Answer 1

The Beta function is defined $$ B(x,y)=\int_0^1 t^{x-1} (1-t)^{y-1}dt=\frac{\Gamma (x)\Gamma (y)}{\Gamma (x+y)} $$ In your case $x=1+\alpha m$ and $y=1+\beta n$ which gives the result you are seeking.