Peculiar Integral Calculation: $\int^1_{0} \bigg( \frac{x}{1-x} \bigg)^t x^{a-1} (1-x)^{b-1} \mathrm{dx}$

Question

I am trying to take the following integral.

$$\int^1_{0} \bigg( \frac{x}{1-x} \bigg)^t x^{a-1} (1-x)^{b-1} \mathrm{dx}$$

where $a, b >0$.

I am at a point where I can only combine the factors so much, and I am at a stopping point. Does anyone have any recommendations on how to proceed possibly using the beta function?

Answer 1

Your integral is$$\int_0^1x^{a+t-1}(1-x)^{b-t-1}\mathrm{d}x=\operatorname{B}(a+t,\,b-t)=\frac{\Gamma(a+t)\Gamma(b-t)}{\Gamma(a+b)}.$$