how to integrate the beta like $\int_{0}^{1} \frac{y^{n-1}(1-y)^{m-n-1}}{1-xy}dy $ How should I find a closed form of $\displaystyle \int_{0}^{1} \frac{y^{n-1}(1-y)^{m-n-1}}{1-xy}dy $ , Any simple methods ?

Assuming there are no convergence issues due to the values of $x$, $m$ and $n$,
$$\begin{eqnarray*} \int_{0}^{1}\frac{y^{n-1}(1-y)^{m-n-1}}{1-xy}\,dy &=& \sum_{k\geq 0}x^k \int_{0}^{1}y^{n+k-1}(1-y)^{m-n-1}\,dy\\&=&\sum_{k\geq 0} x^k \frac{\Gamma(n+k)\Gamma(m-n)}{\Gamma(m+k)}\\&=&B(m-n,n)\cdot\phantom{}_2 F_1(1,n;m;x). \end{eqnarray*}$$
