\[\int_{0}^{\infty}\frac{t^{\alpha-1}}{(x+t)^{\alpha+\beta}}\,\mathrm{d}t=\frac{%
\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}{\Gamma\left(\alpha+\beta%
\right)}\frac{1}{x^{\beta}},\]