How to integrate gamma function

Question

Can anyone tell me how can I integrate this function ?

$\int_0^\infty x \frac{\Gamma(3\alpha) }{\Gamma(\alpha)\Gamma(2\alpha)} x^{(\alpha -1)} (1-x)^{2\alpha-1} \, dx, \alpha>0$

Answer 1

\begin{align} & \int_0^\infty x \frac{\Gamma(3\alpha) }{\Gamma(\alpha)\Gamma(2\alpha)} x^{(\alpha -1)} (1-x)^{2\alpha-1} \, dx \\ &= \frac{\Gamma(3\alpha) } {\Gamma(\alpha)\Gamma(2\alpha)} \int_0^\infty x^{\alpha} (1-x)^{2\alpha-1} \, dx \\ &= \frac{\Gamma(3\alpha) } {\Gamma(\alpha)\Gamma(2\alpha)} \frac{\Gamma(\alpha+1) \Gamma(2\alpha) } {\Gamma(3\alpha+1)} \\ &= \frac{\alpha}{3\alpha} \\ &= \frac13 \end{align}