Let $\beta_i\in (-1/2,0)$, $i=1,2,3,4$. I'm interested in obtaining numerical value of the following integrals: $$ \int_{0<u_1<u_2<u_3<1} (1-u_1)^{\beta_1}(1-u_2)^{\beta_2} (u_3-u_1)^{\beta_3}(u_3-u_2)^{\beta_4} d\mathbf{u} $$ and $$ \int_{0<u_1<u_2<u_3<1} (1-u_2)^{\beta_1}(1-u_3)^{\beta_2} (u_2-u_1)^{\beta_3}(u_3-u_1)^{\beta_4}d\mathbf{u}. $$
I'm able to use MATLAB function "integral3" to compute it, but the time cost is too much for me. The singularities in the integrand seem to slow down the computation substantially.
Although my question in general would be "can anyone help me to compute them efficiently?", one specific question is:
Are the integrals above related to some known special function (e.g, gamma, beta, hypergeometric...), which I could make use of?