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?



Numerical Evaluation of Some Triple Integral involving Negative Powers
In the integral
$$
\int\_0^{u\_2}(1-u\_1)^{b\_1}(u\_3-u\_1)^{b\_3}du\_1
=\sum\_{n=0}^\infty\frac{(-b\_1)\_n}{n!}\int\_0^{u\_2}u\_1^n(u\_3-u\_1)^{b\_3}du\_1
$$
perform the change $u\_1=u\_2(1-x)$; the result is a $\_2F\_1(\dots|u\_3)$ hypergeometric function. The same treatment do for the integration w.r.t. $u\_2$. The finale is an integral of the product of two $\_2F\_1$ functions.


Similar manipulations apply to the second family of triple integrals.


