I came across this integral in a problem I was trying to solve yesterday: $$ \int_{0}^{b0}{\int_{0}^{b1}{\int_{0}^{b2}{\int_{0}^{b3}{\int_{0}^{b4}{\int_{0}^{b5}{\int_{0}^{b6}{\dfrac{(1-x_0)^6}{\sum_{n=0}^{6}2^n{6 \choose n}x_n^n(1-x_n)^{6-n}}dx_6 dx_5 dx_4 dx_3 dx_2 dx_1 dx_0}}}}}}} $$ where $b_n$ is the bound on the integral with respect to $x_n$.
Now, obviously the result of this integral depends on your choice of $b_n$s, but there are a few necessary restraints on the choices: $$ 0<b_n<1, \ \ b_0>b_1>\cdots>b_5>b_6, $$ and $$ \sum_{i=0}^{6} b_i = 1, $$
However, I'm really not experienced enough with integration of this complexity to have any idea where to go from here.
Questions:
Is there anyway to get a sense for the bounds on the output of this integral, or a distribution of possible solutions, using Monte Carlo methods, or some sort of numerical integration technique? Or maybe probabilistic methods? This seems far too complicated to have an analytic solution, but you people are amazing, so maybe I need to have more faith.
How might one even attempt to start tackling an integral of this nature analytically?