Sampling from multivariate customised cumulative distribution function in Matlab

Consider a 5-variate cumulative distribution function (cdf) which I call F.

I want to sample random 5x1 vectors from this cdf in Matlab. F is not a cdf that has been already implemented in Matlab (such as normal, t-student, etc.). Specifically, it is defined as

I have read several question/answers in this and other forums on how to sample from customised probability distribution functions in Matlab. However,

1) Most of them are for univariate cdf, e.g. here. The idea is to apply inverse transform sampling. My problem is a bit more complicated because I would need to "invert"a 5-variate function.

2) Another option could be to use slicesample as suggested here but I don't know how to write the analytical expression of the probability density function in my case.

3) Here's another idea but specific for the bivariate case.

Could you kindly help me to understand how I can proceed?