You need to consider the curse of dimensionality. Adaptive numerical integrations often take hundreds of function evaluations. Depending on the complexity of your functions, a single adaptive numerical integration might take hundreds or more function evaluations.
For example, a quick test of the integral function shows to integrate a simple function like sin(x) over the interval [0, 10*pi], took 390 function evaluations.
Even a very simple and smooth function like exp, over the interval [0, 2] took 150 function evaluations.
So suppose these were a reasonable counts for integral? A 5 level iterated integral could then take on the order of
function evaluations! In either case, we might be talking about something between 75 billion and 9 trillion function evaluations.
So if you have no choice but to do the multiple integration using such a numerical tool, do what you must do, but expect it to be SLOW. Better is to look for alternatives. For example, can you do at least one of those integrals analytically?
I would point out that far too often, I see people doing numerical integration of simple Gaussian density functions, when in fact, that is doable using a function call for the Gaussian CDF (or erfc if you wish to do the transformation yourself.)
Another choice is the use of Monte Carlo integration, which becomes relatively more efficient when you go into a higher number of dimensions.