You might see if Simplify or FullSimplify could make your integral sufficiently simpler to help.
If you are doing five integrals, one for reach of your variables, then you might try doing just doing the first integral, see how much your expression grows, then try the second integral, see how much your expression grows, etc. Mathematica has functions like LeafCount and ByteCount that can tell you how big an expression is. If that shows after one or two integrals that the result seems to be growing exponentially then you know when to give up.
You might try Simplify between each integration. If the results are growing too large then FullSimplify will probably not complete in a reasonable amount of time. If your problem is this complicated then even Simplify probably won't help enough.
If you have an integral of a sum you might turn that into a sum of integrals and do each integral separately, possibly with Simplify, but if your problem is that complicated then this probably won't help enough.
You might consider whether it would be reasonable for you to install more memory. Memory is relatively cheap right now and 16-32GB of memory can sometimes let you ignore concerns about having enough room for a calculation.
You might see if you can increase the "swap space" in your operating system. This is an area on your hard drive that is used to swap things out of memory temporarily so the memory can be reused for further calculations. Using swap space is often 10-20 times slower, but if you can use 4-32GB of swap space in addition to your 4GB of memory and be able to finish the calculation then that might be acceptable.
You might see if you can ask very nicely of someone who has Mathematica and has 32GB of memory and 32-256GB of swap space to let your calculation run for a few hours or a few days and see if they can get your calculation to finish.
Unfortunately there is no reasonable way to determine how large a result will be or how long it will take to complete.