You might be better off in GQ for this one.
The short answer is that, under the Digital Millenium Copyright Act, the usual course of things is for the copyright holder who believes there has been an infringment to ask Youtube to remove the video (a takedown notice). It is probably safe to assume that you don’t qualify for a fair use exception, assuming it is a non-educational video that is not parodying or criticizing the song, so that would be the end of it.
But the short answer is slightly unsatisfying. There is still some risk of legal liability. Individual posters of Youtube videos have been sued. And you might have a significant interest in getting permission to avoid even the threat of takedown.
Unfortunately, the long answer involving permissions and liability is pretty complicated. As best I can tell, the soundtrack is distributed by US-based DRG Records. But often you need to obtain the permission of someone else, or even multiple parties, who might be in other countries in this case. The possible international element also complicates the legal analysis a bit, since international copyright is subject to various treaties depending on the country in question (probably Italy in this case), and probably effects the likelihood of suits (presumably in your favor, but I don’t know).
As a general matter, the means of obtaining permission varies depending on the who holds the copyright. Typically, getting permission is basically a matter of asking for it in writing. Bigger copyright holders will have special offices for just that. Here’s a good place to start. Some have means of instantaneous permission, even over the internet, for a small fee.
But, as mentioned, sometimes there are multiple people from which you would need to seek permission (the rights-holder of the actual sound recording, and the rights-holder of the underlying music, for example). So this can actually get pretty complicated. I would guess it is even more complicated if you have to do this all in Italian. There are businesses that go through all of that for you, but they add an extra layer of fees.
If you don’t get permission, theoretically one can still be sued even if the content is taken down. Youtube is happy to turn over any information they have about you to someone who wants to sue you for the violation. Cite. The copyright statute allows a minimum of $750 in damages regardless of de minimis nature of the violation, but it can be much more if the court considers it “just.” And if done for commercial purposes, you are also potentially criminally liable.
I think that is all basically still true as long as the copyright holder is in the US or Italy. My guess would be that the cost of filing suit (including court costs, lawyer fees, and the cost of finding violations) is probably more than $750 for the rights-holder, so that explains why it rarely if ever happens in cases of individuals posting videos. But it is not totally outside the realm of possibility.
[Note: I have no specialized knowledge in this area and I’m just offering my general observations. Don’t make any judgments about what to do based on this information.]