I'm on mobile so I can't accurately source this information/respond to other posts, but John Green does a vlogbrothers where he talks about quoting too much of a poem in one of his books and his publisher's lawyers sitting down with the estate lawyers to determine how much he could quote before they had to start paying... That being said, probably a lot more expensive to quote Gaga than whoever he did. I wish I had the link to give you, but I've no way to get them on mobile.
I would love to link this video, because I think I remember watching it and it would be a great firsthand account from an author, but I can't for the life of me find it. If anyone can reply to this with a link I'll put it in right away.
(Also: it's generally easier to quote from poems than from songs. Not saying you should do it, but song lyrics are a whole different ballpark.)
Looking for this video, I stumbled across a few other Vlogbrothers videos where copyright is discussed and they've got some good stuff. -T