Looking up in a dictionary, I found as synonyms of nimble, among some words, fast and swift.
So, if they're synonyms, is there any rules or some particular situations, should I use one of these words rather than another?
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Fast is by far the most commonly used of the three. For example, you might be driving a fast car, or driving in the fast lane. If you do a simple job for money, you have made a fast buck. If your waitress returns with your meal sooner than expected, you might say "That was fast!" as a compliment. Fast is almost always used to mean "quick", or to complete in a short amount of time.
Swift means the same thing as fast, but it is not a common word in modern usage. Choosing to say "swift" instead of "fast" feels florid and poetic, as if you were intentionally trying to call back to an older time. Your car is fast; your horse is swift.
Nimble has a broader meaning in common usage, and has less to do with raw speed than with agility. Nimble implies a mastery of complicated actions. A gymnast is nimble. A pickpocket is nimble. A car that can make tight turns at high speed is more nimble than another car that could go faster on a straight course. However, like swift, it is not a frequently used word.
They are just variants. Don't try to pull out differences. "Quick" and "fast" are frequently used adjectives and English has a lot of variants. And authors like variation. Already the Romans said Varietas delectat, variety is pleasant.
Have a look at the synonyms for fast: http://www.freethesaurus.com/fast
Fast should always refer to speed. You have a fast car, for example.
Swift refers to time. For example, a swift verdict from a jury -- they didn't talk faster; it just didn't take very long.
Quick used to mean alive or lively (e.g., Jack be nimble, Jack be lively). But these days "quick" has come to replace swift in reference to time taken for an activity. Thus, quick should not be used as an alternative to "fast" though they are related.