I need help [ with / to / verb+ing ]

Can someone explain to me if those sentences are corrects and what is the difference between them:

• I need help with moving these books.
• I need help moving these books.
• I need help for moving these books.
• I need help to move these books.

These are all understandable and correct

I need help with moving these books.

This is a very complete, fully comprehensible sentence - that is not ambiguous.

I need help [with] moving these books.

Effectively, this is the same as the first sentence using an implied with.

I need help for moving these books.

This sentence is correct, and understandable. However it can sound a little jarred compared to the others - and is perhaps less common.

The main difference with this phrase, is it makes the request a little less personal. In the other sentences, you are making it clear that you will be moving books - and need help to do it. Here, it is more ambiguous about whether you are going to be included in moving the books or not.

Potentially, this kind of phrase could be used for hiring. It is states you have work needing completed, and need help/workers to facilitate that.

That said, you wouldn't get any funny looks by using this phrase - it would be understood.

I need help [in order] to move these books.

This is again a complete sentence, with an optionally implied in order. It makes it clear that you need help so that you can move the books.