Would prefer

[belly_ttt]

I've heard of 3 ways using Would prefer but I wonder which is the best (grammatically right):
I would prefer to stay at home rather than to go out (My teacher taught me so, but he's not a native one and makes errors on occasionally)

I would prefer to stay at home rather than going out(Practical English Usage)
I would prefer to stay at home rather than go out(Practical English Usage)

Since the (Practical English Usage) hasn't made any mention to the first usage, I wonder which is the best way

[JosieW]

I would prefer to stay at home rather than going out.

This is what most English people would say and it sounds more natural.

Yes, you always say: would prefer to . . . . . . .
but: I prefer staying at home rather than going out.

A good book is: English Grammar in Use by Raymond Murphy (Intermediate) - Cambridge University Press - - - See Unit 57