Both are grammatical and idiomatic.
I saw him take the book.
I saw him taking the book.
In both forms the act was witnessed in its entirety. The participle form conveys the sense of action-in-progress, the base form of the verb the sense of the action outside of time.
Changing the tense to the present does not change the grammar; doing so merely changes the "naturalness" of the utterance, since there are few contexts where the present tense makes practical sense.
I see him take the book.
I see him taking the book.
But let's create one.
Corporal, do you work in the 5th police precinct?
-- Yes, I do.
And what is your job?
-- It's a desk job. I'm in the office.
Do you recognize the defendant sitting there?
-- Yes, I do. It's my colleague, Corporal Smith.
And what is Corporal Smith's job?
-- He logs evidence. But we don't have a database for that. We use a physical log book and he writes the entries by hand.
Is it a busy office?
-- Yes, it is. I see him take the book every five minutes to log new evidence throughout the day.
And what do you mean by "take the book"?
-- I mean that I see him taking the book from the countertop, where it's kept so people can consult it, over to his own desk, where he makes the entry and then returns the book to the countertop.
And why doesn't he make the entry right at the countertop?
-- I don't know. You'd have to ask him.
P.S. With atelic verbs the question of "in its entirety" is more complicated. The only way we know that that act of climbing is "complete" is by what is being climbed. Maybe the burglar made it only halfway up the wall, maybe he scaled the wall and down the other side.