Which is the correct words?

He spent one and a half hour / hours taking after his children.
The common way to say that is an hour and a half.

Since one and a half is greater than one you should use the plural hours (pure math).
a half hour
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Note: not "taking after" but "looking after."

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I've done some investigating. It seems that both one and a half hour/hours are common. I guess it could go either way depending on how you parse the phrase.

one (hour) and a half hour.

(one and a half) hours.
Thank you very much!!
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Vincent TeoHe spent one and a half hour / hours taking after his children.
In the context of your question the only one that sounds right to me is "one and a half hours."
And "taking care of" not "taking after."

Note however that in other contexts "one and a half hour" is correct, as in these examples:

A one and a half-hour narrated sightseeing cruise.
A one and a half hour ride taking in the stillness of the early morning.