Root verb after 'had' and 'got'?

Examples:
I had a painter redecorate our house. (have + agent + root of the verb)
I got a painter to redecorate our house. (get + agent + infinitive)

Why are the verbs after 'had' and 'got' take a root form?

Is there other verbs besides 'had' and 'got' that followed by a root verb?