Greetings Rlosch,
Welcome to the Apple Support Communities!
It sounds like you would like to sync iBooks from your iOS devices to your Mac, but are getting an alert in iTunes that says 5 computers already authorized.
When you authorize a computer, you are giving it permission to access the content on your devices. These authorizations only pertain to computers, not iOS devices or iPods. You did mention that you only have one computer, but have you ever authorized other computers or allowed other people to use your Apple ID on their computers?
Information on authorizing your computer in iTunes can be found in the attached article below.
Authorize your computer in iTunes - Apple Support
It sounds like a solution for you would be to deauthorize all computers from iTunes and then reauthorize your one computer again. For information on how to complete this task, please refer to the next attached article.
Deauthorize your computer using iTunes - Apple Support
Cheers,
Joe