IPhone – I am unable to download an audio book that I bought from iTunes to the iPhone 5

1)your friends itunes account was probably registered on your phone when you synced with the computer.

2) you can only be synced with one itunes account at a time. your problem might be that your synced with your friends account so any app that you download with your account wont be able to sync any apps unless you change it to sync with your account(which will delete all apps not purchased with your account.)

Google shows various links that say that it is not possible to automatically re-download Audio Books in iTunes.

For example, this recent one - previously purchased audiobooks not showing up - from 3-Nov-2011:

You only get one download of audiobooks and movies. It is your responsibility to backup your purchases.

Similarly:

• Redownload Audio Book PLEASE!!!
• iTunes Audiobook redownload

A few replies, like this one, include extra info:

• How to back up your iTunes purchases (though that Apple page says "This article has been archived and is no longer updated by Apple.")
• How to contact iTunes Support, to request the ability to re-download your purchase

Conclusion

If you have any past back-ups of your desktop machine where you run iTunes and sync(ed) your device, you should be able to restore your purchase.

Otherwise, your best bet is to contact iTunes Store support.