How can I backup my music library on my iPhone 5 before syncing it to a new computer

I have an iPhone 5 and 795 songs that I would like to keep (most of them I did't purchase them from iTunes) but I recently got a new laptop (finally upgraded to Macbook pro retina display 13'') and I don't want to lose all my tracks. Is there a way I can transfer them into my new computer before I sync it? By the way all my photos and videos were transferred to iPhoto. There was a pop up when I connected my phone with my new mac and I pressed the transfer button. However I didn't sync it yet (I think?). Can I do I transfer them by using AirDrop?

You will not be able to transfer the items purchased or the other Apple ID. You will need to reinstall the data under your Apple ID and pay for the apps. 

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