How to back up iPhone to a new laptop?

Recently my laptop had to be restored and everything in it was lost. There's no way of getting my old data back. Now since I want to connect my iPhone back to my iTunes so I can save and back it up it will only reset everything on my phone. Most of the music I have on there's from CDs that I don't have anymore so it's crucial to save this back on the laptop without it erasing whenever I sync it to my iTunes. Is there any way I can back up my iPhone without losing all of my stuff and still having it saved onto my iTunes?

Added (1). I've also heard of programs to use, but I don't want to pay money for them. I used one before I had to restore this laptop because I was in the same predicament, but I don't remember what the name of the program was.

The useful program need to pay a little, but they really useful.