I bought everything and it shows up in the Breezehome . But the alchemy lab I pay for and it won't be in the house, I bought it a few time and it doesn't go into house!

Sounds like you need the unofficial Dawnguard patch. It's fixed there. If you don't want to install that, you can fix it by typing this into the console.

prid e4ef9disableprid e4ec8enable