PDF or JPEG file saving in Adobe Illustrator

Depends on what you desire your end result to be

one combined image

Make your artboard larger and include the blue item,

2 images so you can for example animate the blue item

make another artboard for the blue item, then file 

What version of AI do you have? It looks like you are not only going beyound the artboard, but also beyond the workspace that that version is capable of handling. How large is the artboard. I would move the artboard up along with its assets and then create another one for the graphic part on the right, export them as a jpg and then combine them in photoshop.