That is NOT normal behavior. A healthy betta will not lay upside down or just float around.
How big is your tank? If the filter is causing him issues, you may want to turn off the filter and see if he can swim any better. His stomach does look pretty big and that MAY cause him to have issues with his swim bladder. If that is the case, then yes, they can swim and float in all sorts of weird ways.
If it is swim bladder disorder then treat him with 1 tespoon of EPSOM salt per gallon. Use PLAIN, UNSCENTED EPSOM. Mae sure it is disolved and slowly acclimate him into it - just as you are supposed to do when you add a fish into new tank water.
If he is not going to the surface for air or having issues swimming to get to the surface - either lower the water level or put him bac into the cup he came in and let the cup float in the tank so that the water will stay warm.
Do not feed him for a day or so.
Are any of his scales sticking out like a pinecone?