What mile said about the battery is true. If you want a powerful laptop the battery life isn't going to be the greatest. A desktop and a cheap laptop would be the best option, but you need to consider 2 things.
1.) space. Do you have room for a desk + a tower/monitor?
2.) performance. If you want a decent laptop you are going to spend at least $300. leaving you with $1200 for a desktop and monitor. Figure $300 for a nice 120hz... Now you are at $900 for a system and that's not very much. Even with a cheaper 60hz monitor... Maybe $950 for a total PC build... It's not going to be the best.
Considering the budget you are working with I would seriously suggest getting a really nice gaming laptop (which will run any program you can throw at it). I have a Toshiba Qosmio gaming laptop that I use for LANs, and while it's not small, it's totally manageable. As mile and I both said, the battery life isn't great, but it's not horrible (just work in power saver or Eco mode when not plugged in). With mine I have it in a laptop backpack and I also have room for all of the following:
Charger
Mouse
Mousepad
Cooling mat
Keyboard
External HDD
Earbuds
Microphone
Extra mouse
Phone charger
Laptop lock
Other small misc items
Again, even with a large 17" gaming laptop in my bag, all of that stuff also fits.