I've used an Elgato Game Capture HD device (and its software) since ~2013, but I should probably upgrade. Everyone seems to really recommend OBS for streaming, though. I should probably give that a try!
OBS Studio is what I use and I'm very impressed with it. It let's me use a hardware encoder and captures gameplay at 1080p 60fps with negligible in-game lag as well as record audio on different tracks and at different levels so I don't have to edit as much.
Not sure how anything could beat it for the price...the price being totally free!
If you are on a windows, press windows + g. Log in and now you have an HD recorder to record your games through your laptop. I use this to record my on screen sometimes and tested it while playing games. Try it out
Personally I use OBS, it took a while to get into the right settings and can be a little temperamental, but it's so customisation and is really easy to use and is even better for if you want to do live streaming! You do need a fairly beefy computer though, as unlike Shadowplay or Crimson ReLive, it is a little heavier on the CPU side.