I think no hardware codecs are available for this hardware (which has solid performance though).
But before shadowplay/quicksync people were using software codecs, and keep using them even now for their superior quality.
Most of programs mentioned here should have a good selection of software codecs. Just some basics on how game window -> youtube video process happens.
Game window -> capturing software (OBS, Fraps, Dxtory, etc) -> codec (x264, h264, mjpeg, lagarith, etc) -> uploading (internet channel) -> youtube (x264 codec, 30 fps, 1280*720).
1. If your upload channel is superior (25 mbs+) - go with a lossless codec (lagarith, huffyuv and other yv12-type codecs), so you can upload video over night (in the best possible quality). You can upload it directly after capturing, no other steps are required
2. For 4+ mbs upload channel - better to use a mjpeg codec, which still has a superior to x264 quality and video should be small enough to upload it over night. Again can upload it directly.
3. For a network channel that is below - only good option is x264 codec, which has several modes. Note that "average bitrate" mode is the worst for x264, as it has several features disabled, which leads to bad compression and quality.
For x264 "CRF 16" (constant rate faction, instead of average bitrate) is a much better mode and considered to have a good quality with a fair file size, feel free to reduce it to 14 or even 12 for a slight better quality with increased size, or increase to 18-20 if your network is very slow. "CRF 0" is a lossless compression with a bit lesser file size and higher quality than mjpeg, but still heavy cpu use (same for all other x264 modes).
For #3 variant (x264) you will have to capture and save the video first using a codec from section #1, then compress it using any video editing software (megui, avidemux, virtualdub, etc).