Your statement is kindof hilarious to me in many ways - You want a software that "records and 'integrates' with drive and dropbox" and "can be shared easily."
If you don't know drive or dropbox then you don't know easy and you don't know sharing hahahahhahaha... both dropbox and drive are so easy its funny
For real though what video are you trying to capture? If you're trying to capture your screen then as others have said - use fraps, dxtory, obs, or any of the dozens of other open source screen capture software. If you're trying to capture from a device some of the open source software allows it but seemingly inhibits the codec to an undesired default. Experiment with the bit rate and the codec to achieve maximum quality with minimal storage impact.
As mentioned above, OBS is free and should provide easy recording. Can I ask what is being recorded or for what purpose?
After the file is recorded, it's quite easy to manually move the file to the desired location (if you got the software installed, for eg. Dropbox or OneDrive, you can save it directly to the folder) and create a link to be shared.