In reply to Arbu:
I use a cardboard box, easy to get one free going out, and just abandon at the final airport.
Coming back, it's pretty easy to get a box, or a sack you can shove the rucksack in and close up with a cable tie. Another option is an IKEA laundry bag which pack small and can live in the bottom of a pack, but I have never done this.
Be interesting to hear how you go on with United.