If shipping via domestic carrier (UPS, etc), either the cardboard and bubble wrap suggested, or you should use the lightest plastic case that fits the rifle. Weight adds to the cost to ship.
My friend borrowed my Plano rifle case to ship his rifle stock to TX. Added extra padding to protect it, and the case was more ridged than cardboard alone. Placed that inside a cardboard shipping box, maybe a little more packing material, and it survived the trip fine.
Make sure you take the insurance, and have detailed "before" pictures in case it's damaged in transit.