use sturdy bones (80 gil from creature comforts) to up the multiplier to 3x (need 36), then use the mechanical (high experience) materials to upgrade.... the best one to use is the one that costs 50,000 from the R+D store since it offers the best gil:experience ratio
most first tier weapons need 50,000 experience to max out, and second tier need 200-400,000 to max out, and final tier need about 1.5 million
http://www.ps3trophies.org/forum/final-fantasy-xiii/35682-final-fantasy-xiii-weapon-list.htmlthis link has a list of all of the weapons and the exact experience needed to max out each tier as well as the catalysts need to upgrade them to the next tier, use it so that you dont use up too many materials when upgrading