XIVDB has updated

[Alkimi]

fastes to get a good new weapon are 12 fates for the atma relic with ilvl100 and then 9x1500 myth for the animus relic.

You still need to complete all 9 atma "trials" to get the new relic to it's highest current stats. Each trial has 4 categories; Enemies, FATEs, Dungeons and Guildleves. It's not going to be quick by any means unless you play an extraordinary amount of hours.

The Leviathan weapons are potentially the fastest of the new ones to get, just epends where both the weapon and the mirror come from.