People who use the shortened form are often convinced they are right because they are being “ironic” and some even claim it’s the original form. But here’s the entry in The American Heritage Dictionary of Idioms:
This expression originated about 1940 in Britain and for a time invariably used couldn’t. About 1960 could was occasionally substituted, and today both versions are used with approximately equal frequency, despite their being antonyms.
“I could care less” just isn’t logically ironic. The people speaking feel irony, but their words don’t convey it. “I’d buy those jeans” could be ironic if you really meant the opposite: you wouldn’t buy those jeans if they were the last pair in the world. But “I could care less” isn’t used to imply its opposite: that you care more. Thus it is not ironic.
“Couldn’t care less” is a strong statement because it says you don’t care at all, zero!
“Could care less,” whatever meaning you take it to have, does not have that crucial message of zero interest which gives the original saying its sting.
See also Michael Quinion on this point: http://www.worldwidewords.org/qa/qa-ico1.htm.