"Indispensible": is it correct?

The Corpus of Contemporary American English reveals some use of indispensible, though it is not mentioned in any of my dictionaries. The use statistics are: 35 for indispensible vs. 1887 for indispensable, so it is overwhelmingly in favour of the latter.

Google ngram confirms that this is not a particularly recent trend, and that the two words were used competitively until 1840, where usage seems to have settled on indispensable.

The -able vs. -ible suffix stems from which Latin conjugation the original verb (these adjectives are all related to verbs) was. 1st conjugation verbs, ending in -are, turn into -able suffixes, whereas 2nd and 3rd conjugation verbs, ending in -ere, turn into -ible. I just looked up a Latin reference, and the original verb was 1st conjugation (dispensare), hence the -able ending is correct.

The spelling differences are neither arbitrary nor irrelevant, especially to linguists.

BBC does make mistakes. I wouldn't be surprised if that headline is revised in a matter of hours or days. Indispensible is clearly a typo in this case, and it is not an uncommon misspelling of indispensable. In my experience, it is rarely frowned upon, and I even thought it was correct until now.

In a similar vein, defensable is also a common misspelling of defensible.