In ordinary American speech the second syllable of dirigible is very likely to be deleted.
That is the fate of most such unstressed syllables with centralized vowels in long words that were inherited from Latin, a syllable-timed language where every syllable, and thus every vowel, was pronounced in the same rhythm, like Modern Spanish.
Modern English, by contrast, is a stress-timed language, where major stressed syllables tend to recur at the same rhythm, no matter how many unstressed syllables there are between then. This leads to consonant loss and revision, vowel reduction and centralization, and syllables reducing to nothing occasionally. In essence, the faster you speak English, the more stuff you leave out.
And that certainly applies to an unstressed vowel between two consonants that can form a consonant cluster. The word dirigible is a rare one (there haven't been any dirigibles since the Hindenburg) and most people only know it from spelling, not pronunciation. So what IS "correct" pronunciation?
If you put in the stressed value for each vowel, you get /'dɪrɪdʒɪbəl/, but nobody would say that. All those /ɪ/ vowels, except the first one, will automatically be changed to muddy, centralized unstressed schwa /ə/, producing /'dɪrədʒəbəl/, with the first syllable /dɪr/ like dear. But in fact the way I pronounce it, and the way I've heard it pronounced, is with the first syllable reduced, too, to a stressed /dər/, as in derby /'dərbi/. So, I've heard /'dərədʒəbəl/, with 4 syllables, all schwa. Deer-idjubble, with initial /ɪ/, sounds bookish and formal, though it does occur.
But the lost syllable is a result of the fact that /r/ and /d/ can form a consonant cluster, as in hard /hard/, or large /lardʒ/. In other words, you don't need an extra vowel in order to pronounce /r/ and /d/ or /dʒ/ together, and when you're spitting out syllables with millisecond neuromuscular timing, it's nice to know that you don't need evrything. So that produces /'dərdʒəbəl/, dropping the unnecessary second schwa.
Since there are no apostrophes in speech, we don't notice how many sounds we're actually leaving out; we all think we're still pronouncing them, after all, and we can all figure out what they're sposta be. And the next word is coming up, and we hafta pay attention.
So, to answer the question,
- Yes, /'dərdʒəbəl/ is good American English pronunciation.
Though dirgible is not the way to spell it. English spelling doesn't represent English pronunciation. Don't be misled (that's mis-led, as in 'fooled', btw -- it's easy to mis-read this word) into thinking that the spelling is the correct pronunciation. It's not. Correct pronunciation, whatever it may be, is pronounced, not spelled, and letters in English spelling are all silent.