What do you mean, out of curiosity, by her support of being rational and having integrity 'at the expense of others'? I can almost see your point, I think, but she was wildly opposed to things like communism precisely because they were carried out 'at the expense of others'. That's why she has that oath (which you might not have encountered yet in the novel) about not living for the sake of any other man, nor asking any man to live for one's own.
While the objections that have been pointed out on this thread are, I think, more or less accurate, much of the opposition I hear in general is related more to the objector's failure to understand what she's saying. Two teachers at my high school, when I asked for their thoughts on her, said they did not like her because (no lie, here - exact quote, both teachers) 'she's very selfish'. People do not often get what she means in referring to 'the virtue of selfishness'. I believe I read on the Ayn Rand Forum a comment to the effect that stealing was selfish - and thus perfectly morally justifiable to Rand. But this is in fact erroneous, as, to her reckoning, stealing presupposes the existence of an object to steal, which in turn presupposes the work someone else put into creating it. Thus, for Rand, things like stealing (or 'looting', as she so frequently puts it) and begging are entirely selfless, because they require one to depend on someone other than oneself. Hence, utterly immoral.
Hope that helps, cows. (Forgive the pun, by the way - 'utterly'...)