The Case for AxI
It seems to me that objections to AxI tend to fall under a few broad categories. Before we get to specific arguments, I'd like to briefly address these general objections, and why they ultimately aren't a problem. (Examples of all the things I mention can be found in the Evidence section.)
1. "He's a vampire. She hates all vampires. Case closed."
It's true that Integra disapproves of vampires in general. However, she's had friendly conversations with Alucard, and been known to voluntarily spend time with him. She obviously sees him as an individual, not a generic hated enemy.
2. "She's a human. He has no respect for humans. Case closed."
While Alucard many not understand humans, it's obvious that he can respect the ones that he's gotten to know. He has friendly conversations with Walter and Integra. It's weakness that he has no respect for, not humanity; he shows far more respect to certain humans (Walter, Integra, the Queen) than any vampire (with the possible exception of Seras).
3. "Alucard and Seras are totally into each other. There's no room for Integra."
Seras and Pip are totally into each other. There's no room for Alucard. And this, my friends, is based not on speculation, but on canon.
4. "Even if there's attraction between the two, they wouldn't act on it."
One aspect of well-developed characters is that they develop over time. The dynamic between Alucard and Integra has progressed from outright enemies to wary respect to a kind of denied closeness to (in the manga, where the plot goes further than the anime) something . It can progress further.
5. "Integra, as the last of the Hellsing line, needs to produce an heir. Alucard can't give her that."
There are two ways around this one (without resorting to dhampirs). First, Integra could get an heir by someone other than Alucard - by marrying someone else, by just having sex with someone else, by artificial insemnation. Second, Alucard could turn Integra into a vampire, eliminating the need for a successor altogether.
For more details, check the case against. With any luck, though, this has succinctly covered all the main points there.
The Case Against
The Witness Stand