harry potter - How did the Dumbledore brothers get away with Ariana's murder?

Some of you might insist that it was not a murder, as we know that a rebounding spell had hit Ariana, as shown in this answer.

However, a death in such a well known family would not go uninvestigated, I presume. Were the Dumbledore brothers investigated? How did they get away with their sister suddenly dropping dead?

I'd think that Ministry of Magic has means to recognize a death caused by magical means. That's how we know that Tom Riddle Snr. was killed by Avada Kedavra. So Ariana's failing health might be a poor excuse.

Answer

Her death was believed to due to illness

Doge stated that implies that Ariana Dumbledore died of some unspecified ailment:

Though Ariana had been in poor health for a long time, the blow, coming so soon after the loss of their mother, had a profound effect on both of her brothers.

—Harry Potter and the Deathly Hallows

According to Rita Skeeter (hardly a reliable source, but in this case I think we can trust her):

Bathilda also parrots the family line on Ariana, calling her “frail” and “delicate.”

—Harry Potter and the Deathly Hallows

In addition, the idea of Ariana's "illness" was widely circulated among those few who knew her:

The tiny number of people who henceforth knew of Ariana’s existence could be counted upon to keep the secret, including her two brothers, who deflected awkward questions with the answer their mother had taught them: “My sister is too frail for school.”

—Harry Potter and the Deathly Hallows

If the Ministry had been inclined to offer even a token investigation (which they might have done) they would quickly have found themselves stymied. The only witnesses were Aberforth and Albus (who would have refused to talk) and Grindelwald (who was in a foreign country and busy becoming a dictator).

Now, if they had procured Albus and Aberforth's wands, they could possibly have used Priori Incantato to determine what spells they had performed. But who is to say that the Ministry can examine someone's wand without sufficient cause, any more than Muggle police in the UK can search someone's home merely because a "fishy" death occurred there. Note that we do not even know whether Priori Incantato (as opposed to Priori Incantatem) can reveal all spells cast by a wand, or simply the most recent.

Similarly, an examination of Ariana's body might have revealed signs of magical death (assuming that no one concealed them), but who is to say the Ministry had enough cause to do so? Given that Ariana's illness had been known for a long time among family friends, it is entirely possible that the Ministry would have thought no more of it than that.

So without evidence, it's hard to say the Ministry would even suspect foul play, let alone sufficient legal grounds to inspect the brothers' wands. And given Albus Dumbledore's brilliance, it's safe to assume that that any evidence would have been covered up very well.