Is it permissible to combine the intention of the Aqiqah and the Udhiyah together on Eid al-Adha, and is the Aqiqah valid?

The jurists differed on the issue of whether one sacrifice suffices for both the Udhiyah (sacrifice of Eid al-Adha) and the Aqiqah (sacrifice for a newborn), with two opinions:

The first: It suffices. This is the view of the Hanafis and one of the two narrations from Imam Ahmad. They likened this to performing the two rak'ahs of "Tahiyyat al-Masjid" (greeting the mosque prayer) along with the Sunnah prayer of the obligatory (Fard) prayer.

The second: It does not suffice. This is the view of the Malikis, Shafi'is, and the other narration from Imam Ahmad. Their argument is that each has a different reason (for its legislation).

What appears to be the case is that a single sacrifice suffices with the intention of both the Aqiqah and the Udhiyah together.