Abu Dawud

788. Commentary:
Regarding the issue of whether «بسم الله» should be recited aloud (jahran) or silently (sirran), the statement of Allamah Ibn al-Qayyim is balanced: “The Prophet sallallahu alayhi wa sallam would sometimes recite it aloud and sometimes silently, but his reciting it silently is more authentically established. It is impossible that the Messenger of Allah sallallahu alayhi wa sallam recited it aloud in all five daily prayers, both during travel and residence, and that this practice remained hidden from the Rightly Guided Caliphs and the rest of the noble Companions radi Allahu anhum ajma‘in. Furthermore, that the people of his city, during the best generations, would also remain unaware of it. This is extremely far-fetched. How could it be that, to establish the aloud recitation of «بسم الله», one relies on ambiguous wording and weak ahadith? In this matter, the authentic ahadith are non-explicit, and those that are explicit are not authentic.” [ذاد المعاد فصل فى هديه صلى الله عليه وسلم فى الصلواة]
For further details, see: [نيل الاوطار و سبل السلام]
The position of Shaykh al-Albani rahimahullah is also that «بسم الله» should be recited silently. See: [صفة الصلواة النبى صلي الله عليه وسلم ص 96]
And this is the preferred view.