It is narrated from Sayyiduna Anas (may Allah be pleased with him) that the Noble Prophet (peace and blessings be upon him), Abu Bakr, Umar, and Uthman (may Allah be pleased with them all) would all begin the recitation in prayer with «الحمد لله رب العالمين». Imam Darimi (may Allah have mercy on him) said: This is what we say, and I do not consider it correct to say «بسم الله الرحمٰن الرحيم» aloud.
Hadith Referenceسنن دارمي / من كتاب الصللاة / 1274
Hadith Takhrijیہ حدیث صحیح متفق علیہ ہے۔ دیکھئے: [بخاري 743] ، [مسلم 399] ، [مسند أبى يعلی 2881] ، و [ابن حبان 1798]
Brief Explanation
(Commentary on Hadith 1273)
When «الحمد للّٰه رب العالمين» would begin the recitation, they would certainly say "Bismillah" quietly (sirran). Therefore, the correct view is that "Bismillah" should be recited quietly, and the recitation should begin from «الحمد للّٰه رب العالمين». And if sometimes "Bismillah" is recited aloud (jahran), then there is no harm in it; it is established from some Companions (radi Allahu anhum) that they did so.
Some people consider it obligatory to recite "Bismillah" aloud before Surah al-Fatihah in prayer, which is not correct. The aforementioned hadith is completely contrary to their statement.