If I am at home with no work commitments, is it better for me to pray the 5 daily prayers individually? IAdditionally, did the Prophet and Ahlulbayt give preference to praying them individually (e.g. Imam Hussain not delaying his prayers even in Karbala)? What narrations from our Imams do we have on the permissibility of joining? The reason I ask is because why did Allah prescribe the 5 prayers to the Prophet (PBUH), if the Prophet was going to join them into 3 separate times anyways?

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Scholars have different opinions on whether it’s better to join them or separate them. A number of scholars such as Ayatollah Sistani state that it’s best to pray the Asr about 20 minutes after Dhuhr (probably to do the Nawafel). The Ahlulbayt have recommended us to separate between the prayers, but many scholars understand that separating between them means that you do some Ta’qibat between the prayers (and don’t just get up to do Asr immediately after finishing Dhuhr). Yes, Sunnis believe that the time of Asr is a while after Dhuhr (sometimes 4 hours in the summer months). In our school of thought we don’t wait that long.

Therefore, what’s best for you to do is to do so supplications/Ta’qibat/Nawafel (for about 15-20 minutes) and then offer the Asr.
Also remember that the Qur’an gives main 3 times for Salat not 5.