What is the ruling regarding one who bows (rukūʿ) before completing the recitation of the Qur’an, or rises from prostration (sujūd) before completing the supplication (duʿāʾ), or remains for a long time between the two prostrations to complete the supplication?

It is not legislated for the worshipper to recite the Quran during bowing (rukūʿ) or to supplicate during rising from prostration (sujūd). Rather, the legislated remembrance (dhikr) during the transition is Takbir (saying "Allahu Akbar"). Whoever performs some supplications along with Takbir while rising from prostration has performed a legislated remembrance in an improper place. This does not invalidate the prayer, but it is contrary to what is most preferable. It is sunnah for him to perform the prostration of forgetfulness (sujūd al-sahw) if he did it out of forgetfulness.