Praise be to Allah.
The hadith was narrated by at-Tirmidhi (3521) from Layth ibn Abi Sulaym. Al-Haafiz Ibn Hajar (may Allah have mercy on him) said in Taqreeb at-Tahdheeb (2/464): He is sadooq (sincere) but he muddled things a great deal, and his hadith cannot be proven sound, so it is to be rejected. End quote.
The hadith was classed as da‘eef (weak) by Shaykh al-Albaani (may Allah have mercy on him) in Da‘eef at-Tirmidhi and elsewhere.
However the du‘aa’ mentioned in the hadith is proven from another hadith, in a longer form than what is mentioned here:
It was narrated from ‘Aa’ishah (may Allah be pleased with her) that the Messenger of Allah (blessings and peace of Allah be upon him) taught her this du‘aa’: “Allaahumma inni as’aluka min al-khayri kullihi, ‘aajilihi wa aajilihi, ma ‘alimtu minhu wa maa lam a‘lam. Wa a‘oodhu bika min al-sharri kullihi, ‘aajilihi wa aajilihi, ma ‘alimtu minhu wa maa lam a‘lam. Allaahumma inni as’aluka min khayri ma sa’alaka ‘abduka wa nabiyyuka Muhammad sall-Allahu ‘alayhi wa sallam, wa a‘oodhu bika min sharri ma ‘aadha minhu ‘abduka wa nabiyyuka. Allaahumma inni as’aluka al-jannah wa ma qarraba ilayha min qawlin aw ‘amal, wa a‘oodhu bika min al-naar wa ma qarraba ilayha min qawlin aw ‘amal, wa as’aluka an taj‘al kulla qadaa’in taqdihi li khayraan (O Allaah, I ask You for all that is good, in this world and in the Hereafter, what I know of it and what I do not know. O Allaah, I seek refuge with You from all evil, in this world and in the Hereafter, what I know of it and what I do not know. O Allaah, I ask You for the good that Your slave and Prophet Muhammad (blessings and peace of Allah be upon him) has asked You for, and I seek refuge with You from the evil from which Your slave and Prophet sought refuge. O Allaah, I ask You for Paradise and for that which brings one closer to it, in word and deed, and I seek refuge in You from Hell and from that which brings one closer to it, in word and deed. And I ask You to make every decree that You decree concerning me good).”
This hadith was narrated by Ahmad in his Musnad (24498) and Ibn Maajah (3846). It was classed as saheeh by al-Albaani in Saheeh al-Jaami‘ (1276).
However, if a du‘aa’ is good and appropriate, and sound in meaning, it is permissible to recite it even if it is narrated in a da‘eef hadith, and even if it is not narrated in a hadith or report at all. A person may choose to offer in his prayer (salaah) any du‘aa’ he likes, asking for the good of this world and the hereafter, choosing the most appropriate words, whilst paying attention to the fact that du‘aa’s which are proven to have been narrated from the Prophet (blessings and peace of Allah be upon him) are more appropriate and bring greater barakah (blessing), although it is not stipulated that one’s du‘aa’s should be narrated from him.
The Prophet (blessings and peace of Allah be upon him) said: “… then let him choose whatever du‘aa’ he likes and say that.” Narrated by al-Bukhaari (835) and Muslim (402).
And he (blessings and peace of Allah be upon him) said: “…and as for prostrating, strive in du‘aa’, for it is deserving of a response (from your Lord).” Narrated by Muslim (479).
For more information on the ruling on offering du‘aa’ in general and specific du‘aa’s from the Qur’an and Sunnah, please see the answer to question no. 102600.
And Allah knows best.