Trying to answer the question in two parts:
- Refuting the existence of Prārabdha for Jeevanmukta
- Difference between Jīvanamukti Videhamukti is not absolute
- Reason for rebirth of mentioned Jīvanamukta Rishis
Note: This answer is based on Advaita philosophy of Adi Shankaracharya
According to Adi Shankaracharya, talking about the existence of Prārabdha is meaningless for those who attains the Brahman since all three types of Karmas gets destroyed after attaining the absolute and any of Karmas doesn't remain effective to them. In other words, Prārabdha also verily ceases to exist after attaining the absolute truth. The purpose of mentioning the the existence of Prārabdha for Jīvanamukta is to convince the ignorant only.
Quoting from Aparoksha Anubhuti from Vedanta Shastra Library:
The theory one hears of from the scripture, that Prarabdha does not lose its hold upon one even after the origination of the knowledge
of Atman, is now being refuted.
After the origination of the knowledge of Reality, Prarabdha verily ceases to exist, inasmuch as the body and the like become
non-existent; just as a dream does not exist on waking.
That Karma which is done in a previous life is known as Prarabdha (which produces the present life). But such Karma cannot take the
place of Prarabdha (for a man of knowledge), as he has no other birth
(being free from ego).
Just as the body in a dream is superimposed (and therefore illusory), so is also this body. How could there be any birth of the
superimposed (body), and in the absence of birth (of the body) where
is the room for that (i.e., Prarabdha) at all?
The Vedanta texts declare ignorance to be verily the material (cause) of the phenomenal world just as earth is of a jar. That
(ignorance) being destroyed, where can the universe subsist?
Just as a person out of confusion perceives only the snake leaving aside the rope, so does an ignorant person see only the phenomenal
world without knowing the reality.
The real nature of the rope being known, the appearance of the snake no longer persists; so the substratum being known, the
phenomenal world disappears completely.
The body also being within the phenomenal world (and therefore unreal), how could Prarabdha exist? It is, therefore, for the
understanding of the ignorant alone that the Shruti speaks of
"And all the actions of a man perish when he realizes that (Atman) which is both the higher and the lower". Here the clear use of the
plural by the Shruti is to negate Prarabdha as well.
If the ignorant still arbitrarily maintain this, they will not only involve themselves into two absurdities but will also run the
risk of forgoing the Vedantic conclusion. So one should accept those
Shrutis alone from which proceeds true knowledge.
This is also explained logically in the text Vivekchudamani verse Nos. 445-463. Quoting several of them:
Prarabdha work is certainly very strong for the man of realisation, and is spent only by the actual experience of its fruit;
while the actions previously accumulated and those yet to come are
destroyed by the fire of perfect knowledge. But none of the three at
all affects those who, realising their identity with Brahman, are
always living absorbed in that idea. They are verily the transcendent
For the sage who lives in his own Self as Brahman, the One without a second, devoid of identification with the limiting adjuncts,
the question of the existence of Prarabdha work is meaningless, like
the question of a man who has awakened from sleep having any
connection with the objects seen in the dream-state.
Prarabdha work can be maintained only so long as one lives identified with the body. But no one admits that the man of
realisation ever identifies himself with the body. Hence Prarabdha
work should be rejected in his case.
The attributing of Prarabdha work to the body even is certainly an error. How can something that is superimposed (on another) have any
existence, and how can that which is unreal have a birth ? And how can
that which has not been born at all, die ? So how can Prarabdha work
exist for something that is unreal ?
462-463. "If the effects of ignorance are destroyed with their root by
knowledge, then how does the body live?" - it is to convince those
fools who entertain a doubt like this, that the Shrutis, from a
relative standpoint, hypothesise Prarabdha work, but not for proving
the reality of the body etc., of the man of realisation.
Further clarity on why people think that Jivanmukti is not Videhamukti:
546-547. The sun which appears to be, but is not actually, swallowed
by Rahu, is said to be swallowed, on account of delusion, by people,
not knowing the real nature of the sun. Similarly, ignorant people
look upon the perfect knower of Brahman, who is wholly rid of bondages
of the body etc., as possessed of the body, seeing but an appearance
Clarifying the definition of mukti:
- For the giving up of the body is not Liberation, nor that of the staff and the water-bowl; but Liberation consists in the destruction
of the heart’s knot which is Nescience.
32nd sutra of 3rd Pada of 3rd Adhyaya of Brahmasutra disclose the reason behind rebirth of Rishis who have already attained the knowledge of Brahman.
According to it, these Rishis are having divine mission and until it gets accomplished they reborn but yet never comes under the influence of ignorance. This case is equated with that of existence of Prarabdha karma of Jivanmukta.
Quoting Brahmasutra 3.3.32 with Adi Shankaracharya's commentary from wisdom library.
यावदधिकारमवस्थितिराधिकारिकाणाम् ॥ ३२ ॥
yāvadadhikāramavasthitirādhikārikāṇām || 32 ||
- Of those who have a mission to fulfil (there is corporeal) existence, so long as the mission is not fulfilled.
Rishi Apantaratama was born again as Vyasa. Sanatkumara was born as
Skanda. So also other Rishis like Vasishtha and Narada were born
again. Now these Rishis had attained the knowledge of Brahman, and yet
they had to be reborn. If that is so, what is the utility of such
knowledge of Brahman?—says the opponent. This Sutra refutes it and
says that ordinarily a person after attaining Knowledge is not reborn.
But the case of those who have a divine mission to fulfil is
different. Those perfected sages have one or more births until their
mission is fulfilled, after which they are not born again. But then
they never come under the sway of ignorance although they may be
reborn. Their case is analogous to that of a Jivanmukta, who even
after attaining Knowledge continues his corporeal existence as long as
the Prarabdha Karma lasts. The divine mission of these people is comparable to the Prarabdha Karma.