The philosophy in Upaniṣads has accepted the existence of rebirth. An aphorism of Upaniṣad states that-
'sasyamiva pacyate mṛtyuḥ, sasyamiva jāyate punaḥ.'
[Living beings are repeatedly born and expire like the crops in the field.]
Dr. Sarvapalli Radhakrishnan has said, 'the belief in rebirth exists at least from the time of Upaniṣads. This is a naturally developed idea of the Vedic and Brahmanical literature, but this idea is clearly expressed in Upaniṣads. Nimbarka, a prominent Acharya of Vedānta explained rebirth with its causes. He states that 'soul is imperishable and indestructible. Until the soul is veiled with ignorance, the cycle of birth and death continues. When the ignorance completely vanishes, the chain of rebirth gets ceased.
Sānkhya and Yoga Philosophy
According to Sāmkhya philosophers, the soul has to repeatedly procure different bodies, in order to experience the misery, due to the previously acquired karmas.
Ātmano bhogāyatanaśarīraṁ
The body is the medium of experience of misery thus, the soul experiences pain and pleasure through the body. The soul transmigrates from one form of life to another and procures a newer body there, through liṅga śarīra (subtle thumb sized body). Until and unless, viveka khyāti (soul is different from matter) is obtained, this cycle continues.
In Pātanjala Yoga Bhāṣya, it has been said 'sati mule tadvipāko jatyāyurbhogaḥ', i.e. as far as there is an existence of kleśa, the birth, life span and experience of fruition of karma would continue.
Nyāya- Vaiśeṣika
Nyāya- Vaiśeṣikas also believe in the eternality of soul. Rebirth is proved on the basis of the belief of eternal existence of soul. The soul has to undergo rebirth, in order to experience the results of the previously acquired karmas. As it is said -
'pūrvakṛtaphalānubhavanāt tadutpattiḥ'
Due to adṛṣṭa, the soul transmigrates in the world.
Mimānsā Philosophy
Mīmānsakas do believe in the existence of rebirth. They also believe in the soul, the other world etc. With yajña (ritual of sacrifice), a factor by name Adṛṣṭa emerges, which is the controller all the life activities.
Gītā
Gītā is called as the essence of all the upaniṣads. Amongst the Indian philosophies, the Gītā is a prominent text which represents the Indian culture. It clearly expresses a belief in the concept of rebirth. The author writes-
'jātasya hi druvaṃ mṛtyuḥ, druvaṃ janma mṛtasya ca'
[The born would definitely die and the dead being would definitely take birth again.]
As the person leaves off the old clothes and wears new ones, similarly-the soul abandons the previous body and procures a new one.
vāsānsi jīrṇāni yathā vihāya, navāni gṛhṇāti naroparāṇi
tathā sarīrāṇi vihāya jīrṇānyanyāni sanyāti navāni dehī.
As the body passes through different stages like childhood, adulthood and old age, similarly, the soul adopts a new body and different states. A self-restraint person does not get deluded in those states.
Buddhism
Buddhists are non-believers in the existence of the soul. Since, they do not believe in the independent existence of the soul, they accept the soul as a stream of psyche (santati pravāḥ). This philosophy believes in the concept of rebirth. Lord Buddha accepted that the thorn that pricked his foot was the result of his killing a being, in one of his previous births.
ita ekanavatikalpe śaktyā me puruṣo hataḥ
Tena karmavipākena pāde viddhosmi bhikṣavaḥ.
Jain
The Jain philosophy believes in the existence of the soul and karma. In the theory of karma, the concept of rebirth owns an important place. Lord Mahavira has said - 'The attachment and aversion are the seeds of karma and karma is the prime cause of life and death. Rebirth can occur in the case of only those souls, which are bound by karma. The soul devoid of karma gets relieved, from the cycle of birth and death. Emphasizing on the concept of rebirth, Jinabhadragaṇi Kshamasraman writes-
bālaṃ śarīraṃ dehāntarapuvvaṃ indiya imattāo.
juvadeho bālādiva sa jassa deho sa dehī tti.
As the body of an adult is the successive state of the body of a child, similarly, the body of a child is the successive state of previous birth's body. The one, who is endowed with these states, is ' the soul'. The memory of previous birth makes a strong ground for the belief in the existence of previous birth. Lord Mahavira made his disciples recall their previous births, in order to make them firm in monkhood and self-restraint life.
When Meghakumar intended to revert from his self-restraint life and monkhood, Lord Mahavira made him realise his previous births.
Terms like ‘potta parihar' in the Jain āgamas also indicate towards acceptance of rebirth. While answering Gośalaka, Lord Mahavira had said 'These beings in sesame seeds would be reborn in these plants. There are several other illustrations of rebirth in the Jain āgamas like, Bhagavatī, Ācārāṅga, Uttarādhyayana etc.