Astika schools are those which accept Vedas as valid proof and do not depend whether they are authored or eternal.
Out of the six astika darshanas of sanatana dharma, four do not agree that the vedas are authorless and eternal. All these six schools, however, accepted Vedas as valid proof (Shabda Pramana) and authortative. Thus they are considered astika darshana as opposed to Buddhism/Charvaka etc.
Samkhya & Yoga believes that the Vedas are non eternal. The Nyaya & Vaisheshika believe that the Vedas are authored by God and non eternal. Only the Mimamsa and Vedanta schools believe that Vedas are Apauruṣeyā and eternal.
Now what are the arguments put forth by these schools and how are they refuted? For this we have to refer to Mímáṃsá by the holy sage Jaimini.
The detailed arguments are provided in the sarva-darshana-samgraha. Please read from page 187 onwards. It is around 10 pages.
Nyaya: how can the Vedas be said to be underived from any personal author, when there is no evidence to establish this? Would you
maintain that they have no personal author because, although there is
an unbroken line of tradition, there is no remembrance of any
author...The sentences of the Veda must have originated from a
personal author, since they have the character of sentences like those
of Kálidása and other writers. And, again, the sentences of the Veda
have been composed by a competent person, since, while they possess
authority, they have, at the same time, the character of sentences,
like those of Manu and other sages.
Mimamsaka: I shall now [says the Mímáṃsaka] clear up the whole question. What is meant by this paurusheyatva ["derivation from a
personal author"] which it is sought to prove? ...For the sentences of
the Veda are universally defined to be sentences which prove things
that are not provable by other evidence. But if you could establish
that these Vedic sentences only prove what is provable by other
evidence, this definition would be at once contradicted, just as if a
man were to say that his mother was a barren woman.
And it ends with
Therefore as the Veda is thus proved to have not originated from any
personal author, and as the minutest germ of suspicion against it is
thus absolutely destroyed, we hold it as satisfactorily demonstrated
that it has a self-established authority in all matters relating to
Remember, Nyaya argues that the Vedas are authored by God so it is faultless. Mimansa does not accept Ishvara and Vedanta does not accept God to be an author of the Vedas. The arguments are that Mimansa and Vedanta hold sound is eternal and the vedas are sounds and the vibrations of the air "manifest" the always existing sound. Thus vedas do not have authors in the traditional sense. Therefore, even Ishvara, who is faultness and infalliable, is not considered to be an author. This is also argued extensively in the above texts and the references therein.
The two answers below provide more details.