If so, why don't Muslims accept the Bible?
The Qur'an commands Muslims to believe in the revelations given to all of God's prophets. These prophets include Moses, David, and Jesus (peace be upon them). Muslims must believe in the revelations received by such prophets. Books associated with them are called by the names Torah, Psalms, and Gospel. Muslims, therefore believe in the original Torah, Psalms and Gospel.
The Bible contains books by the same names and other books besides. Muslims are not commanded to believe in the other books. To be more specific, the Bible contains 66 books in the Protestant version, and 73 in the Catholic version. In the Bible, consisting 66 or 73 books, the Torah is represented as 5 books, the Psalms as 1, and the Gospel in 4 versions. This makes a total of 10 books within the Bible. Muslims are not required to believe in any more than the ten books. But that assumes that the 10 books referred to are the same as what the Qur'an refers to when the Qur'an speaks of the Torah, Psalms and Gospel.
A further check proves that these books changed contents over time although their names remained the same. To understand this concept, consider that a book like the Encyclopedia Britannica changes its contents and size several times throughout its history, yet is called by the same name. Obviously the Encyclopedia Britannica is not the same as it used to be. A closer check shows that something similar occurred with the Torah, Psalms, and Gospel.
For a long time many people believed that Moses wrote the five books called Torah in the Bible. But now it is widely known that Moses could not have written those books in their entirety. One evidence for this is that Chapter 34 of the book of Deuteronomy describes the death and burial of Moses, and how the people mourned for him after his death. Obviously, Moses did not write that passage. Biblical scholars also acknowledge that the present composition of the Psalms cannot in its entirety be traced back to David.
The Gospels is a more complex matter because the Bible contains not one but four. There is only one true Gospel which Jesus preached, but these are represented in four versions each according to a different author. The Qur'an requires Muslims to believe in the original Gospel that Jesus preached before the four versions were written. Everyone agrees that the four versions were written long after Jesus.