This paper presents two new fast discrete cosine transform computation algorithms: radix-3 and radix-6 algorithm. They are superior to the conventional radix-2 algorithm as they require less computational complexity. Besides, they provide a wider choice of the sequence length for which the DCT can be realized and support the prime-factor-decomposed computation algorithm to realize the 2m3n-point DCT. Furthermore, a mixed-radix algorithm is proposed such that an optimal performance can be achieved by applying the proposed radix-3 and radix-6 and the well-developed radix-2 decomposition techniques in a proper sequence. This mixed-radix algorithm not only possesses the advantages mentioned above. Compared with the prime-factor-decomposed algorithm, this mixed-radix algorithm i) requires less computational effort and ii) saves complicated data routing and mapping procedures.