Can the integral above be expressed in terms of elementary functions alone?
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Refer to Integrals of differential binomials and Chebyshev's criterion .
Let y = x 6 , then the integral becomes 6 1 ∫ y − 4 / 5 ( 3 + y 2 ) 1 / 2 d y which is in the form of F ( x ) = ∫ x m ( a + b x n ) p d x where m = − 5 4 , a = 3 , b = 1 , n = 2 , p = 2 1 .
It can be evaluated in terms of elementary functions if and only if at one of the numbers p , n m + 1 , n m + 1 + p is an integer. By inspection, none of these are integers. Hence, it can't be integrated in terms of elementary functions alone.