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Integrate[(c^3 - x^3)^(-1 - n)/x^2, x] ==
((1 - x^3/c^3)^n*(-2*c^3*Hypergeometric2F1[-1/3, n, 2/3, x^3/c^3] + x^3*Hypergeometric2F1[2/3, 1 + n, 5/3, x^3/c^3]))/(2*c^6*x*(c^3 - x^3)^n)

            3    3 -1 - n
          (c  - x )
Integrate[---------------, x] == 
                 2
                x

       3                                         3
      x  n      3                     1      2  x
((1 - --)  (-2 c  Hypergeometric2F1[-(-), n, -, --] + 
       3                              3      3   3
      c                                         c
 
                                         3
       3                   2         5  x
      x  Hypergeometric2F1[-, 1 + n, -, --])) / 
                           3         3   3
                                        c
 
      6     3    3 n
  (2 c  x (c  - x ) )