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Integrate[(c^3 - x^3)^(-1 - n), x] ==
-((((-c + x)/((-1 + (-1)^(2/3))*c))^(1 + n)* (-((-1)^(2/3)*c) + x)* ((c - (-1)^(2/3)*x)/((1 + (-1)^(1/3))*c))^(1 + n)* (c^3 - x^3)^(-1 - n)*AppellF1[-n, 1 + n, 1 + n, 1 - n, ((-1)^(1/3)*(c + (-1)^(1/3)*x))/ ((1 + (-1)^(1/3))*c), ((-1)^(2/3)*c - x)/ ((-1 + (-1)^(2/3))*c)])/n)



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

         -c + x      1 + n        2/3
-(((----------------)      (-((-1)    c) + x) 
              2/3
    (-1 + (-1)   ) c
 
                2/3
        c - (-1)    x  1 + n   3    3 -1 - n
      (---------------)      (c  - x )       
                1/3
       (1 + (-1)   ) c
 
      AppellF1[-n, 1 + n, 1 + n, 1 - n, 
 
           1/3          1/3          2/3
       (-1)    (c + (-1)    x)   (-1)    c - x
       -----------------------, ----------------]) / n
                    1/3                   2/3
           (1 + (-1)   ) c      (-1 + (-1)   ) c
 
    )

Time to compute: 0.43 second

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