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Integrate[1/Sqrt[(2/3)*x^3 + c], x] ==
((-1)^(1/6)*2^(2/3)*3^(1/12)*c^(1/3)* Sqrt[(-1)^(5/6)*(-1 + ((-2/3)^(1/3)*x)/c^(1/3))]* Sqrt[1 + ((-2/3)^(1/3)*x)/c^(1/3) + ((-2/3)^(2/3)*x^2)/c^(2/3)]* EllipticF[ArcSin[ Sqrt[-(-1)^(5/6) - (I*(-2/3)^(1/3)*x)/c^(1/3)]/ 3^(1/4)], (-1)^(1/3)])/Sqrt[c + (2*x^3)/3]

                1
Integrate[--------------, x] == 
               2  3
          Sqrt[- x  + c]
               3

     1/6  2/3  1/12  1/3
((-1)    2    3     c    
 
                          2  1/3
                       (-(-))    x
             5/6          3
    Sqrt[(-1)    (-1 + -----------)] 
                           1/3
                          c
 
                2  1/3        2  2/3  2
             (-(-))    x   (-(-))    x
                3             3
    Sqrt[1 + ----------- + ------------] 
                 1/3            2/3
                c              c
 
                                          2  1/3
                                     I (-(-))    x
                               5/6        3
                     Sqrt[-(-1)    - -------------]
                                          1/3
                                         c
    EllipticF[ArcSin[------------------------------], 
                                   1/4
                                  3
 
                             3
         1/3              2 x
     (-1)   ]) / Sqrt[c + ----]
                           3