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Integrate[1/(-b + a*x^3), x] ==
-(2*Sqrt[3]*ArcTan[(1 + (2*a^(1/3)*x)/b^(1/3))/ Sqrt[3]] - 2*Log[-b^(1/3) + a^(1/3)*x] + Log[b^(2/3) + a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/ (6*a^(1/3)*b^(2/3))

              1
Integrate[---------, x] == 
                  3
          -b + a x

                          1/3
                       2 a    x
                   1 + --------
                          1/3
                         b
-(2 Sqrt[3] ArcTan[------------] - 
                     Sqrt[3]
 
             1/3    1/3
     2 Log[-b    + a    x] + 
 
          2/3    1/3  1/3      2/3  2
     Log[b    + a    b    x + a    x ]) / 
 
      1/3  2/3
  (6 a    b   )