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Integrate[1/(x^(1/4) + x^(1/3)), x] ==
-12*x^(1/12) + 6*x^(1/6) - 4*x^(1/4) + 3*x^(1/3) - (12*x^(5/12))/5 + 2*Sqrt[x] - (12*x^(7/12))/7 + (3*x^(2/3))/2 + 12*Log[1 + x^(1/12)]

               1
Integrate[-----------, x] == 
           1/4    1/3
          x    + x

                                           5/12
     1/12      1/6      1/4      1/3   12 x
-12 x     + 6 x    - 4 x    + 3 x    - -------- + 
                                          5
 
                  7/12      2/3
              12 x       3 x                  1/12
  2 Sqrt[x] - -------- + ------ + 12 Log[1 + x    ]
                 7         2