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Integrate[x/(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 - (4*x^(3/4))/3 + (6*x^(5/6))/5 - (12*x^(11/12))/11 + x - (12*x^(13/12))/13 + (6*x^(7/6))/7 - (4*x^(5/4))/5 + (3*x^(4/3))/4 - (12*x^(17/12))/17 + (2*x^(3/2))/3 - (12*x^(19/12))/19 + (3*x^(5/3))/5 + 12*Log[1 + x^(1/12)]

               x
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      3/4      5/6
              12 x       3 x      4 x      6 x
  2 Sqrt[x] - -------- + ------ - ------ + ------ - 
                 7         2        3        5
 
      11/12           13/12      7/6      5/4
  12 x            12 x        6 x      4 x
  --------- + x - --------- + ------ - ------ + 
     11              13         7        5
 
     4/3       17/12      3/2       19/12      5/3
  3 x      12 x        2 x      12 x        3 x
  ------ - --------- + ------ - --------- + ------ + 
    4         17         3         19         5
 
              1/12
  12 Log[1 + x    ]