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Integrate[x*Log[x]*Log[1 + x]^4, x] ==
(234 + 744*x - 15*x^2 - 180*x*Log[x] + 6*x^2*Log[x] - 564*Log[1 + x] - 540*x*Log[1 + x] + 24*x^2*Log[1 + x] + 180*Log[x]*Log[1 + x] + 168*x*Log[x]*Log[1 + x] - 12*x^2*Log[x]* Log[1 + x] + 186*Log[1 + x]^2 + 168*x*Log[1 + x]^2 - 18*x^2*Log[1 + x]^2 + 84*Log[-x]*Log[1 + x]^2 - 84*Log[x]*Log[1 + x]^2 - 72*x*Log[x]*Log[1 + x]^2 + 12*x^2*Log[x]* Log[1 + x]^2 - 32*Log[1 + x]^3 - 24*x*Log[1 + x]^3 + 8*x^2*Log[1 + x]^3 - 24*Log[-x]*Log[1 + x]^3 + 24*Log[x]*Log[1 + x]^3 + 16*x*Log[x]*Log[1 + x]^3 - 8*x^2*Log[x]* Log[1 + x]^3 + 2*Log[1 + x]^4 - 2*x^2*Log[1 + x]^4 + 4*Log[-x]*Log[1 + x]^4 - 4*Log[x]*Log[1 + x]^4 + 4*x^2*Log[x]*Log[1 + x]^4 + 180*PolyLog[2, -x] + 8*Log[1 + x]* (21 - 9*Log[1 + x] + 2*Log[1 + x]^2)* PolyLog[2, 1 + x] - 168*PolyLog[3, 1 + x] + 144*Log[1 + x]*PolyLog[3, 1 + x] - 48*Log[1 + x]^2*PolyLog[3, 1 + x] - 144*PolyLog[4, 1 + x] + 96*Log[1 + x]* PolyLog[4, 1 + x] - 96*PolyLog[5, 1 + x])/8



                             4
Integrate[x Log[x] Log[1 + x] , x] == 

                   2                     2
(234 + 744 x - 15 x  - 180 x Log[x] + 6 x  Log[x] - 
 
    564 Log[1 + x] - 540 x Log[1 + x] + 
 
        2
    24 x  Log[1 + x] + 180 Log[x] Log[1 + x] + 
 
    168 x Log[x] Log[1 + x] - 
 
        2                                   2
    12 x  Log[x] Log[1 + x] + 186 Log[1 + x]  + 
 
                    2       2           2
    168 x Log[1 + x]  - 18 x  Log[1 + x]  + 
 
                         2                       2
    84 Log[-x] Log[1 + x]  - 84 Log[x] Log[1 + x]  - 
 
                          2
    72 x Log[x] Log[1 + x]  + 
 
        2                  2                3
    12 x  Log[x] Log[1 + x]  - 32 Log[1 + x]  - 
 
                   3      2           3
    24 x Log[1 + x]  + 8 x  Log[1 + x]  - 
 
                         3                       3
    24 Log[-x] Log[1 + x]  + 24 Log[x] Log[1 + x]  + 
 
                          3
    16 x Log[x] Log[1 + x]  - 
 
       2                  3               4
    8 x  Log[x] Log[1 + x]  + 2 Log[1 + x]  - 
 
       2           4                       4
    2 x  Log[1 + x]  + 4 Log[-x] Log[1 + x]  - 
 
                       4      2                  4
    4 Log[x] Log[1 + x]  + 4 x  Log[x] Log[1 + x]  + 
 
    180 PolyLog[2, -x] + 
 
                                                  2
    8 Log[1 + x] (21 - 9 Log[1 + x] + 2 Log[1 + x] ) 
 
     PolyLog[2, 1 + x] - 168 PolyLog[3, 1 + x] + 
 
    144 Log[1 + x] PolyLog[3, 1 + x] - 
 
                 2
    48 Log[1 + x]  PolyLog[3, 1 + x] - 
 
    144 PolyLog[4, 1 + x] + 
 
    96 Log[1 + x] PolyLog[4, 1 + x] - 
 
    96 PolyLog[5, 1 + x]) / 8

Time to compute: 0.66 second

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