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Integrate[(x^3*Log[x]*Log[1 + x]^2)/(1 + x), x] ==
(-2600*x + 157*x^2 - 16*x^3 + 1020*x*Log[x] - 114*x^2*Log[x] + 16*x^3*Log[x] + 1580*Log[1 + x] + 1380*x*Log[1 + x] - 168*x^2*Log[1 + x] + 32*x^3*Log[1 + x] - 1020*Log[x]*Log[1 + x] - 792*x*Log[x]*Log[1 + x] + 180*x^2*Log[x]* Log[1 + x] - 48*x^3*Log[x]*Log[1 + x] - 294*Log[1 + x]^2 - 216*x*Log[1 + x]^2 + 54*x^2*Log[1 + x]^2 - 24*x^3*Log[1 + x]^2 - 396*Log[-x]*Log[1 + x]^2 + 396*Log[x]*Log[1 + x]^2 + 216*x*Log[x]* Log[1 + x]^2 - 108*x^2*Log[x]*Log[1 + x]^2 + 72*x^3*Log[x]*Log[1 + x]^2 + 72*Log[-x]*Log[1 + x]^3 - 72*Log[x]*Log[1 + x]^3 - 1020*PolyLog[2, -x] + 72*Log[1 + x]* (-11 + 3*Log[1 + x])*PolyLog[2, 1 + x] + 792*PolyLog[3, 1 + x] - 432*Log[1 + x]* PolyLog[3, 1 + x] + 432*PolyLog[4, 1 + x])/216



           3                  2
          x  Log[x] Log[1 + x]
Integrate[---------------------, x] == 
                  1 + x

                2       3
(-2600 x + 157 x  - 16 x  + 1020 x Log[x] - 
 
         2              3
    114 x  Log[x] + 16 x  Log[x] + 1580 Log[1 + x] + 
 
                             2
    1380 x Log[1 + x] - 168 x  Log[1 + x] + 
 
        3
    32 x  Log[1 + x] - 1020 Log[x] Log[1 + x] - 
 
    792 x Log[x] Log[1 + x] + 
 
         2
    180 x  Log[x] Log[1 + x] - 
 
        3                                   2
    48 x  Log[x] Log[1 + x] - 294 Log[1 + x]  - 
 
                    2       2           2
    216 x Log[1 + x]  + 54 x  Log[1 + x]  - 
 
        3           2                         2
    24 x  Log[1 + x]  - 396 Log[-x] Log[1 + x]  + 
 
                         2
    396 Log[x] Log[1 + x]  + 
 
                           2
    216 x Log[x] Log[1 + x]  - 
 
         2                  2
    108 x  Log[x] Log[1 + x]  + 
 
        3                  2
    72 x  Log[x] Log[1 + x]  + 
 
                         3                       3
    72 Log[-x] Log[1 + x]  - 72 Log[x] Log[1 + x]  - 
 
    1020 PolyLog[2, -x] + 
 
    72 Log[1 + x] (-11 + 3 Log[1 + x]) 
 
     PolyLog[2, 1 + x] + 792 PolyLog[3, 1 + x] - 
 
    432 Log[1 + x] PolyLog[3, 1 + x] + 
 
    432 PolyLog[4, 1 + x]) / 216

Time to compute: 0.44 second

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