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Integrate[(x^2*Log[x]*Log[1 + x])/(1 + x), x] ==
(-11*x + x^2 + 6*x*Log[x] - x^2*Log[x] + 5*Log[1 + x] + 4*x*Log[1 + x] - x^2*Log[1 + x] - 6*Log[x]*Log[1 + x] - 4*x*Log[x]*Log[1 + x] + 2*x^2*Log[x]*Log[1 + x] - 2*Log[-x]*Log[1 + x]^2 + 2*Log[x]*Log[1 + x]^2 - 6*PolyLog[2, -x] - 4*Log[1 + x]*PolyLog[2, 1 + x] + 4*PolyLog[3, 1 + x])/4 

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

          2                 2
(-11 x + x  + 6 x Log[x] - x  Log[x] + 5 Log[1 + x] + 
 
                      2
    4 x Log[1 + x] - x  Log[1 + x] - 
 
    6 Log[x] Log[1 + x] - 4 x Log[x] Log[1 + x] + 
 
       2                                         2
    2 x  Log[x] Log[1 + x] - 2 Log[-x] Log[1 + x]  + 
 
                       2
    2 Log[x] Log[1 + x]  - 6 PolyLog[2, -x] - 
 
    4 Log[1 + x] PolyLog[2, 1 + x] + 
 
    4 PolyLog[3, 1 + x]) / 4
Time to compute: 0.14 second
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