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Integrate[x^2*Log[x]*Log[1 + x]^2, x] ==
(-137*x + 19*x^2 - 4*x^3 + 66*x*Log[x] - 15*x^2*Log[x] + 4*x^3*Log[x] + 71*Log[1 + x] + 48*x*Log[1 + x] - 15*x^2*Log[1 + x] + 8*x^3*Log[1 + x] - 66*Log[x]*Log[1 + x] - 36*x*Log[x]*Log[1 + x] + 18*x^2*Log[x]*Log[1 + x] - 12*x^3*Log[x]*Log[1 + x] - 6*Log[1 + x]^2 - 6*x^3*Log[1 + x]^2 - 18*Log[-x]*Log[1 + x]^2 + 18*Log[x]*Log[1 + x]^2 + 18*x^3*Log[x]* Log[1 + x]^2 - 66*PolyLog[2, -x] - 36*Log[1 + x]*PolyLog[2, 1 + x] + 36*PolyLog[3, 1 + x])/54



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

              2      3                     2
(-137 x + 19 x  - 4 x  + 66 x Log[x] - 15 x  Log[x] + 
 
       3
    4 x  Log[x] + 71 Log[1 + x] + 48 x Log[1 + x] - 
 
        2                 3
    15 x  Log[1 + x] + 8 x  Log[1 + x] - 
 
    66 Log[x] Log[1 + x] - 36 x Log[x] Log[1 + x] + 
 
        2
    18 x  Log[x] Log[1 + x] - 
 
        3                                 2
    12 x  Log[x] Log[1 + x] - 6 Log[1 + x]  - 
 
       3           2                        2
    6 x  Log[1 + x]  - 18 Log[-x] Log[1 + x]  + 
 
                        2
    18 Log[x] Log[1 + x]  + 
 
        3                  2
    18 x  Log[x] Log[1 + x]  - 66 PolyLog[2, -x] - 
 
    36 Log[1 + x] PolyLog[2, 1 + x] + 
 
    36 PolyLog[3, 1 + x]) / 54

Time to compute: 0.24 second

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