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Integrate[x^3*Log[x]*Log[1 + b*x]^2, x] ==
(6960*b*x - 1068*b^2*x^2 + 296*b^3*x^3 - 81*b^4*x^4 - 3600*b*x*Log[x] + 936*b^2*x^2*Log[x] - 336*b^3*x^3*Log[x] + 108*b^4*x^4*Log[x] - 3360*Log[1 + b*x] - 2160*b*x*Log[1 + b*x] + 648*b^2*x^2*Log[1 + b*x] - 336*b^3*x^3* Log[1 + b*x] + 216*b^4*x^4*Log[1 + b*x] + 3600*Log[x]*Log[1 + b*x] + 1728*b*x*Log[x]* Log[1 + b*x] - 864*b^2*x^2*Log[x]*Log[1 + b*x] + 576*b^3*x^3*Log[x]*Log[1 + b*x] - 432*b^4*x^4*Log[x]*Log[1 + b*x] + 216*Log[1 + b*x]^2 - 216*b^4*x^4*Log[1 + b*x]^2 - 864*Log[x]*Log[1 + b*x]^2 + 864*b^4*x^4*Log[x]* Log[1 + b*x]^2 + 864*Log[-(b*x)]*Log[1 + b*x]^2 + 3600*PolyLog[2, -(b*x)] + 1728*Log[1 + b*x]* PolyLog[2, 1 + b*x] - 1728*PolyLog[3, 1 + b*x])/ (3456*b^4)

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

                  2  2        3  3       4  4
(6960 b x - 1068 b  x  + 296 b  x  - 81 b  x  - 
 
                           2  2
    3600 b x Log[x] + 936 b  x  Log[x] - 
 
         3  3               4  4
    336 b  x  Log[x] + 108 b  x  Log[x] - 
 
    3360 Log[1 + b x] - 2160 b x Log[1 + b x] + 
 
         2  2                     3  3
    648 b  x  Log[1 + b x] - 336 b  x  Log[1 + b x] + 
 
         4  4
    216 b  x  Log[1 + b x] + 
 
    3600 Log[x] Log[1 + b x] + 
 
    1728 b x Log[x] Log[1 + b x] - 
 
         2  2
    864 b  x  Log[x] Log[1 + b x] + 
 
         3  3
    576 b  x  Log[x] Log[1 + b x] - 
 
         4  4
    432 b  x  Log[x] Log[1 + b x] + 
 
                    2        4  4             2
    216 Log[1 + b x]  - 216 b  x  Log[1 + b x]  - 
 
                           2
    864 Log[x] Log[1 + b x]  + 
 
         4  4                    2
    864 b  x  Log[x] Log[1 + b x]  + 
 
                                2
    864 Log[-(b x)] Log[1 + b x]  + 
 
    3600 PolyLog[2, -(b x)] + 
 
    1728 Log[1 + b x] PolyLog[2, 1 + b x] - 
 
                                       4
    1728 PolyLog[3, 1 + b x]) / (3456 b )