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Integrate[x^3*Log[3 + x]*Log[1 + b*x]^2, x] ==
-(-10800*b - 13608*b^2 - 46656*b^3 - 6960*b*x - 9864*b^2*x - 21816*b^3*x - 61236*b^4*x + 1068*b^2*x^2 + 1872*b^3*x^2 + 3402*b^4*x^2 - 296*b^3*x^3 - 444*b^4*x^3 + 81*b^4*x^4 + 10800*b*Log[3 + x] + 8424*b^2*Log[3 + x] + 9072*b^3*Log[3 + x] + 8748*b^4*Log[3 + x] + 3600*b*x*Log[3 + x] - 936*b^2*x^2*Log[3 + x] + 336*b^3*x^3*Log[3 + x] - 108*b^4*x^4*Log[3 + x] + 3360*Log[1 + b*x] + 7056*b*Log[1 + b*x] + 18792*b^2*Log[1 + b*x] + 58320*b^3*Log[1 + b*x] + 2160*b*x*Log[1 + b*x] + 4320*b^2*x*Log[1 + b*x] + 12960*b^3*x*Log[1 + b*x] + 58320*b^4*x* Log[1 + b*x] - 648*b^2*x^2*Log[1 + b*x] - 1728*b^3*x^2*Log[1 + b*x] - 5832*b^4*x^2* Log[1 + b*x] + 336*b^3*x^3*Log[1 + b*x] + 1008*b^4*x^3*Log[1 + b*x] - 216*b^4*x^4* Log[1 + b*x] - 5184*b*Log[3 + x]*Log[1 + b*x] - 7776*b^2*Log[3 + x]*Log[1 + b*x] - 15552*b^3*Log[3 + x]*Log[1 + b*x] - 34992*b^4*Log[3 + x]*Log[1 + b*x] - 1728*b*x*Log[3 + x]*Log[1 + b*x] + 864*b^2*x^2*Log[3 + x]*Log[1 + b*x] - 576*b^3*x^3*Log[3 + x]*Log[1 + b*x] + 432*b^4*x^4*Log[3 + x]*Log[1 + b*x] - 216*Log[1 + b*x]^2 - 864*b*Log[1 + b*x]^2 - 3888*b^2*Log[1 + b*x]^2 - 23328*b^3* Log[1 + b*x]^2 - 23328*b^4*x*Log[1 + b*x]^2 + 3888*b^4*x^2*Log[1 + b*x]^2 - 864*b^4*x^3*Log[1 + b*x]^2 + 216*b^4*x^4*Log[1 + b*x]^2 + 864*Log[3 + x]* Log[1 + b*x]^2 - 864*b^4*x^4*Log[3 + x]* Log[1 + b*x]^2 - 864*Log[(b*(3 + x))/(-1 + 3*b)]* Log[1 + b*x]^2 + 69984*b^4* Log[(b*(3 + x))/(-1 + 3*b)]*Log[1 + b*x]^2 - 3600*Log[3 + x]*Log[(1 + b*x)/(1 - 3*b)] + 5184*b*Log[3 + x]*Log[(1 + b*x)/(1 - 3*b)] + 7776*b^2*Log[3 + x]*Log[(1 + b*x)/(1 - 3*b)] + 15552*b^3*Log[3 + x]*Log[(1 + b*x)/(1 - 3*b)] + 34992*b^4*Log[3 + x]*Log[(1 + b*x)/(1 - 3*b)] + 144*(-25 + 36*b + 54*b^2 + 108*b^3 + 243*b^4)* PolyLog[2, (b*(3 + x))/(-1 + 3*b)] + 1728*(-1 + 81*b^4)*Log[1 + b*x]* PolyLog[2, (1 + b*x)/(1 - 3*b)] + 1728*PolyLog[3, (1 + b*x)/(1 - 3*b)] - 139968*b^4*PolyLog[3, (1 + b*x)/(1 - 3*b)])/ (3456*b^4)



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

                    2          3
-(-10800 b - 13608 b  - 46656 b  - 6960 b x - 
 
           2            3            4
     9864 b  x - 21816 b  x - 61236 b  x + 
 
           2  2         3  2         4  2
     1068 b  x  + 1872 b  x  + 3402 b  x  - 
 
          3  3        4  3       4  4
     296 b  x  - 444 b  x  + 81 b  x  + 
 
                                2
     10800 b Log[3 + x] + 8424 b  Log[3 + x] + 
 
           3                    4
     9072 b  Log[3 + x] + 8748 b  Log[3 + x] + 
 
                                2  2
     3600 b x Log[3 + x] - 936 b  x  Log[3 + x] + 
 
          3  3                   4  4
     336 b  x  Log[3 + x] - 108 b  x  Log[3 + x] + 
 
     3360 Log[1 + b x] + 7056 b Log[1 + b x] + 
 
            2                       3
     18792 b  Log[1 + b x] + 58320 b  Log[1 + b x] + 
 
                                   2
     2160 b x Log[1 + b x] + 4320 b  x Log[1 + b x] + 
 
            3
     12960 b  x Log[1 + b x] + 
 
            4
     58320 b  x Log[1 + b x] - 
 
          2  2
     648 b  x  Log[1 + b x] - 
 
           3  2
     1728 b  x  Log[1 + b x] - 
 
           4  2
     5832 b  x  Log[1 + b x] + 
 
          3  3
     336 b  x  Log[1 + b x] + 
 
           4  3
     1008 b  x  Log[1 + b x] - 
 
          4  4
     216 b  x  Log[1 + b x] - 
 
     5184 b Log[3 + x] Log[1 + b x] - 
 
           2
     7776 b  Log[3 + x] Log[1 + b x] - 
 
            3
     15552 b  Log[3 + x] Log[1 + b x] - 
 
            4
     34992 b  Log[3 + x] Log[1 + b x] - 
 
     1728 b x Log[3 + x] Log[1 + b x] + 
 
          2  2
     864 b  x  Log[3 + x] Log[1 + b x] - 
 
          3  3
     576 b  x  Log[3 + x] Log[1 + b x] + 
 
          4  4
     432 b  x  Log[3 + x] Log[1 + b x] - 
 
                     2                     2
     216 Log[1 + b x]  - 864 b Log[1 + b x]  - 
 
           2             2          3             2
     3888 b  Log[1 + b x]  - 23328 b  Log[1 + b x]  - 
 
            4               2
     23328 b  x Log[1 + b x]  + 
 
           4  2             2
     3888 b  x  Log[1 + b x]  - 
 
          4  3             2
     864 b  x  Log[1 + b x]  + 
 
          4  4             2
     216 b  x  Log[1 + b x]  + 
 
                                2
     864 Log[3 + x] Log[1 + b x]  - 
 
          4  4                        2
     864 b  x  Log[3 + x] Log[1 + b x]  - 
 
             b (3 + x)              2
     864 Log[---------] Log[1 + b x]  + 
             -1 + 3 b
 
            4     b (3 + x)              2
     69984 b  Log[---------] Log[1 + b x]  - 
                  -1 + 3 b
 
                         1 + b x
     3600 Log[3 + x] Log[-------] + 
                         1 - 3 b
 
                           1 + b x
     5184 b Log[3 + x] Log[-------] + 
                           1 - 3 b
 
           2                1 + b x
     7776 b  Log[3 + x] Log[-------] + 
                            1 - 3 b
 
            3                1 + b x
     15552 b  Log[3 + x] Log[-------] + 
                             1 - 3 b
 
            4                1 + b x
     34992 b  Log[3 + x] Log[-------] + 
                             1 - 3 b
 
                           2        3        4
     144 (-25 + 36 b + 54 b  + 108 b  + 243 b ) 
 
                 b (3 + x)
      PolyLog[2, ---------] + 
                 -1 + 3 b
 
                    4
     1728 (-1 + 81 b ) Log[1 + b x] 
 
                 1 + b x
      PolyLog[2, -------] + 
                 1 - 3 b
 
                     1 + b x
     1728 PolyLog[3, -------] - 
                     1 - 3 b
 
             4            1 + b x            4
     139968 b  PolyLog[3, -------]) / (3456 b )
                          1 - 3 b

Time to compute: 1.33 second

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