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Integrate[x^2*Log[a + b*x]*Log[c + d*x]^2, x] ==
(-132*a*b^2*c^2*d - 72*a^2*b*c*d^2 - 274*b^3*c^2*d*x - 132*a*b^2*c*d^2*x - 104*a^2*b*d^3*x + 38*b^3*c*d^2*x^2 + 19*a*b^2*d^3*x^2 - 8*b^3*d^3*x^3 + 132*a*b^2*c^2*d*Log[a + b*x] + 30*a^2*b*c*d^2*Log[a + b*x] + 8*a^3*d^3*Log[a + b*x] + 132*b^3*c^2*d*x* Log[a + b*x] - 30*b^3*c*d^2*x^2*Log[a + b*x] + 8*b^3*d^3*x^3*Log[a + b*x] + 142*b^3*c^3*Log[c + d*x] + 102*a*b^2*c^2*d* Log[c + d*x] + 96*a^2*b*c*d^2*Log[c + d*x] + 96*b^3*c^2*d*x*Log[c + d*x] + 72*a*b^2*c*d^2*x* Log[c + d*x] + 96*a^2*b*d^3*x*Log[c + d*x] - 30*b^3*c*d^2*x^2*Log[c + d*x] - 30*a*b^2*d^3*x^2*Log[c + d*x] + 16*b^3*d^3*x^3*Log[c + d*x] - 72*a*b^2*c^2*d*Log[a + b*x]*Log[c + d*x] - 36*a^2*b*c*d^2*Log[a + b*x]*Log[c + d*x] - 24*a^3*d^3*Log[a + b*x]*Log[c + d*x] - 72*b^3*c^2*d*x*Log[a + b*x]*Log[c + d*x] + 36*b^3*c*d^2*x^2*Log[a + b*x]*Log[c + d*x] - 24*b^3*d^3*x^3*Log[a + b*x]*Log[c + d*x] - 12*b^3*c^3*Log[c + d*x]^2 - 18*a*b^2*c^2*d* Log[c + d*x]^2 - 36*a^2*b*c*d^2*Log[c + d*x]^2 - 36*a^2*b*d^3*x*Log[c + d*x]^2 + 18*a*b^2*d^3*x^2*Log[c + d*x]^2 - 12*b^3*d^3*x^3*Log[c + d*x]^2 + 36*b^3*c^3*Log[a + b*x]*Log[c + d*x]^2 + 36*b^3*d^3*x^3*Log[a + b*x]*Log[c + d*x]^2 - 36*b^3*c^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]* Log[c + d*x]^2 + 36*a^3*d^3* Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x]^2 - 132*b^3*c^3*Log[a + b*x]* Log[(b*(c + d*x))/(b*c - a*d)] + 72*a*b^2*c^2*d*Log[a + b*x]* Log[(b*(c + d*x))/(b*c - a*d)] + 36*a^2*b*c*d^2*Log[a + b*x]* Log[(b*(c + d*x))/(b*c - a*d)] + 24*a^3*d^3*Log[a + b*x]* Log[(b*(c + d*x))/(b*c - a*d)] + 12*(-11*b^3*c^3 + 6*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 2*a^3*d^3)*PolyLog[2, (d*(a + b*x))/ (-(b*c) + a*d)] - 72*(b^3*c^3 - a^3*d^3)* Log[c + d*x]*PolyLog[2, (b*(c + d*x))/ (b*c - a*d)] + 72*b^3*c^3* PolyLog[3, (b*(c + d*x))/(b*c - a*d)] - 72*a^3*d^3*PolyLog[3, (b*(c + d*x))/(b*c - a*d)])/ (108*b^3*d^3)



           2                          2
Integrate[x  Log[a + b x] Log[c + d x] , x] == 

         2  2         2      2        3  2
(-132 a b  c  d - 72 a  b c d  - 274 b  c  d x - 
 
           2    2          2    3         3    2  2
    132 a b  c d  x - 104 a  b d  x + 38 b  c d  x  + 
 
          2  3  2      3  3  3
    19 a b  d  x  - 8 b  d  x  + 
 
           2  2
    132 a b  c  d Log[a + b x] + 
 
        2      2
    30 a  b c d  Log[a + b x] + 
 
       3  3
    8 a  d  Log[a + b x] + 
 
         3  2
    132 b  c  d x Log[a + b x] - 
 
        3    2  2
    30 b  c d  x  Log[a + b x] + 
 
       3  3  3
    8 b  d  x  Log[a + b x] + 
 
         3  3
    142 b  c  Log[c + d x] + 
 
           2  2
    102 a b  c  d Log[c + d x] + 
 
        2      2
    96 a  b c d  Log[c + d x] + 
 
        3  2
    96 b  c  d x Log[c + d x] + 
 
          2    2
    72 a b  c d  x Log[c + d x] + 
 
        2    3
    96 a  b d  x Log[c + d x] - 
 
        3    2  2
    30 b  c d  x  Log[c + d x] - 
 
          2  3  2
    30 a b  d  x  Log[c + d x] + 
 
        3  3  3
    16 b  d  x  Log[c + d x] - 
 
          2  2
    72 a b  c  d Log[a + b x] Log[c + d x] - 
 
        2      2
    36 a  b c d  Log[a + b x] Log[c + d x] - 
 
        3  3
    24 a  d  Log[a + b x] Log[c + d x] - 
 
        3  2
    72 b  c  d x Log[a + b x] Log[c + d x] + 
 
        3    2  2
    36 b  c d  x  Log[a + b x] Log[c + d x] - 
 
        3  3  3
    24 b  d  x  Log[a + b x] Log[c + d x] - 
 
        3  3             2
    12 b  c  Log[c + d x]  - 
 
          2  2               2
    18 a b  c  d Log[c + d x]  - 
 
        2      2             2
    36 a  b c d  Log[c + d x]  - 
 
        2    3               2
    36 a  b d  x Log[c + d x]  + 
 
          2  3  2             2
    18 a b  d  x  Log[c + d x]  - 
 
        3  3  3             2
    12 b  d  x  Log[c + d x]  + 
 
        3  3                          2
    36 b  c  Log[a + b x] Log[c + d x]  + 
 
        3  3  3                          2
    36 b  d  x  Log[a + b x] Log[c + d x]  - 
 
        3  3     d (a + b x)               2
    36 b  c  Log[------------] Log[c + d x]  + 
                 -(b c) + a d
 
        3  3     d (a + b x)               2
    36 a  d  Log[------------] Log[c + d x]  - 
                 -(b c) + a d
 
         3  3                  b (c + d x)
    132 b  c  Log[a + b x] Log[-----------] + 
                                b c - a d
 
          2  2                    b (c + d x)
    72 a b  c  d Log[a + b x] Log[-----------] + 
                                   b c - a d
 
        2      2                  b (c + d x)
    36 a  b c d  Log[a + b x] Log[-----------] + 
                                   b c - a d
 
        3  3                  b (c + d x)
    24 a  d  Log[a + b x] Log[-----------] + 
                               b c - a d
 
             3  3        2  2        2      2
    12 (-11 b  c  + 6 a b  c  d + 3 a  b c d  + 
 
          3  3             d (a + b x)
       2 a  d ) PolyLog[2, ------------] - 
                           -(b c) + a d
 
         3  3    3  3
    72 (b  c  - a  d ) Log[c + d x] 
 
                b (c + d x)
     PolyLog[2, -----------] + 
                 b c - a d
 
        3  3            b (c + d x)
    72 b  c  PolyLog[3, -----------] - 
                         b c - a d
 
        3  3            b (c + d x)           3  3
    72 a  d  PolyLog[3, -----------]) / (108 b  d )
                         b c - a d

Time to compute: 1.33 second

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