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Integrate[Log[a + x + b*x]*Log[c - x + d*x]^2, x] ==
x*Log[a + (1 + b)*x]*Log[c + (-1 + d)*x]^2 - ((c + (-1 + d)*x)*(2 - 2*Log[c + (-1 + d)*x] + Log[c + (-1 + d)*x]^2))/((1 + b)*(-1 + d)) - b*(((c + (-1 + d)*x)*(2 - 2*Log[c + (-1 + d)*x] + Log[c + (-1 + d)*x]^2))/((1 + b)*(-1 + d)) - (a*(Log[-(((-1 + d)*(a + x + b*x))/(a + c + b*c - a*d))]*Log[c + (-1 + d)*x]^2 + 2*Log[c + (-1 + d)*x]*PolyLog[2, ((1 + b)*(c + (-1 + d)*x))/(a + c + b*c - a*d)] - 2*PolyLog[3, ((1 + b)*(c + (-1 + d)*x))/(a + c + b*c - a*d)]))/(1 + b)^2) + (a*(Log[-(((-1 + d)*(a + x + b*x))/(a + c + b*c - a*d))]*Log[c + (-1 + d)*x]^2 + 2*Log[c + (-1 + d)*x]*PolyLog[2, ((1 + b)*(c + (-1 + d)*x))/(a + c + b*c - a*d)] - 2*PolyLog[3, ((1 + b)*(c + (-1 + d)*x))/(a + c + b*c - a*d)]))/(1 + b)^2 + 2*((-a + a*d - 2*x - 2*b*x + 2*d*x + 2*b*d*x - c*Log[c + (-1 + d)*x] - b*c*Log[c + (-1 + d)*x] + x*Log[c + (-1 + d)*x] + b*x*Log[c + (-1 + d)*x] - d*x*Log[c + (-1 + d)*x] - b*d*x*Log[c + (-1 + d)*x] + Log[a + x + b*x]* (-((-1 + d)*(a + x + b*x)) + (-1 + d)*(a + x + b*x)*Log[c + (-1 + d)*x] + (a + c + b*c - a*d)* Log[((1 + b)*(c + (-1 + d)*x))/(a + c + b*c - a*d)]) + (a + c + b*c - a*d)* PolyLog[2, -(((-1 + d)*(a + x + b*x))/ (a + c + b*c - a*d))])/ ((1 + b)*(-1 + d)^2) - (c*((Log[a + x + b*x] - Log[-(((-1 + d)*(a + x + b*x))/(a + c + b*c - a*d))])*Log[c + (-1 + d)*x]^2 - 2*Log[c + (-1 + d)*x]*PolyLog[2, ((1 + b)*(c + (-1 + d)*x))/(a + c + b*c - a*d)] + 2*PolyLog[3, ((1 + b)*(c + (-1 + d)*x))/(a + c + b*c - a*d)]))/(2*(-1 + d)^2)) - 2*d*((-a + a*d - 2*x - 2*b*x + 2*d*x + 2*b*d*x - c*Log[c + (-1 + d)*x] - b*c*Log[c + (-1 + d)*x] + x*Log[c + (-1 + d)*x] + b*x*Log[c + (-1 + d)*x] - d*x*Log[c + (-1 + d)*x] - b*d*x*Log[c + (-1 + d)*x] + Log[a + x + b*x]* (-((-1 + d)*(a + x + b*x)) + (-1 + d)*(a + x + b*x)*Log[c + (-1 + d)*x] + (a + c + b*c - a*d)* Log[((1 + b)*(c + (-1 + d)*x))/(a + c + b*c - a*d)]) + (a + c + b*c - a*d)* PolyLog[2, -(((-1 + d)*(a + x + b*x))/ (a + c + b*c - a*d))])/ ((1 + b)*(-1 + d)^2) - (c*((Log[a + x + b*x] - Log[-(((-1 + d)*(a + x + b*x))/(a + c + b*c - a*d))])*Log[c + (-1 + d)*x]^2 - 2*Log[c + (-1 + d)*x]*PolyLog[2, ((1 + b)*(c + (-1 + d)*x))/(a + c + b*c - a*d)] + 2*PolyLog[3, ((1 + b)*(c + (-1 + d)*x))/(a + c + b*c - a*d)]))/(2*(-1 + d)^2)) 

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

                                        2
x Log[a + (1 + b) x] Log[c + (-1 + d) x]  - 
 
  ((c + (-1 + d) x) (2 - 2 Log[c + (-1 + d) x] + 
 
                          2
       Log[c + (-1 + d) x] )) / ((1 + b) (-1 + d)) - 
 
  b (((c + (-1 + d) x) 
 
        (2 - 2 Log[c + (-1 + d) x] + 
 
                             2
          Log[c + (-1 + d) x] )) / ((1 + b) (-1 + d))\
 
                  (-1 + d) (a + x + b x)
      - (a (Log[-(----------------------)] 
                    a + c + b c - a d
 
                              2
           Log[c + (-1 + d) x]  + 
 
          2 Log[c + (-1 + d) x] 
 
                      (1 + b) (c + (-1 + d) x)
           PolyLog[2, ------------------------] - 
                         a + c + b c - a d
 
                       (1 + b) (c + (-1 + d) x)
          2 PolyLog[3, ------------------------])) / 
                          a + c + b c - a d
 
             2              (-1 + d) (a + x + b x)
      (1 + b) ) + (a (Log[-(----------------------)] 
                              a + c + b c - a d
 
                           2
        Log[c + (-1 + d) x]  + 
 
       2 Log[c + (-1 + d) x] 
 
                   (1 + b) (c + (-1 + d) x)
        PolyLog[2, ------------------------] - 
                      a + c + b c - a d
 
                    (1 + b) (c + (-1 + d) x)
       2 PolyLog[3, ------------------------])) / 
                       a + c + b c - a d
 
          2
   (1 + b)  + 2 ((-a + a d - 2 x - 2 b x + 2 d x + 
 
        2 b d x - c Log[c + (-1 + d) x] - 
 
        b c Log[c + (-1 + d) x] + 
 
        x Log[c + (-1 + d) x] + 
 
        b x Log[c + (-1 + d) x] - 
 
        d x Log[c + (-1 + d) x] - 
 
        b d x Log[c + (-1 + d) x] + 
 
        Log[a + x + b x] 
 
         (-((-1 + d) (a + x + b x)) + 
 
           (-1 + d) (a + x + b x) 
 
            Log[c + (-1 + d) x] + 
 
           (a + c + b c - a d) 
 
                (1 + b) (c + (-1 + d) x)
            Log[------------------------]) + 
                   a + c + b c - a d
 
        (a + c + b c - a d) 
 
                      (-1 + d) (a + x + b x)
         PolyLog[2, -(----------------------)]) / 
                        a + c + b c - a d
 
                       2
      ((1 + b) (-1 + d) ) - 
 
     (c ((Log[a + x + b x] - 
 
                   (-1 + d) (a + x + b x)
             Log[-(----------------------)]) 
                     a + c + b c - a d
 
                              2
           Log[c + (-1 + d) x]  - 
 
          2 Log[c + (-1 + d) x] 
 
                      (1 + b) (c + (-1 + d) x)
           PolyLog[2, ------------------------] + 
                         a + c + b c - a d
 
                       (1 + b) (c + (-1 + d) x)
          2 PolyLog[3, ------------------------])) / 
                          a + c + b c - a d
 
                 2
      (2 (-1 + d) )) - 
 
  2 d ((-a + a d - 2 x - 2 b x + 2 d x + 2 b d x - 
 
        c Log[c + (-1 + d) x] - 
 
        b c Log[c + (-1 + d) x] + 
 
        x Log[c + (-1 + d) x] + 
 
        b x Log[c + (-1 + d) x] - 
 
        d x Log[c + (-1 + d) x] - 
 
        b d x Log[c + (-1 + d) x] + 
 
        Log[a + x + b x] 
 
         (-((-1 + d) (a + x + b x)) + 
 
           (-1 + d) (a + x + b x) 
 
            Log[c + (-1 + d) x] + 
 
           (a + c + b c - a d) 
 
                (1 + b) (c + (-1 + d) x)
            Log[------------------------]) + 
                   a + c + b c - a d
 
        (a + c + b c - a d) 
 
                      (-1 + d) (a + x + b x)
         PolyLog[2, -(----------------------)]) / 
                        a + c + b c - a d
 
                       2
      ((1 + b) (-1 + d) ) - 
 
     (c ((Log[a + x + b x] - 
 
                   (-1 + d) (a + x + b x)
             Log[-(----------------------)]) 
                     a + c + b c - a d
 
                              2
           Log[c + (-1 + d) x]  - 
 
          2 Log[c + (-1 + d) x] 
 
                      (1 + b) (c + (-1 + d) x)
           PolyLog[2, ------------------------] + 
                         a + c + b c - a d
 
                       (1 + b) (c + (-1 + d) x)
          2 PolyLog[3, ------------------------])) / 
                          a + c + b c - a d
 
                 2
      (2 (-1 + d) ))
Time to compute: 1.25 second
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