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Integrate[x^n*Log[a + b*x + c*x^2], x] ==
(x^n*(-2*x + (2^(1 - n)*a*(1 + n)* (((c*x)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x))^n* Hypergeometric2F1[-n, -n, 1 - n, (b - Sqrt[b^2 - 4*a*c])/ (b - Sqrt[b^2 - 4*a*c] + 2*c*x)] - ((c*x)/(b - Sqrt[b^2 - 4*a*c] + 2*c*x))^n* Hypergeometric2F1[-n, -n, 1 - n, (b + Sqrt[b^2 - 4*a*c])/ (b + Sqrt[b^2 - 4*a*c] + 2*c*x)]))/ (Sqrt[b^2 - 4*a*c]*n* ((c*x)/(b - Sqrt[b^2 - 4*a*c] + 2*c*x))^n* ((c*x)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x))^n) + (2^(-1 - n)*b*(1 + n)*((-b + Sqrt[b^2 - 4*a*c])* ((c*x)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x))^n* Hypergeometric2F1[-n, -n, 1 - n, (b - Sqrt[b^2 - 4*a*c])/ (b - Sqrt[b^2 - 4*a*c] + 2*c*x)] + (b + Sqrt[b^2 - 4*a*c])* ((c*x)/(b - Sqrt[b^2 - 4*a*c] + 2*c*x))^n* Hypergeometric2F1[-n, -n, 1 - n, (b + Sqrt[b^2 - 4*a*c])/ (b + Sqrt[b^2 - 4*a*c] + 2*c*x)]))/ (c*Sqrt[b^2 - 4*a*c]*n* ((c*x)/(b - Sqrt[b^2 - 4*a*c] + 2*c*x))^n* ((c*x)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x))^n) + (1 + n)*x*Log[a + x*(b + c*x)]))/(1 + n)^2



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

  n           1 - n
(x  (-2 x + (2      a (1 + n) 
 
                       c x              n
         ((----------------------------)  
                     2
           b + Sqrt[b  - 4 a c] + 2 c x
 
            Hypergeometric2F1[-n, -n, 1 - n, 
 
                           2
                 b - Sqrt[b  - 4 a c]
             ----------------------------] - 
                       2
             b - Sqrt[b  - 4 a c] + 2 c x
 
                        c x              n
           (----------------------------)  
                      2
            b - Sqrt[b  - 4 a c] + 2 c x
 
            Hypergeometric2F1[-n, -n, 1 - n, 
 
                           2
                 b + Sqrt[b  - 4 a c]
             ----------------------------])) / 
                       2
             b + Sqrt[b  - 4 a c] + 2 c x
 
              2
       (Sqrt[b  - 4 a c] n 
 
                      c x              n
         (----------------------------)  
                    2
          b - Sqrt[b  - 4 a c] + 2 c x
 
                      c x              n
         (----------------------------) ) + 
                    2
          b + Sqrt[b  - 4 a c] + 2 c x
 
        -1 - n
      (2       b (1 + n) 
 
                      2
         ((-b + Sqrt[b  - 4 a c]) 
 
                         c x              n
            (----------------------------)  
                       2
             b + Sqrt[b  - 4 a c] + 2 c x
 
            Hypergeometric2F1[-n, -n, 1 - n, 
 
                           2
                 b - Sqrt[b  - 4 a c]
             ----------------------------] + 
                       2
             b - Sqrt[b  - 4 a c] + 2 c x
 
                      2
           (b + Sqrt[b  - 4 a c]) 
 
                         c x              n
            (----------------------------)  
                       2
             b - Sqrt[b  - 4 a c] + 2 c x
 
            Hypergeometric2F1[-n, -n, 1 - n, 
 
                           2
                 b + Sqrt[b  - 4 a c]
             ----------------------------])) / 
                       2
             b + Sqrt[b  - 4 a c] + 2 c x
 
                2
       (c Sqrt[b  - 4 a c] n 
 
                      c x              n
         (----------------------------)  
                    2
          b - Sqrt[b  - 4 a c] + 2 c x
 
                      c x              n
         (----------------------------) ) + 
                    2
          b + Sqrt[b  - 4 a c] + 2 c x
 
                                                2
      (1 + n) x Log[a + x (b + c x)])) / (1 + n)

Time to compute: 1.86 second

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