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(x^(1 + m)*(-1 + Hypergeometric2F1[1, 1 + m, 2 + m, -((a*x)/b)] + (1 + m)*Log[b + a*x]))/(1 + m)^2

Integrate[x^m*Log[b + a*x], x] ==
(x^(1 + m)*(-1 + Hypergeometric2F1[1, 1 + m, 2 + m, -((a*x)/b)] + (1 + m)*Log[b + a*x]))/(1 + m)^2 

           m
Integrate[x  Log[b + a x], x] == 

  1 + m
(x      (-1 + Hypergeometric2F1[1, 1 + m, 2 + m, 
 
         a x                                     2
       -(---)] + (1 + m) Log[b + a x])) / (1 + m)
          b
Time to compute: 0.07 second
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