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

          Log[b + a x]
Integrate[------------, x] == 
                m
               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: 0.09 second
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