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-((ExpIntegralE[-n, -Log[a*x]]*Log[a*x]^(1 + n))/a)

Integrate[Log[a*x]^n, x] ==
-((ExpIntegralE[-n, -Log[a*x]]*Log[a*x]^(1 + n))/a) 

                  n
Integrate[Log[a x] , x] == 

                                      1 + n
  ExpIntegralE[-n, -Log[a x]] Log[a x]
-(-----------------------------------------)
                      a
Time to compute: 0.03 second
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