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

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

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

                                            1 + n
-(ExpIntegralE[-n, -((1 + m) Log[x])] Log[x]     )
Time to compute: 0.05 second
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