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


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



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

                                        1 + n
-(Gamma[1 + n, -((1 + m) Log[x])] Log[x]      
 
                       -1 - n
    (-((1 + m) Log[x]))      )

Time to compute: 0.05 second

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