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(a^(b^(-1) + x)*b + (1 + b*x)*ExpIntegralEi[((1 + b*x)*Log[a])/b]*(b - Log[a]))/(a^b^(-1)*b^3*(1 + b*x))

Integrate[(a^x*x)/(1 + b*x)^2, x] ==
(a^(b^(-1) + x)*b + (1 + b*x)*ExpIntegralEi[ ((1 + b*x)*Log[a])/b]*(b - Log[a]))/ (a^b^(-1)*b^3*(1 + b*x)) 

              x
             a  x
Integrate[----------, x] == 
                   2
          (1 + b x)

  1/b + x
(a        b + (1 + b x) 
 
                   (1 + b x) Log[a]
     ExpIntegralEi[----------------] (b - Log[a])) / 
                          b
 
    1/b  3
  (a    b  (1 + b x))
Time to compute: 0.07 second
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