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(x*ExpIntegralEi[(3*(a + Log[b*x^2]))/2])/(2*b*E^((3*a)/2)*Sqrt[b*x^2])

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

                2
               x
Integrate[-------------, x] == 
                     2
          a + Log[b x ]

                              2
                3 (a + Log[b x ])
x ExpIntegralEi[-----------------]
                        2
----------------------------------
          (3 a)/2         2
     2 b E        Sqrt[b x ]
Time to compute: 0.04 second
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