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Integrate[x^2/(1 + Sqrt[b + a*x]), x] ==
(30*a*(-1 + b)*x - 15*a^2*x^2 + 4*Sqrt[b + a*x]* (15 + 8*b^2 + 5*a*x + 3*a^2*x^2 - b*(25 + 4*a*x)) - 60*(-1 + b)^2* ArcTanh[Sqrt[b + a*x]] - 30*(-1 + b)^2* Log[-1 + b + a*x])/(30*a^3)

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

                       2  2
(30 a (-1 + b) x - 15 a  x  + 
 
                             2              2  2
    4 Sqrt[b + a x] (15 + 8 b  + 5 a x + 3 a  x  - 
 
       b (25 + 4 a x)) - 
 
               2
    60 (-1 + b)  ArcTanh[Sqrt[b + a x]] - 
 
               2                           3
    30 (-1 + b)  Log[-1 + b + a x]) / (30 a )