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Sqrt[b + a*x^2]*((b*x)/(8*a) + x^3/4) - (b^2*Log[2*(a*x + Sqrt[a]*Sqrt[b + a*x^2])])/(8*a^(3/2))

Integrate[x^2*Sqrt[b + a*x^2], x] ==
Sqrt[b + a*x^2]*((b*x)/(8*a) + x^3/4) - (b^2*Log[2*(a*x + Sqrt[a]*Sqrt[b + a*x^2])])/ (8*a^(3/2)) 

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

                       3
            2   b x   x
Sqrt[b + a x ] (--- + --) - 
                8 a   4
 
   2                                  2
  b  Log[2 (a x + Sqrt[a] Sqrt[b + a x ])]
  ----------------------------------------
                      3/2
                   8 a
Time to compute: 0.08 second
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