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


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



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

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

Time to compute: 0.10 second

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