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Sqrt[a - b*x^2]*((5*a*x)/8 - (b*x^3)/4) + (3*a^2*ArcTan[(Sqrt[b]*x)/Sqrt[a - b*x^2]])/(8*Sqrt[b])


Integrate[(a - b*x^2)^(3/2), x] ==
Sqrt[a - b*x^2]*((5*a*x)/8 - (b*x^3)/4) + (3*a^2*ArcTan[(Sqrt[b]*x)/Sqrt[a - b*x^2]])/ (8*Sqrt[b])



                  2 3/2
Integrate[(a - b x )   , x] == 

                           3
            2   5 a x   b x
Sqrt[a - b x ] (----- - ----) + 
                  8      4
 
     2          Sqrt[b] x
  3 a  ArcTan[--------------]
                          2
              Sqrt[a - b x ]
  ---------------------------
           8 Sqrt[b]

Time to compute: 0.11 second

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