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Integrate[Sqrt[a + x]*(-b + x^2)^(3/2), x] ==
(Sqrt[a + x]*(2*(b - x^2)*(-8*a^3 + 29*a*b + 6*a^2*x + 77*b*x - 5*a*x^2 - 35*x^3) - (8*(a + x)*(-((Sqrt[-a - Sqrt[b]]*(-4*a^4 + 15*a^2*b + 21*b^2)*(-b + x^2))/(a + x)^2) - (I*(4*a^5 + 4*a^4*Sqrt[b] - 15*a^3*b - 15*a^2*b^(3/2) - 21*a*b^2 - 21*b^(5/2))* Sqrt[(-Sqrt[b] + x)/(a + x)]* Sqrt[(Sqrt[b] + x)/(a + x)]*EllipticE[ I*ArcSinh[Sqrt[-a - Sqrt[b]]/Sqrt[a + x]], (a - Sqrt[b])/(a + Sqrt[b])])/Sqrt[a + x] + (I*Sqrt[b]*(4*a^4 + a^3*Sqrt[b] - 15*a^2*b - 33*a*b^(3/2) - 21*b^2)*Sqrt[(-Sqrt[b] + x)/ (a + x)]*Sqrt[(Sqrt[b] + x)/(a + x)]* EllipticF[I*ArcSinh[Sqrt[-a - Sqrt[b]]/ Sqrt[a + x]], (a - Sqrt[b])/ (a + Sqrt[b])])/Sqrt[a + x]))/ Sqrt[-a - Sqrt[b]]))/(315*Sqrt[-b + x^2])



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

                      2
(Sqrt[a + x] (2 (b - x ) 
 
            3               2                   2
       (-8 a  + 29 a b + 6 a  x + 77 b x - 5 a x  - 
 
             3
         35 x ) - (8 (a + x) 
 
         (-((Sqrt[-a - Sqrt[b]] 
 
                     4       2         2         2
                (-4 a  + 15 a  b + 21 b ) (-b + x ))\
 
                        2
               / (a + x) ) - 
 
                  5      4               3
           (I (4 a  + 4 a  Sqrt[b] - 15 a  b - 
 
                    2  3/2         2       5/2
                15 a  b    - 21 a b  - 21 b   ) 
 
                   -Sqrt[b] + x       Sqrt[b] + x
              Sqrt[------------] Sqrt[-----------] 
                      a + x              a + x
 
              EllipticE[I 
 
                        Sqrt[-a - Sqrt[b]]
                ArcSinh[------------------], 
                           Sqrt[a + x]
 
               a - Sqrt[b]
               -----------]) / Sqrt[a + x] + 
               a + Sqrt[b]
 
           (I Sqrt[b] 
 
                  4    3               2
              (4 a  + a  Sqrt[b] - 15 a  b - 
 
                      3/2       2       -Sqrt[b] + x
                33 a b    - 21 b ) Sqrt[------------] 
                                           a + x
 
                   Sqrt[b] + x
              Sqrt[-----------] 
                      a + x
 
              EllipticF[I 
 
                        Sqrt[-a - Sqrt[b]]
                ArcSinh[------------------], 
                           Sqrt[a + x]
 
               a - Sqrt[b]
               -----------]) / Sqrt[a + x])) / 
               a + Sqrt[b]
 
                                              2
       Sqrt[-a - Sqrt[b]])) / (315 Sqrt[-b + x ])

Time to compute: 4.33 second

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