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 ])