\(\int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx\) [766]

Optimal result

Integrand size = 29, antiderivative size = 771 \[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=\frac {\sqrt {3+b} (c-d) \sqrt {c+d} (5 b c+3 d) E\left (\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right )|\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{4 b (b c-3 d) f}+\frac {\sqrt {c+d} \left (18 b c d-9 d^2+b^2 \left (3 c^2+4 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(3+b) d},\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right ),\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{4 b^2 \sqrt {3+b} d f}+\frac {(b c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \sqrt {3+b \sin (e+f x)}}-\frac {(5 b c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f \sqrt {3+b \sin (e+f x)}}+\frac {(3+b)^{3/2} (5 b c-3 d+2 b d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{4 b^2 \sqrt {c+d} f}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {3+b \sin (e+f x)}} \]

[Out]

1/4*(6*a*b*c*d-a^2*d^2+b^2*(3*c^2+4*d^2))*EllipticPi((a+b)^(1/2)*(c+d*sin(f*x+e))^(1/2)/(c+d)^(1/2)/(a+b*sin(f
*x+e))^(1/2),b*(c+d)/(a+b)/d,((a-b)*(c+d)/(a+b)/(c-d))^(1/2))*sec(f*x+e)*(a+b*sin(f*x+e))*(c+d)^(1/2)*(-(-a*d+
b*c)*(1-sin(f*x+e))/(c+d)/(a+b*sin(f*x+e)))^(1/2)*((-a*d+b*c)*(1+sin(f*x+e))/(c-d)/(a+b*sin(f*x+e)))^(1/2)/b^2
/d/f/(a+b)^(1/2)+1/4*(c-d)*(a*d+5*b*c)*EllipticE((a+b)^(1/2)*(c+d*sin(f*x+e))^(1/2)/(c+d)^(1/2)/(a+b*sin(f*x+e
))^(1/2),((a-b)*(c+d)/(a+b)/(c-d))^(1/2))*sec(f*x+e)*(a+b*sin(f*x+e))*(a+b)^(1/2)*(c+d)^(1/2)*(-(-a*d+b*c)*(1-
sin(f*x+e))/(c+d)/(a+b*sin(f*x+e)))^(1/2)*((-a*d+b*c)*(1+sin(f*x+e))/(c-d)/(a+b*sin(f*x+e)))^(1/2)/b/(-a*d+b*c
)/f-1/2*b*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/f/(a+b*sin(f*x+e))^(1/2)+1/4*(a+b)^(3/2)*(-a*d+5*b*c+2*b*d)*Ellipt
icF((c+d)^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(c+d*sin(f*x+e))^(1/2),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*sec
(f*x+e)*(c+d*sin(f*x+e))*((-a*d+b*c)*(1-sin(f*x+e))/(a+b)/(c+d*sin(f*x+e)))^(1/2)*(-(-a*d+b*c)*(1+sin(f*x+e))/
(a-b)/(c+d*sin(f*x+e)))^(1/2)/b^2/f/(c+d)^(1/2)+1/2*(-a*d+b*c)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/f/(a+b*sin(f*
x+e))^(1/2)-1/4*(a*d+5*b*c)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/f/(a+b*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 2.29 (sec) , antiderivative size = 784, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2900, 3126, 3140, 3132, 2890, 3077, 2897, 3075} \[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=\frac {\sqrt {c+d} \left (-a^2 d^2+6 a b c d+b^2 \left (3 c^2+4 d^2\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{4 b^2 d f \sqrt {a+b}}+\frac {(a+b)^{3/2} (-a d+5 b c+2 b d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{4 b^2 f \sqrt {c+d}}+\frac {\sqrt {a+b} (c-d) \sqrt {c+d} (a d+5 b c) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{4 b f (b c-a d)}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}+\frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {(a d+5 b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f \sqrt {a+b \sin (e+f x)}} \]

[In]

Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2),x]

[Out]

(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(5*b*c + a*d)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c
+ d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - S
in[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f
*x]))]*(a + b*Sin[e + f*x]))/(4*b*(b*c - a*d)*f) + (Sqrt[c + d]*(6*a*b*c*d - a^2*d^2 + b^2*(3*c^2 + 4*d^2))*El
lipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e
+ f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*
(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f
*x]))/(4*b^2*Sqrt[a + b]*d*f) + ((b*c - a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*f*Sqrt[a + b*Sin[e + f*
x]]) - ((5*b*c + a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*f*Sqrt[a + b*Sin[e + f*x]]) + ((a + b)^(3/2)*(
5*b*c - a*d + 2*b*d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f
*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c +
d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]
))/(4*b^2*Sqrt[c + d]*f) - (b*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(2*f*Sqrt[a + b*Sin[e + f*x]])

Rule 2890

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[
2*((a + b*Sin[e + f*x])/(d*f*Rt[(a + b)/(c + d), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c -
d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticPi
[b*((c + d)/(d*(a + b))), ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]])],
(a - b)*((c + d)/((a + b)*(c - d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2
, 0] && NeQ[c^2 - d^2, 0] && PosQ[(a + b)/(c + d)]

Rule 2897

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Si
mp[2*((c + d*Sin[e + f*x])/(f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 - Sin[e +
f*x])/((a + b)*(c + d*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 + Sin[e + f*x])/((a - b)*(c + d*Sin[e + f*x]))
)]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], (a + b)*((c -
d)/((a - b)*(c + d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c
^2 - d^2, 0] && PosQ[(c + d)/(a + b)]

Rule 2900

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-b)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^n/(f*(m + n))), x] + Dist[1/(d*(m + n)
), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n - 1)*Simp[a^2*c*d*(m + n) + b*d*(b*c*(m - 1) + a*d
*n) + (a*d*(2*b*c + a*d)*(m + n) - b*d*(a*c - b*d*(m + n - 1)))*Sin[e + f*x] + b*d*(b*c*n + a*d*(2*m + n - 1))
*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c
^2 - d^2, 0] && LtQ[0, m, 2] && LtQ[-1, n, 2] && NeQ[m + n, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3075

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin
[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A*(c - d)*((a + b*Sin[e + f*x])/(f*(b*c - a*d)^2*Rt[(a + b)/(c +
d), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d)
)*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticE[ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Sin
[e + f*x]]/Sqrt[a + b*Sin[e + f*x]])], (a - b)*((c + d)/((a + b)*(c - d)))], x] /; FreeQ[{a, b, c, d, e, f, A,
B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]

Rule 3077

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*s
in[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[(A - B)/(a - b), Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e
+ f*x]]), x], x] - Dist[(A*b - a*B)/(a - b), Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin
[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2
- d^2, 0] && NeQ[A, B]

Rule 3126

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(d*(n + 1)
*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) +
(c*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*
c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n +
1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2
, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3132

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(((a_.) + (b_.)*sin[(e_.) + (f_.
)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[C/b^2, Int[Sqrt[a + b*Sin[e + f
*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Dist[1/b^2, Int[(A*b^2 - a^2*C + b*(b*B - 2*a*C)*Sin[e + f*x])/((a + b
*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a
*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3140

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) +
(f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(Sqrt[c + d*Sin[
e + f*x]]/(d*f*Sqrt[a + b*Sin[e + f*x]])), x] + Dist[1/(2*d), Int[(1/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Si
n[e + f*x]]))*Simp[2*a*A*d - C*(b*c - a*d) - 2*(a*c*C - d*(A*b + a*B))*Sin[e + f*x] + (2*b*B*d - C*(b*c + a*d)
)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0
] && NeQ[c^2 - d^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}+\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (\frac {1}{2} d \left (4 a^2 c-b^2 c+3 a b d\right )+d \left (3 a b c+2 a^2 d+b^2 d\right ) \sin (e+f x)+\frac {3}{2} b d (b c+a d) \sin ^2(e+f x)\right )}{(a+b \sin (e+f x))^{3/2}} \, dx}{2 d} \\ & = \frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {\int \frac {-\frac {1}{4} b \left (a^2-b^2\right ) d \left (4 a c^2+b c d+a d^2\right )-\frac {1}{2} b \left (a^2-b^2\right ) d \left (2 b c^2+3 a c d+b d^2\right ) \sin (e+f x)-\frac {1}{4} b \left (a^2-b^2\right ) d^2 (5 b c+a d) \sin ^2(e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{b \left (a^2-b^2\right ) d} \\ & = \frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {(5 b c+a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f \sqrt {a+b \sin (e+f x)}}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {\int \frac {-\frac {1}{4} b \left (a^2-b^2\right ) d^2 \left (8 a^2 c^2-5 b^2 c^2+6 a b c d+3 a^2 d^2\right )-\frac {1}{2} b \left (a^2-b^2\right ) d^2 \left (5 a^2 c d+b^2 c d+3 a b \left (c^2+d^2\right )\right ) \sin (e+f x)-\frac {1}{4} b \left (a^2-b^2\right ) d^2 \left (6 a b c d-a^2 d^2+b^2 \left (3 c^2+4 d^2\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{2 b \left (a^2-b^2\right ) d^2} \\ & = \frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {(5 b c+a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f \sqrt {a+b \sin (e+f x)}}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {\int \frac {-\frac {1}{4} b^3 \left (a^2-b^2\right ) d^2 \left (8 a^2 c^2-5 b^2 c^2+6 a b c d+3 a^2 d^2\right )+\frac {1}{4} a^2 b \left (a^2-b^2\right ) d^2 \left (6 a b c d-a^2 d^2+b^2 \left (3 c^2+4 d^2\right )\right )+b \left (-\frac {1}{2} b^2 \left (a^2-b^2\right ) d^2 \left (5 a^2 c d+b^2 c d+3 a b \left (c^2+d^2\right )\right )+\frac {1}{2} a b \left (a^2-b^2\right ) d^2 \left (6 a b c d-a^2 d^2+b^2 \left (3 c^2+4 d^2\right )\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{2 b^3 \left (a^2-b^2\right ) d^2}+\frac {\left (6 a b c d-a^2 d^2+b^2 \left (3 c^2+4 d^2\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{8 b^2} \\ & = \frac {\sqrt {c+d} \left (6 a b c d-a^2 d^2+b^2 \left (3 c^2+4 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{4 b^2 \sqrt {a+b} d f}+\frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {(5 b c+a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f \sqrt {a+b \sin (e+f x)}}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {((a+b) (b c-a d) (5 b c+a d)) \int \frac {1+\sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{8 b}+\frac {((a+b) (b c-a d) (5 b c-a d+2 b d)) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{8 b^2} \\ & = \frac {\sqrt {a+b} (c-d) \sqrt {c+d} (5 b c+a d) E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{4 b (b c-a d) f}+\frac {\sqrt {c+d} \left (6 a b c d-a^2 d^2+b^2 \left (3 c^2+4 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{4 b^2 \sqrt {a+b} d f}+\frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {(5 b c+a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f \sqrt {a+b \sin (e+f x)}}+\frac {(a+b)^{3/2} (5 b c-a d+2 b d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{4 b^2 \sqrt {c+d} f}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1855\) vs. \(2(771)=1542\).

Time = 13.62 (sec) , antiderivative size = 1855, normalized size of antiderivative = 2.41 \[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=-\frac {d \cos (e+f x) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}+\frac {-\frac {4 (-b c+3 d) \left (24 c^2+7 b c d+9 d^2\right ) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{\sqrt {2}}\right ),\frac {2 (-b c+3 d)}{(3+b) (-c+d)}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (3+b \sin (e+f x))}{-b c+3 d}} \sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{(3+b) (c+d) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-4 (-b c+3 d) \left (8 b c^2+36 c d+4 b d^2\right ) \left (\frac {\sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{\sqrt {2}}\right ),\frac {2 (-b c+3 d)}{(3+b) (-c+d)}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (3+b \sin (e+f x))}{-b c+3 d}} \sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{(3+b) (c+d) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {\sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} \operatorname {EllipticPi}\left (\frac {-b c+3 d}{(3+b) d},\arcsin \left (\frac {\sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{\sqrt {2}}\right ),\frac {2 (-b c+3 d)}{(3+b) (-c+d)}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (3+b \sin (e+f x))}{-b c+3 d}} \sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{(3+b) d \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )+2 \left (-5 b c d-3 d^2\right ) \left (\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d \sqrt {3+b \sin (e+f x)}}+\frac {\sqrt {\frac {3-b}{3+b}} (3+b) \cos \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {3-b}{3+b}} \sin \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{\sqrt {\frac {3+b \sin (e+f x)}{3+b}}}\right )|\frac {2 (-b c+3 d)}{(3-b) (c+d)}\right ) \sqrt {c+d \sin (e+f x)}}{b d \sqrt {\frac {(3+b) \cos ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{3+b \sin (e+f x)}} \sqrt {3+b \sin (e+f x)} \sqrt {\frac {3+b \sin (e+f x)}{3+b}} \sqrt {\frac {(3+b) (c+d \sin (e+f x))}{(c+d) (3+b \sin (e+f x))}}}-\frac {2 (-b c+3 d) \left (\frac {((3+b) c+3 d) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{\sqrt {2}}\right ),\frac {2 (-b c+3 d)}{(3+b) (-c+d)}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (3+b \sin (e+f x))}{-b c+3 d}} \sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{(3+b) (c+d) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {(b c+3 d) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} \operatorname {EllipticPi}\left (\frac {-b c+3 d}{(3+b) d},\arcsin \left (\frac {\sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{\sqrt {2}}\right ),\frac {2 (-b c+3 d)}{(3+b) (-c+d)}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (3+b \sin (e+f x))}{-b c+3 d}} \sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{(3+b) d \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{b d}\right )}{8 f} \]

[In]

Integrate[Sqrt[3 + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2),x]

[Out]

-1/2*(d*Cos[e + f*x]*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/f + ((-4*(-(b*c) + 3*d)*(24*c^2 + 7*b*
c*d + 9*d^2)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/
2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e + f
*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*
Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*(c + d)*Sqrt[3 + b*S
in[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - 4*(-(b*c) + 3*d)*(8*b*c^2 + 36*c*d + 4*b*d^2)*((Sqrt[((c + d)*Cot[(-e
+ Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x])
)/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqr
t[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*
x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*(c + d)*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*
x]]) - (Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + 3*d)/((3 + b)*d), ArcSin[Sqrt
[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 +
b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e +
f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 +
b)*d*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])) + 2*(-5*b*c*d - 3*d^2)*((Cos[e + f*x]*Sqrt[c + d*Sin
[e + f*x]])/(d*Sqrt[3 + b*Sin[e + f*x]]) + (Sqrt[(3 - b)/(3 + b)]*(3 + b)*Cos[(-e + Pi/2 - f*x)/2]*EllipticE[A
rcSin[(Sqrt[(3 - b)/(3 + b)]*Sin[(-e + Pi/2 - f*x)/2])/Sqrt[(3 + b*Sin[e + f*x])/(3 + b)]], (2*(-(b*c) + 3*d))
/((3 - b)*(c + d))]*Sqrt[c + d*Sin[e + f*x]])/(b*d*Sqrt[((3 + b)*Cos[(-e + Pi/2 - f*x)/2]^2)/(3 + b*Sin[e + f*
x])]*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[(3 + b*Sin[e + f*x])/(3 + b)]*Sqrt[((3 + b)*(c + d*Sin[e + f*x]))/((c + d)*
(3 + b*Sin[e + f*x]))]) - (2*(-(b*c) + 3*d)*((((3 + b)*c + 3*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c
+ d)]*EllipticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]
], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2
- f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*
x]))/(-(b*c) + 3*d)])/((3 + b)*(c + d)*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - ((b*c + 3*d)*Sqrt[
((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + 3*d)/((3 + b)*d), ArcSin[Sqrt[((-3 - b)*Cs
c[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))
]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*
c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*d*Sqrt[3
+ b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])))/(b*d)))/(8*f)

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 11.78 (sec) , antiderivative size = 247614, normalized size of antiderivative = 321.16

method	result	size
default	\(\text {Expression too large to display}\)	\(247614\)

[In]

int((a+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F(-1)]

Timed out. \[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=\text {Timed out} \]

[In]

integrate((a+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=\int \sqrt {a + b \sin {\left (e + f x \right )}} \left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

integrate((a+b*sin(f*x+e))**(1/2)*(c+d*sin(f*x+e))**(3/2),x)

[Out]

Integral(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))**(3/2), x)

Maxima [F]

\[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=\int { \sqrt {b \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]

[In]

integrate((a+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^(3/2), x)

Giac [F]

\[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=\int { \sqrt {b \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]

[In]

integrate((a+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=\int \sqrt {a+b\,\sin \left (e+f\,x\right )}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]

[In]

int((a + b*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^(3/2),x)

[Out]

int((a + b*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^(3/2), x)