\(\int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx\) [765]

Optimal result

Integrand size = 29, antiderivative size = 861 \[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx=\frac {\sqrt {3+b} (c-d) \sqrt {c+d} \left (42 b c d-27 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right )|\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{24 b^2 (b c-3 d) f}-\frac {\sqrt {c+d} \left (45 b c d^2-27 d^3-3 b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(3+b) d},\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right ),\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{8 b^3 \sqrt {3+b} d f}-\frac {\left (42 b c d-27 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{24 b f \sqrt {3+b \sin (e+f x)}}-\frac {(13 b c-9 d) d \cos (e+f x) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{12 b f}-\frac {d^2 \cos (e+f x) (3+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {(3+b)^{3/2} \left (27 d^2-18 b d (2 c+d)+b^2 \left (33 c^2+26 c d+16 d^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{24 b^3 \sqrt {c+d} f} \]

[Out]

-1/8*(5*a^2*b*c*d^2-a^3*d^3-a*b^2*d*(15*c^2+4*d^2)-5*b^3*(c^3+4*c*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*si
n(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^3/d/f/(a+b)^(1/2)+1/24*(c-d)*(14*a*b*c*d-3*a^2*d^2+b^2*(33*c^2+16*d^2))*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^2/(-a*d+b*c)/f+1/24*(a+b)^(3/2)*(3*a^2*d^2-6*a*b*d*(2*c+d
)+b^2*(33*c^2+26*c*d+16*d^2))*EllipticF((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^3/f/(c+d)^(1/2)-1/3*d^2*cos(f*x+e)*(a+b*si
n(f*x+e))^(3/2)*(c+d*sin(f*x+e))^(1/2)/b/f-1/24*(14*a*b*c*d-3*a^2*d^2+b^2*(33*c^2+16*d^2))*cos(f*x+e)*(c+d*sin
(f*x+e))^(1/2)/b/f/(a+b*sin(f*x+e))^(1/2)-1/12*d*(-3*a*d+13*b*c)*cos(f*x+e)*(a+b*sin(f*x+e))^(1/2)*(c+d*sin(f*
x+e))^(1/2)/b/f

Rubi [A] (verified)

Time = 2.23 (sec) , antiderivative size = 888, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2872, 3128, 3140, 3132, 2890, 3077, 2897, 3075} \[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx=-\frac {\cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)} d^2}{3 b f}-\frac {(13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} d}{12 b f}+\frac {\sqrt {a+b} (c-d) \sqrt {c+d} \left (\left (33 c^2+16 d^2\right ) b^2+14 a c d b-3 a^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{24 b^2 (b c-a d) f}+\frac {(a+b)^{3/2} \left (\left (33 c^2+26 d c+16 d^2\right ) b^2-6 a d (2 c+d) b+3 a^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{24 b^3 \sqrt {c+d} f}-\frac {\left (\left (33 c^2+16 d^2\right ) b^2+14 a c d b-3 a^2 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{24 b f \sqrt {a+b \sin (e+f x)}}-\frac {\sqrt {c+d} \left (-5 \left (c^3+4 d^2 c\right ) b^3-a d \left (15 c^2+4 d^2\right ) b^2+5 a^2 c d^2 b-a^3 d^3\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 b^3 \sqrt {a+b} f d} \]

[In]

Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2),x]

[Out]

(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(14*a*b*c*d - 3*a^2*d^2 + b^2*(33*c^2 + 16*d^2))*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 - 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]))/(24*b^2*(b*c - a*d)*f) - (Sqrt[c + d]*(5*a^2*b
*c*d^2 - a^3*d^3 - a*b^2*d*(15*c^2 + 4*d^2) - 5*b^3*(c^3 + 4*c*d^2))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSi
n[(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]))/(8*b^3*Sqrt[a + b]*d*f) - ((14*a*
b*c*d - 3*a^2*d^2 + b^2*(33*c^2 + 16*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(24*b*f*Sqrt[a + b*Sin[e + f
*x]]) - (d*(13*b*c - 3*a*d)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(12*b*f) - (d^2*Co
s[e + f*x]*(a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]])/(3*b*f) + ((a + b)^(3/2)*(3*a^2*d^2 - 6*a*b*d*
(2*c + d) + b^2*(33*c^2 + 26*c*d + 16*d^2))*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]))/(24*b^3*Sqrt[c + d]*f)

Rule 2872

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Dist[1/
(d*(m + n)), Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d*(m + n) + b^2*(b*c*(m - 2) + a
*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n
- 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
&& NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m]
|| (EqQ[a, 0] && NeQ[c, 0])))

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 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 3128

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

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 {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {\int \frac {\sqrt {a+b \sin (e+f x)} \left (\frac {1}{2} \left (a d^3+3 b c \left (2 c^2+d^2\right )\right )+d \left (9 b c^2-a c d+2 b d^2\right ) \sin (e+f x)+\frac {1}{2} d^2 (13 b c-3 a d) \sin ^2(e+f x)\right )}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 b} \\ & = -\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{12 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {\int \frac {\frac {1}{4} d \left (13 b^2 c^2 d+a^2 d^3+a b \left (24 c^3+22 c d^2\right )\right )-\frac {1}{2} d \left (a^2 c d^2-a b d \left (23 c^2+7 d^2\right )-b^2 \left (12 c^3+19 c d^2\right )\right ) \sin (e+f x)+\frac {1}{4} d^2 \left (14 a b c d-3 a^2 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) \sin ^2(e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{6 b d} \\ & = -\frac {\left (14 a b c d-3 a^2 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{24 b f \sqrt {a+b \sin (e+f x)}}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{12 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {\int \frac {-\frac {1}{4} d^2 \left (a^3 d^3+b^3 c \left (33 c^2+16 d^2\right )-a b^2 d \left (45 c^2+16 d^2\right )-a^2 b c \left (48 c^2+61 d^2\right )\right )+\frac {1}{2} d^2 \left (13 b^3 c^2 d+a^3 c d^2+a^2 b d \left (32 c^2+15 d^2\right )+a b^2 c \left (15 c^2+44 d^2\right )\right ) \sin (e+f x)-\frac {3}{4} d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c 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}{12 b d^2} \\ & = -\frac {\left (14 a b c d-3 a^2 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{24 b f \sqrt {a+b \sin (e+f x)}}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{12 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {\int \frac {-\frac {1}{4} b^2 d^2 \left (a^3 d^3+b^3 c \left (33 c^2+16 d^2\right )-a b^2 d \left (45 c^2+16 d^2\right )-a^2 b c \left (48 c^2+61 d^2\right )\right )+\frac {3}{4} a^2 d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right )+b \left (\frac {1}{2} b d^2 \left (13 b^3 c^2 d+a^3 c d^2+a^2 b d \left (32 c^2+15 d^2\right )+a b^2 c \left (15 c^2+44 d^2\right )\right )+\frac {3}{2} a d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c 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}{12 b^3 d^2}-\frac {\left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{16 b^3} \\ & = -\frac {\sqrt {c+d} \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c 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))}{8 b^3 \sqrt {a+b} d f}-\frac {\left (14 a b c d-3 a^2 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{24 b f \sqrt {a+b \sin (e+f x)}}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{12 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {\left (-\frac {1}{4} b^2 d^2 \left (a^3 d^3+b^3 c \left (33 c^2+16 d^2\right )-a b^2 d \left (45 c^2+16 d^2\right )-a^2 b c \left (48 c^2+61 d^2\right )\right )+\frac {3}{4} a^2 d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right )-b \left (\frac {1}{2} b d^2 \left (13 b^3 c^2 d+a^3 c d^2+a^2 b d \left (32 c^2+15 d^2\right )+a b^2 c \left (15 c^2+44 d^2\right )\right )+\frac {3}{2} a d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{12 (a-b) b^3 d^2}-\frac {\left (-a b \left (\frac {1}{2} b d^2 \left (13 b^3 c^2 d+a^3 c d^2+a^2 b d \left (32 c^2+15 d^2\right )+a b^2 c \left (15 c^2+44 d^2\right )\right )+\frac {3}{2} a d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right )\right )+b \left (-\frac {1}{4} b^2 d^2 \left (a^3 d^3+b^3 c \left (33 c^2+16 d^2\right )-a b^2 d \left (45 c^2+16 d^2\right )-a^2 b c \left (48 c^2+61 d^2\right )\right )+\frac {3}{4} a^2 d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right )\right )\right ) \int \frac {1+\sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{12 (a-b) b^3 d^2} \\ & = \frac {\sqrt {a+b} (c-d) \sqrt {c+d} \left (14 a b c d-3 a^2 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) 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))}{24 b^2 (b c-a d) f}-\frac {\sqrt {c+d} \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c 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))}{8 b^3 \sqrt {a+b} d f}-\frac {\left (14 a b c d-3 a^2 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{24 b f \sqrt {a+b \sin (e+f x)}}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{12 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {(a+b)^{3/2} \left (3 a^2 d^2-6 a b d (2 c+d)+b^2 \left (33 c^2+26 c d+16 d^2\right )\right ) \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))}{24 b^3 \sqrt {c+d} f} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1945\) vs. \(2(861)=1722\).

Time = 8.69 (sec) , antiderivative size = 1945, normalized size of antiderivative = 2.26 \[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx=\frac {-\frac {4 (-b c+3 d) \left (144 b c^3+59 b^2 c^2 d+174 b c d^2-9 d^3+16 b^2 d^3\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 (48 b^2 c^3+276 b c^2 d-36 c d^2+76 b^2 c d^2+84 b d^3\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 (-33 b^2 c^2 d-42 b c d^2+27 d^3-16 b^2 d^3\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 )}{48 b f}+\frac {\sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \left (-\frac {d (13 b c+3 d) \cos (e+f x)}{12 b}-\frac {1}{6} d^2 \sin (2 (e+f x))\right )}{f} \]

[In]

Integrate[Sqrt[3 + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2),x]

[Out]

((-4*(-(b*c) + 3*d)*(144*b*c^3 + 59*b^2*c^2*d + 174*b*c*d^2 - 9*d^3 + 16*b^2*d^3)*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]]) -
4*(-(b*c) + 3*d)*(48*b^2*c^3 + 276*b*c^2*d - 36*c*d^2 + 76*b^2*c*d^2 + 84*b*d^3)*((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]]) -
(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*(-33*b^2*c^2*d - 42*b*c*d^2 + 27*d^3 - 16*b^2*d^3)*((C
os[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[ArcSin[(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*Sq
rt[((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)*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]])))/(b*d)))/(48*b*f) + (Sqrt[3 + b*Sin[e +
f*x]]*Sqrt[c + d*Sin[e + f*x]]*(-1/12*(d*(13*b*c + 3*d)*Cos[e + f*x])/b - (d^2*Sin[2*(e + f*x)])/6))/f

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 18.53 (sec) , antiderivative size = 364669, normalized size of antiderivative = 423.54

method	result	size
default	\(\text {Expression too large to display}\)	\(364669\)

[In]

int((a+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F(-1)]

Timed out. \[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate((a+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate((a+b*sin(f*x+e))**(1/2)*(c+d*sin(f*x+e))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx=\int { \sqrt {b \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]

[In]

integrate((a+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^(5/2), x)

Giac [F]

\[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx=\int { \sqrt {b \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]

[In]

integrate((a+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(5/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx=\int \sqrt {a+b\,\sin \left (e+f\,x\right )}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]

[In]

int((a + b*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^(5/2),x)

[Out]

int((a + b*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^(5/2), x)