3.765 \(\int \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx\)

Optimal. Leaf size=888 \[ -\frac {\cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)} d^2}{3 b f}-\frac {(13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} d}{12 b f}+\frac {\sqrt {a+b} (c-d) \sqrt {c+d} \left (\left (33 c^2+16 d^2\right ) b^2+14 a c d b-3 a^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{24 b^2 (b c-a d) f}+\frac {(a+b)^{3/2} \left (\left (33 c^2+26 d c+16 d^2\right ) b^2-6 a d (2 c+d) b+3 a^2 d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{24 b^3 \sqrt {c+d} f}-\frac {\left (\left (33 c^2+16 d^2\right ) b^2+14 a c d b-3 a^2 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{24 b f \sqrt {a+b \sin (e+f x)}}-\frac {\sqrt {c+d} \left (-5 \left (c^3+4 d^2 c\right ) b^3-a d \left (15 c^2+4 d^2\right ) b^2+5 a^2 c d^2 b-a^3 d^3\right ) \Pi \left (\frac {b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 b^3 \sqrt {a+b} f d} \]

[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]  time = 3.39, antiderivative size = 888, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2793, 3049, 3061, 3053, 2811, 2998, 2818, 2996} \[ -\frac {\cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)} d^2}{3 b f}-\frac {(13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} d}{12 b f}+\frac {\sqrt {a+b} (c-d) \sqrt {c+d} \left (\left (33 c^2+16 d^2\right ) b^2+14 a c d b-3 a^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{24 b^2 (b c-a d) f}+\frac {(a+b)^{3/2} \left (\left (33 c^2+26 d c+16 d^2\right ) b^2-6 a d (2 c+d) b+3 a^2 d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{24 b^3 \sqrt {c+d} f}-\frac {\left (\left (33 c^2+16 d^2\right ) b^2+14 a c d b-3 a^2 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{24 b f \sqrt {a+b \sin (e+f x)}}-\frac {\sqrt {c+d} \left (-5 \left (c^3+4 d^2 c\right ) b^3-a d \left (15 c^2+4 d^2\right ) b^2+5 a^2 c d^2 b-a^3 d^3\right ) \Pi \left (\frac {b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 b^3 \sqrt {a+b} f d} \]

Antiderivative was successfully verified.

[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 2793

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[(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 2811

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])*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))])/(d*f*
Rt[(a + b)/(c + d), 2]*Cos[e + f*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] && PosQ[(a + b)/(c + d)]

Rule 2818

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])*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))])/(f*(b*c - a*d)*Rt[(c + d)/(a
+ b), 2]*Cos[e + f*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] && PosQ[(c + d)/(a + b)]

Rule 2996

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])*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))])/(f*(b*c - a*d)^2*Rt[(a + b)/(c + d), 2]*Cos[e + f*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] && EqQ[A, B] && PosQ[(a + b)/(c + d)]

Rule 2998

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 3049

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 3053

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 3061

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*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])/((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]

Rubi steps

\begin {align*} \int \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx &=-\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 ) \Pi \left (\frac {b (c+d)}{(a+b) d};\sin ^{-1}\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 (\sin ^{-1}\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 ) \Pi \left (\frac {b (c+d)}{(a+b) d};\sin ^{-1}\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 ) F\left (\sin ^{-1}\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*}

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Mathematica [B]  time = 7.14, size = 1978, normalized size = 2.23 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/48*((-4*(-(b*c) + a*d)*(-48*a*b*c^3 - 59*b^2*c^2*d - 58*a*b*c*d^2 + a^2*d^3 - 16*b^2*d^3)*Sqrt[((c + d)*Cot
[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f
*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + 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*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2
- f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e
+ f*x]]) - 4*(-(b*c) + a*d)*(-48*b^2*c^3 - 92*a*b*c^2*d + 4*a^2*c*d^2 - 76*b^2*c*d^2 - 28*a*b*d^3)*((Sqrt[((c
+ d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*S
in[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + 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*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e
+ Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + 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) + a*d)/((a + b)*d),
ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) +
a*d))/((a + 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*(a
+ b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) +
a*d)])/((a + b)*d*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])) + 2*(33*b^2*c^2*d + 14*a*b*c*d^2 - 3*a^2
*d^3 + 16*b^2*d^3)*((Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d*Sqrt[a + b*Sin[e + f*x]]) + (Sqrt[(a - b)/(a +
b)]*(a + b)*Cos[(-e + Pi/2 - f*x)/2]*EllipticE[ArcSin[(Sqrt[(a - b)/(a + b)]*Sin[(-e + Pi/2 - f*x)/2])/Sqrt[(a
+ b*Sin[e + f*x])/(a + b)]], (2*(-(b*c) + a*d))/((a - b)*(c + d))]*Sqrt[c + d*Sin[e + f*x]])/(b*d*Sqrt[((a +
b)*Cos[(-e + Pi/2 - f*x)/2]^2)/(a + b*Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/(a + b
)]*Sqrt[((a + b)*(c + d*Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x]))]) - (2*(-(b*c) + a*d)*((((a + b)*c + a*d
)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]
^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + 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*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a -
b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]
]*Sqrt[c + d*Sin[e + f*x]]) - ((b*c + a*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b
*c) + a*d)/((a + b)*d), ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]
/Sqrt[2]], (2*(-(b*c) + a*d))/((a + 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*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Si
n[e + f*x]))/(-(b*c) + a*d)])/((a + b)*d*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])))/(b*d)))/(b*f) +
(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*(-1/12*(d*(13*b*c + a*d)*Cos[e + f*x])/b - (d^2*Sin[2*(e +
f*x)])/6))/f

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 17.17, size = 404501, normalized size = 455.52 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {a+b\,\sin \left (e+f\,x\right )}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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