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

Optimal. Leaf size=894 \[ -\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} b^2}{3 d f}+\frac {(3 b c-13 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} b}{12 d f}-\frac {\left (-\left (\left (3 c^2-16 d^2\right ) b^2\right )+14 a c d b+33 a^2 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)} b}{24 d^2 f \sqrt {a+b \sin (e+f x)}}+\frac {\sqrt {a+b} (c-d) \sqrt {c+d} \left (-\left (\left (3 c^2-16 d^2\right ) b^2\right )+14 a c d b+33 a^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{24 d^2 (b c-a d) f}+\frac {\sqrt {c+d} \left (\left (c^3+4 d^2 c\right ) b^3-5 a d \left (c^2-4 d^2\right ) b^2+15 a^2 c d^2 b+5 a^3 d^3\right ) \Pi \left (\frac {b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 \sqrt {a+b} d^3 f b}+\frac {(a+b)^{3/2} \left (-\left (\left (3 c^2-2 d c-16 d^2\right ) b^2\right )+6 a d (2 c+3 d) b+15 a^2 d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{24 d^2 \sqrt {c+d} f b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 3.27, antiderivative size = 894, 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} \[ -\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} b^2}{3 d f}+\frac {(3 b c-13 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} b}{12 d f}-\frac {\left (-\left (3 c^2-16 d^2\right ) b^2+14 a c d b+33 a^2 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)} b}{24 d^2 f \sqrt {a+b \sin (e+f x)}}+\frac {\sqrt {a+b} (c-d) \sqrt {c+d} \left (-\left (3 c^2-16 d^2\right ) b^2+14 a c d b+33 a^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{24 d^2 (b c-a d) f}+\frac {\sqrt {c+d} \left (\left (c^3+4 d^2 c\right ) b^3-5 a d \left (c^2-4 d^2\right ) b^2+15 a^2 c d^2 b+5 a^3 d^3\right ) \Pi \left (\frac {b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 \sqrt {a+b} d^3 f b}+\frac {(a+b)^{3/2} \left (-\left (3 c^2-2 d c-16 d^2\right ) b^2+6 a d (2 c+3 d) b+15 a^2 d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{24 d^2 \sqrt {c+d} f b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(14*a*b*c*d + 33*a^2*d^2 - b^2*(3*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*d^2*(b*c - a*d)*f) + (Sqrt[c + d]*(15*a^2*
b*c*d^2 + 5*a^3*d^3 - 5*a*b^2*d*(c^2 - 4*d^2) + 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*Sqrt[a + b]*d^3*f) - (b*(14*
a*b*c*d + 33*a^2*d^2 - b^2*(3*c^2 - 16*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(24*d^2*f*Sqrt[a + b*Sin[e
+ f*x]]) + (b*(3*b*c - 13*a*d)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(12*d*f) + ((a
+ b)^(3/2)*(15*a^2*d^2 + 6*a*b*d*(2*c + 3*d) - b^2*(3*c^2 - 2*c*d - 16*d^2))*EllipticF[ArcSin[(Sqrt[c + d]*Sq
rt[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*d^2*Sqrt[c + d]*f) - (b^2*Cos[e + f*x]*Sqrt[
a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2))/(3*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 (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)} \, dx &=-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}+\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (\frac {1}{2} \left (b^3 c+6 a^3 d+3 a b^2 d\right )-b \left (a b c-9 a^2 d-2 b^2 d\right ) \sin (e+f x)-\frac {1}{2} b^2 (3 b c-13 a d) \sin ^2(e+f x)\right )}{\sqrt {a+b \sin (e+f x)}} \, dx}{3 d}\\ &=\frac {b (3 b c-13 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{12 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}+\frac {\int \frac {\frac {1}{4} b \left (b^3 c^2+24 a^3 c d+22 a b^2 c d+13 a^2 b d^2\right )+\frac {1}{2} b \left (23 a^2 b c d+7 b^3 c d+12 a^3 d^2-a b^2 \left (c^2-19 d^2\right )\right ) \sin (e+f x)+\frac {1}{4} b^2 \left (14 a b c d+33 a^2 d^2-b^2 \left (3 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 {b \left (14 a b c d+33 a^2 d^2-b^2 \left (3 c^2-16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{24 d^2 f \sqrt {a+b \sin (e+f x)}}+\frac {b (3 b c-13 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{12 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}+\frac {\int \frac {\frac {1}{4} b \left (48 a^4 c d^2+25 a^2 b^2 c d^2+59 a^3 b d^3+b^4 \left (3 c^3-16 c d^2\right )-a b^3 \left (15 c^2 d-16 d^3\right )\right )+\frac {1}{2} b \left (b^4 c^2 d+37 a^3 b c d^2+24 a^4 d^3-a^2 b^2 d \left (16 c^2-51 d^2\right )+a b^3 c \left (3 c^2+20 d^2\right )\right ) \sin (e+f x)+\frac {3}{4} b^2 \left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 d \left (c^2-4 d^2\right )+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 {b \left (14 a b c d+33 a^2 d^2-b^2 \left (3 c^2-16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{24 d^2 f \sqrt {a+b \sin (e+f x)}}+\frac {b (3 b c-13 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{12 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}+\frac {\int \frac {-\frac {3}{4} a^2 b^2 \left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 d \left (c^2-4 d^2\right )+b^3 \left (c^3+4 c d^2\right )\right )+\frac {1}{4} b^3 \left (48 a^4 c d^2+25 a^2 b^2 c d^2+59 a^3 b d^3+b^4 \left (3 c^3-16 c d^2\right )-a b^3 \left (15 c^2 d-16 d^3\right )\right )+b \left (\frac {1}{2} b^2 \left (b^4 c^2 d+37 a^3 b c d^2+24 a^4 d^3-a^2 b^2 d \left (16 c^2-51 d^2\right )+a b^3 c \left (3 c^2+20 d^2\right )\right )-\frac {3}{2} a b^2 \left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 d \left (c^2-4 d^2\right )+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 (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 d \left (c^2-4 d^2\right )+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 d^2}\\ &=\frac {\sqrt {c+d} \left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 d \left (c^2-4 d^2\right )+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 \sqrt {a+b} d^3 f}-\frac {b \left (14 a b c d+33 a^2 d^2-b^2 \left (3 c^2-16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{24 d^2 f \sqrt {a+b \sin (e+f x)}}+\frac {b (3 b c-13 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{12 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}+\frac {\left (-\frac {3}{4} a^2 b^2 \left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 d \left (c^2-4 d^2\right )+b^3 \left (c^3+4 c d^2\right )\right )+\frac {1}{4} b^3 \left (48 a^4 c d^2+25 a^2 b^2 c d^2+59 a^3 b d^3+b^4 \left (3 c^3-16 c d^2\right )-a b^3 \left (15 c^2 d-16 d^3\right )\right )-b \left (\frac {1}{2} b^2 \left (b^4 c^2 d+37 a^3 b c d^2+24 a^4 d^3-a^2 b^2 d \left (16 c^2-51 d^2\right )+a b^3 c \left (3 c^2+20 d^2\right )\right )-\frac {3}{2} a b^2 \left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 d \left (c^2-4 d^2\right )+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^2 \left (b^4 c^2 d+37 a^3 b c d^2+24 a^4 d^3-a^2 b^2 d \left (16 c^2-51 d^2\right )+a b^3 c \left (3 c^2+20 d^2\right )\right )-\frac {3}{2} a b^2 \left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 d \left (c^2-4 d^2\right )+b^3 \left (c^3+4 c d^2\right )\right )\right )+b \left (-\frac {3}{4} a^2 b^2 \left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 d \left (c^2-4 d^2\right )+b^3 \left (c^3+4 c d^2\right )\right )+\frac {1}{4} b^3 \left (48 a^4 c d^2+25 a^2 b^2 c d^2+59 a^3 b d^3+b^4 \left (3 c^3-16 c d^2\right )-a b^3 \left (15 c^2 d-16 d^3\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+33 a^2 d^2-b^2 \left (3 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 d^2 (b c-a d) f}+\frac {\sqrt {c+d} \left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 d \left (c^2-4 d^2\right )+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 \sqrt {a+b} d^3 f}-\frac {b \left (14 a b c d+33 a^2 d^2-b^2 \left (3 c^2-16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{24 d^2 f \sqrt {a+b \sin (e+f x)}}+\frac {b (3 b c-13 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{12 d f}+\frac {(a+b)^{3/2} \left (15 a^2 d^2+6 a b d (2 c+3 d)-b^2 \left (3 c^2-2 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 d^2 \sqrt {c+d} f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}\\ \end {align*}

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Mathematica [B]  time = 7.10, size = 1979, normalized size = 2.21 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-4*(-(b*c) + a*d)*(-(b^3*c^2) + 48*a^3*c*d + 58*a*b^2*c*d + 59*a^2*b*d^2 + 16*b^3*d^2)*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*Sqr
t[((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)*(-4*a*b^2*c^2 + 92*a^2*b*c*d + 28*b^3*c*d + 48*a^3*d^2 + 76*a*b^2*d^2)*((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 + P
i/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*Si
n[e + f*x]]) - (Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + a*d)/((a + b)*d), Arc
Sin[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*(3*b^3*c^2 - 14*a*b^2*c*d - 33*a^2*b*d^2
- 16*b^3*d^2)*((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)*C
os[(-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)]*S
qrt[((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)*Sq
rt[((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 + P
i/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]]*Sq
rt[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)]/Sqr
t[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]])))/(b*d)))/(48*d*f) + (
Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*(-1/12*(b*(b*c + 13*a*d)*Cos[e + f*x])/d - (b^2*Sin[2*(e + f
*x)])/6))/f

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fricas [F]  time = 35.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (b^{2} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2}\right )} \sqrt {b \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)*sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) +
c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 19.75, size = 410016, normalized size = 458.63 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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