\(\int \frac {(3+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{3/2}} \, dx\) [781]

Optimal result

Integrand size = 29, antiderivative size = 762 \[ \int \frac {(3+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {\sqrt {3+b} \left (12 b c d-18 d^2-b^2 \left (3 c^2-d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right )|\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{(b c-3 d) d^2 \sqrt {c+d} f}-\frac {b (3 b c-15 d) \sqrt {c+d} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(3+b) d},\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right ),\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{\sqrt {3+b} d^3 f}+\frac {2 (b c-3 d)^2 \cos (e+f x) \sqrt {3+b \sin (e+f x)}}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b \left (12 b c d-18 d^2-b^2 \left (3 c^2-d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d^2 \left (c^2-d^2\right ) f \sqrt {3+b \sin (e+f x)}}-\frac {(3+b)^{3/2} (6 d-b (3 c+d)) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{d^2 (c+d)^{3/2} f} \]

[Out]

-(4*a*b*c*d-2*a^2*d^2-b^2*(3*c^2-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)*(-(-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/(c+d)^
(1/2)-b*(-5*a*d+3*b*c)*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)/d^3/f/(a+b)^(1/2)-(a+b
)^(3/2)*(2*a*d-b*(3*c+d))*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)/d^2/(c+d)^(3/2)/f+2*(-a*d+b*c)^2*cos(f*x+e)*(a+b
*sin(f*x+e))^(1/2)/d/(c^2-d^2)/f/(c+d*sin(f*x+e))^(1/2)+b*(4*a*b*c*d-2*a^2*d^2-b^2*(3*c^2-d^2))*cos(f*x+e)*(c+
d*sin(f*x+e))^(1/2)/d^2/(c^2-d^2)/f/(a+b*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 1.68 (sec) , antiderivative size = 780, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2871, 3140, 3132, 2890, 3077, 2897, 3075} \[ \int \frac {(3+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {\sqrt {a+b} \left (-2 a^2 d^2+4 a b c d-\left (b^2 \left (3 c^2-d^2\right )\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{d^2 f \sqrt {c+d} (b c-a d)}+\frac {b \left (-2 a^2 d^2+4 a b c d-\left (b^2 \left (3 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d^2 f \left (c^2-d^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {b \sqrt {c+d} (3 b c-5 a d) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{d^3 f \sqrt {a+b}}-\frac {(a+b)^{3/2} (2 a d-b (3 c+d)) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{d^2 f (c+d)^{3/2}}+\frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}} \]

[In]

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

[Out]

-((Sqrt[a + b]*(4*a*b*c*d - 2*a^2*d^2 - b^2*(3*c^2 - 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]))/(d^2*Sqrt[c + d]*(b*c - a*d)*f)) - (b*Sqrt[c + d]*(3*b*c - 5*a*d)*Ell
ipticPi[(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]))/(Sqrt[a + b]*d^3*f) + (2*(b*c - a*d)^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(d*(c^2 - d^2)*f*Sqrt[c + d
*Sin[e + f*x]]) + (b*(4*a*b*c*d - 2*a^2*d^2 - b^2*(3*c^2 - d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d^2*(
c^2 - d^2)*f*Sqrt[a + b*Sin[e + f*x]]) - ((a + b)^(3/2)*(2*a*d - b*(3*c + d))*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]))/(d^2*(c + d)^(3/2)*f)

Rule 2871

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((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 - 3)*(c + d*Sin[e
+ f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 +
c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 +
d*n*(2*a*b*c - d*(a^2 + b^2)))*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] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

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 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 {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}-\frac {2 \int \frac {\frac {1}{2} \left (b^3 c^2-a^3 c d-3 a b^2 c d+3 a^2 b d^2\right )+\frac {1}{2} \left (a^2 b c d-b^3 c d-a^3 d^2-a b^2 \left (2 c^2-3 d^2\right )\right ) \sin (e+f x)+\frac {1}{2} b \left (4 a b c d-2 a^2 d^2-b^2 \left (3 c^2-d^2\right )\right ) \sin ^2(e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{d \left (c^2-d^2\right )} \\ & = \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b \left (4 a b c d-2 a^2 d^2-b^2 \left (3 c^2-d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d^2 \left (c^2-d^2\right ) f \sqrt {a+b \sin (e+f x)}}-\frac {\int \frac {\frac {1}{2} \left (-2 a^4 c d^2+4 a^3 b d^3-a b^3 d \left (5 c^2-d^2\right )+b^4 \left (3 c^3-c d^2\right )\right )+\left (b^4 c^2 d+2 a^3 b c d^2-a^4 d^3+a b^3 c \left (3 c^2-5 d^2\right )-6 a^2 b^2 d \left (c^2-d^2\right )\right ) \sin (e+f x)+\frac {1}{2} b^3 (3 b c-5 a d) \left (c^2-d^2\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{d^2 \left (c^2-d^2\right )} \\ & = \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b \left (4 a b c d-2 a^2 d^2-b^2 \left (3 c^2-d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d^2 \left (c^2-d^2\right ) f \sqrt {a+b \sin (e+f x)}}-\frac {(b (3 b c-5 a d)) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{2 d^2}-\frac {\int \frac {-\frac {1}{2} a^2 b^3 (3 b c-5 a d) \left (c^2-d^2\right )+\frac {1}{2} b^2 \left (-2 a^4 c d^2+4 a^3 b d^3-a b^3 d \left (5 c^2-d^2\right )+b^4 \left (3 c^3-c d^2\right )\right )+b \left (-a b^3 (3 b c-5 a d) \left (c^2-d^2\right )+b \left (b^4 c^2 d+2 a^3 b c d^2-a^4 d^3+a b^3 c \left (3 c^2-5 d^2\right )-6 a^2 b^2 d \left (c^2-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}{b^2 d^2 \left (c^2-d^2\right )} \\ & = -\frac {b \sqrt {c+d} (3 b c-5 a d) \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))}{\sqrt {a+b} d^3 f}+\frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b \left (4 a b c d-2 a^2 d^2-b^2 \left (3 c^2-d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d^2 \left (c^2-d^2\right ) f \sqrt {a+b \sin (e+f x)}}-\frac {\left (-\frac {1}{2} a^2 b^3 (3 b c-5 a d) \left (c^2-d^2\right )+\frac {1}{2} b^2 \left (-2 a^4 c d^2+4 a^3 b d^3-a b^3 d \left (5 c^2-d^2\right )+b^4 \left (3 c^3-c d^2\right )\right )-b \left (-a b^3 (3 b c-5 a d) \left (c^2-d^2\right )+b \left (b^4 c^2 d+2 a^3 b c d^2-a^4 d^3+a b^3 c \left (3 c^2-5 d^2\right )-6 a^2 b^2 d \left (c^2-d^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{(a-b) b^2 d^2 \left (c^2-d^2\right )}+\frac {\left (-a b \left (-a b^3 (3 b c-5 a d) \left (c^2-d^2\right )+b \left (b^4 c^2 d+2 a^3 b c d^2-a^4 d^3+a b^3 c \left (3 c^2-5 d^2\right )-6 a^2 b^2 d \left (c^2-d^2\right )\right )\right )+b \left (-\frac {1}{2} a^2 b^3 (3 b c-5 a d) \left (c^2-d^2\right )+\frac {1}{2} b^2 \left (-2 a^4 c d^2+4 a^3 b d^3-a b^3 d \left (5 c^2-d^2\right )+b^4 \left (3 c^3-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}{(a-b) b^2 d^2 \left (c^2-d^2\right )} \\ & = -\frac {\sqrt {a+b} \left (4 a b c d-2 a^2 d^2-b^2 \left (3 c^2-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))}{d^2 \sqrt {c+d} (b c-a d) f}-\frac {b \sqrt {c+d} (3 b c-5 a d) \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))}{\sqrt {a+b} d^3 f}+\frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b \left (4 a b c d-2 a^2 d^2-b^2 \left (3 c^2-d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d^2 \left (c^2-d^2\right ) f \sqrt {a+b \sin (e+f x)}}-\frac {(a+b)^{3/2} (2 a d-b (3 c+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))}{d^2 (c+d)^{3/2} f} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1964\) vs. \(2(762)=1524\).

Time = 8.49 (sec) , antiderivative size = 1964, normalized size of antiderivative = 2.58 \[ \int \frac {(3+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {2 \left (b^2 c^2 \cos (e+f x)-6 b c d \cos (e+f x)+9 d^2 \cos (e+f x)\right ) \sqrt {3+b \sin (e+f x)}}{d \left (-c^2+d^2\right ) f \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {4 (-b c+3 d) \left (-b^3 c^2-54 c d-6 b^2 c d+36 b d^2+b^3 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 (-12 b^2 c^2+18 b c d-2 b^3 c d-54 d^2+18 b^2 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 (3 b^3 c^2-12 b^2 c d+18 b d^2-b^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 )}{2 (c-d) d (c+d) f} \]

[In]

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

[Out]

(-2*(b^2*c^2*Cos[e + f*x] - 6*b*c*d*Cos[e + f*x] + 9*d^2*Cos[e + f*x])*Sqrt[3 + b*Sin[e + f*x]])/(d*(-c^2 + d^
2)*f*Sqrt[c + d*Sin[e + f*x]]) - ((-4*(-(b*c) + 3*d)*(-(b^3*c^2) - 54*c*d - 6*b^2*c*d + 36*b*d^2 + b^3*d^2)*Sq
rt[((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 + P
i/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]]*Sq
rt[c + d*Sin[e + f*x]]) - 4*(-(b*c) + 3*d)*(-12*b^2*c^2 + 18*b*c*d - 2*b^3*c*d - 54*d^2 + 18*b^2*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*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*(3*b^3*c^2 - 12*b^2*c*d + 18*b*d^
2 - b^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[ArcSin[(Sqrt[(3 - b)/(3 + b)]*Sin[(-e + Pi/2 - f*x)/2])/Sqrt[(3 + b*S
in[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)]*Sqr
t[((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)*Cs
c[(-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)))/(2*(c - d)*d*(
c + d)*f)

Maple [B] (warning: unable to verify)

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 35.49 (sec) , antiderivative size = 762903, normalized size of antiderivative = 1001.19

method	result	size
default	\(\text {Expression too large to display}\)	\(762903\)

[In]

int((a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F(-1)]

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

[In]

integrate((a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

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

[In]

integrate((a+b*sin(f*x+e))**(5/2)/(c+d*sin(f*x+e))**(3/2),x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate((a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e) + a)^(5/2)/(d*sin(f*x + e) + c)^(3/2), x)

Giac [F]

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

[In]

integrate((a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e) + a)^(5/2)/(d*sin(f*x + e) + c)^(3/2), x)

Mupad [F(-1)]

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

[In]

int((a + b*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(3/2),x)

[Out]

int((a + b*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(3/2), x)