\(\int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^3} \, dx\) [760]

Optimal result

Integrand size = 27, antiderivative size = 512 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^3} \, dx=\frac {(b c-3 d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b \left (9-b^2\right ) f (3+b \sin (e+f x))^2}+\frac {3 (b c-3 d) \left (6 b c+9 d-3 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 b \left (9-b^2\right )^2 f (3+b \sin (e+f x))}+\frac {3 (b c-3 d) \left (6 b c+9 d-3 b^2 d\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{4 b^2 \left (9-b^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (108 b c d^2+243 d^3+9 b^2 d \left (7 c^2-5 d^2\right )+b^4 d \left (11 c^2+8 d^2\right )-6 b^3 c \left (3 c^2+11 d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 b^3 \left (9-b^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {(b c-3 d) \left (108 b c d-84 b^3 c d+243 d^2+18 b^2 \left (4 c^2-3 d^2\right )+b^4 \left (4 c^2+15 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 (3-b)^2 b^3 (3+b)^3 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

1/2*(-a*d+b*c)^2*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/b/(a^2-b^2)/f/(a+b*sin(f*x+e))^2+3/4*(-a*d+b*c)*(a^2*d+2*a*
b*c-3*b^2*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/b/(a^2-b^2)^2/f/(a+b*sin(f*x+e))-3/4*(-a*d+b*c)*(a^2*d+2*a*b*c-
3*b^2*d)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(
1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/b^2/(a^2-b^2)^2/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-1/4*(4*a^3*b*c*d
^2+3*a^4*d^3+a^2*b^2*d*(7*c^2-5*d^2)+b^4*d*(11*c^2+8*d^2)-2*a*b^3*c*(3*c^2+11*d^2))*(sin(1/2*e+1/4*Pi+1/2*f*x)
^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x
+e))/(c+d))^(1/2)/b^3/(a^2-b^2)^2/f/(c+d*sin(f*x+e))^(1/2)-1/4*(-a*d+b*c)*(4*a^3*b*c*d-28*a*b^3*c*d+3*a^4*d^2+
2*a^2*b^2*(4*c^2-3*d^2)+b^4*(4*c^2+15*d^2))*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*Elli
pticPi(cos(1/2*e+1/4*Pi+1/2*f*x),2*b/(a+b),2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/(a-b)^2/b^3
/(a+b)^3/f/(c+d*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {2871, 3134, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^3} \, dx=\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}+\frac {3 \left (a^2 d+2 a b c-3 b^2 d\right ) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 b f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))}+\frac {3 \left (a^2 d+2 a b c-3 b^2 d\right ) (b c-a d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 b^2 f \left (a^2-b^2\right )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (3 a^4 d^2+4 a^3 b c d+2 a^2 b^2 \left (4 c^2-3 d^2\right )-28 a b^3 c d+b^4 \left (4 c^2+15 d^2\right )\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{4 b^3 f (a-b)^2 (a+b)^3 \sqrt {c+d \sin (e+f x)}}+\frac {\left (3 a^4 d^3+4 a^3 b c d^2+a^2 b^2 d \left (7 c^2-5 d^2\right )-2 a b^3 c \left (3 c^2+11 d^2\right )+b^4 d \left (11 c^2+8 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{4 b^3 f \left (a^2-b^2\right )^2 \sqrt {c+d \sin (e+f x)}} \]

[In]

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

[Out]

((b*c - a*d)^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2) + (3*(b*c - a
*d)*(2*a*b*c + a^2*d - 3*b^2*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*b*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x
])) + (3*(b*c - a*d)*(2*a*b*c + a^2*d - 3*b^2*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e
+ f*x]])/(4*b^2*(a^2 - b^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((4*a^3*b*c*d^2 + 3*a^4*d^3 + a^2*b^2*d
*(7*c^2 - 5*d^2) + b^4*d*(11*c^2 + 8*d^2) - 2*a*b^3*c*(3*c^2 + 11*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c
+ d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*b^3*(a^2 - b^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) + ((b*c - a*d)*(4*
a^3*b*c*d - 28*a*b^3*c*d + 3*a^4*d^2 + 2*a^2*b^2*(4*c^2 - 3*d^2) + b^4*(4*c^2 + 15*d^2))*EllipticPi[(2*b)/(a +
b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*b^3*(a + b)^3*f*Sqrt[
c + d*Sin[e + f*x]])

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

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 2884

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(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] && GtQ[c + d, 0]

Rule 2886

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/
(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 3081

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

Rule 3134

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + D
ist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*
(b*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(
b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*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] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3138

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) +
(f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Dist[C/(b*d), Int[Sqrt[a + b*Sin[e + f*x]]
, x], x] - Dist[1/(b*d), Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[e + f*x], x]/(Sqrt[a + b*Sin[e +
f*x]]*(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*} \text {integral}& = \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\int \frac {\frac {1}{2} \left (-4 a b c^3+9 b^2 c^2 d-6 a b c d^2+a^2 d^3\right )-\left (a^2 c d^2+2 a b d \left (2 c^2+d^2\right )-b^2 \left (c^3+6 c d^2\right )\right ) \sin (e+f x)-\frac {1}{2} d \left (2 a b c d+3 a^2 d^2-b^2 \left (c^2+4 d^2\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}} \, dx}{2 b \left (a^2-b^2\right )} \\ & = \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {3 (b c-a d) \left (2 a b c+a^2 d-3 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 b \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {\int \frac {\frac {1}{4} (b c-a d) \left (8 a^2 b c^3+4 b^3 c^3-30 a b^2 c^2 d+9 a^2 b c d^2+15 b^3 c d^2+a^3 d^3-7 a b^2 d^3\right )+\frac {1}{2} d (b c-a d) \left (5 a^2 b c^2+b^3 c^2-a^3 c d-11 a b^2 c d+2 a^2 b d^2+4 b^3 d^2\right ) \sin (e+f x)+\frac {3}{4} d (b c-a d)^2 \left (2 a b c+a^2 d-3 b^2 d\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{2 b \left (a^2-b^2\right )^2 (b c-a d)} \\ & = \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {3 (b c-a d) \left (2 a b c+a^2 d-3 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 b \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}-\frac {\int \frac {-\frac {1}{4} d (b c-a d) \left (2 a^2 b^2 c^3+3 a^4 c d^2+a^3 b d \left (3 c^2+d^2\right )-7 a b^3 d \left (3 c^2+d^2\right )+b^4 c \left (4 c^2+15 d^2\right )\right )+\frac {1}{4} d (b c-a d) \left (6 a b^3 c^3-7 a^2 b^2 c^2 d-11 b^4 c^2 d-4 a^3 b c d^2+22 a b^3 c d^2-3 a^4 d^3+5 a^2 b^2 d^3-8 b^4 d^3\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{2 b^2 \left (a^2-b^2\right )^2 d (b c-a d)}+\frac {\left (3 (b c-a d) \left (2 a b c+a^2 d-3 b^2 d\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{8 b^2 \left (a^2-b^2\right )^2} \\ & = \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {3 (b c-a d) \left (2 a b c+a^2 d-3 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 b \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {\left (4 a^3 b c d^2+3 a^4 d^3+a^2 b^2 d \left (7 c^2-5 d^2\right )+b^4 d \left (11 c^2+8 d^2\right )-2 a b^3 c \left (3 c^2+11 d^2\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{8 b^3 \left (a^2-b^2\right )^2}+\frac {\left ((b c-a d) \left (4 a^3 b c d-28 a b^3 c d+3 a^4 d^2+2 a^2 b^2 \left (4 c^2-3 d^2\right )+b^4 \left (4 c^2+15 d^2\right )\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{8 b^3 \left (a^2-b^2\right )^2}+\frac {\left (3 (b c-a d) \left (2 a b c+a^2 d-3 b^2 d\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{8 b^2 \left (a^2-b^2\right )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \\ & = \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {3 (b c-a d) \left (2 a b c+a^2 d-3 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 b \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {3 (b c-a d) \left (2 a b c+a^2 d-3 b^2 d\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{4 b^2 \left (a^2-b^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (\left (4 a^3 b c d^2+3 a^4 d^3+a^2 b^2 d \left (7 c^2-5 d^2\right )+b^4 d \left (11 c^2+8 d^2\right )-2 a b^3 c \left (3 c^2+11 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{8 b^3 \left (a^2-b^2\right )^2 \sqrt {c+d \sin (e+f x)}}+\frac {\left ((b c-a d) \left (4 a^3 b c d-28 a b^3 c d+3 a^4 d^2+2 a^2 b^2 \left (4 c^2-3 d^2\right )+b^4 \left (4 c^2+15 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{8 b^3 \left (a^2-b^2\right )^2 \sqrt {c+d \sin (e+f x)}} \\ & = \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {3 (b c-a d) \left (2 a b c+a^2 d-3 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 b \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {3 (b c-a d) \left (2 a b c+a^2 d-3 b^2 d\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{4 b^2 \left (a^2-b^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (4 a^3 b c d^2+3 a^4 d^3+a^2 b^2 d \left (7 c^2-5 d^2\right )+b^4 d \left (11 c^2+8 d^2\right )-2 a b^3 c \left (3 c^2+11 d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 b^3 \left (a^2-b^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {(b c-a d) \left (4 a^3 b c d-28 a b^3 c d+3 a^4 d^2+2 a^2 b^2 \left (4 c^2-3 d^2\right )+b^4 \left (4 c^2+15 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 (a-b)^2 b^3 (a+b)^3 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.52 (sec) , antiderivative size = 1088, normalized size of antiderivative = 2.12 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^3} \, dx=\frac {\sqrt {c+d \sin (e+f x)} \left (\frac {-b^2 c^2 \cos (e+f x)+6 b c d \cos (e+f x)-9 d^2 \cos (e+f x)}{2 b \left (-9+b^2\right ) (3+b \sin (e+f x))^2}-\frac {9 \left (-2 b^2 c^2 \cos (e+f x)+3 b c d \cos (e+f x)+b^3 c d \cos (e+f x)+9 d^2 \cos (e+f x)-3 b^2 d^2 \cos (e+f x)\right )}{4 b \left (-9+b^2\right )^2 (3+b \sin (e+f x))}\right )}{f}+\frac {-\frac {2 \left (144 b c^3+8 b^3 c^3-162 b^2 c^2 d+135 b c d^2+21 b^3 c d^2-27 d^3-15 b^2 d^3\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(3+b) \sqrt {c+d \sin (e+f x)}}-\frac {2 i \left (180 b c^2 d+4 b^3 c^2 d-108 c d^2-132 b^2 c d^2+72 b d^3+16 b^3 d^3\right ) \cos (e+f x) \left ((b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )+3 d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right ) \sqrt {\frac {d-d \sin (e+f x)}{c+d}} \sqrt {-\frac {d+d \sin (e+f x)}{c-d}} (-b c+3 d+b (c+d \sin (e+f x)))}{b (b c-3 d) d^2 \sqrt {-\frac {1}{c+d}} (3+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}-\frac {2 i \left (-18 b^2 c^2 d+27 b c d^2+9 b^3 c d^2+81 d^3-27 b^2 d^3\right ) \cos (e+f x) \cos (2 (e+f x)) \left (2 b (b c-3 d) (c-d) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )+d \left (2 (3+b) (b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )-\left (-18+b^2\right ) d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right )\right ) \sqrt {\frac {d-d \sin (e+f x)}{c+d}} \sqrt {-\frac {d+d \sin (e+f x)}{c-d}} (-b c+3 d+b (c+d \sin (e+f x)))}{b^2 (b c-3 d) d \sqrt {-\frac {1}{c+d}} (3+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \left (-2 c^2+d^2+4 c (c+d \sin (e+f x))-2 (c+d \sin (e+f x))^2\right ) \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}}{16 (-3+b)^2 b (3+b)^2 f} \]

[In]

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

[Out]

(Sqrt[c + d*Sin[e + f*x]]*((-(b^2*c^2*Cos[e + f*x]) + 6*b*c*d*Cos[e + f*x] - 9*d^2*Cos[e + f*x])/(2*b*(-9 + b^
2)*(3 + b*Sin[e + f*x])^2) - (9*(-2*b^2*c^2*Cos[e + f*x] + 3*b*c*d*Cos[e + f*x] + b^3*c*d*Cos[e + f*x] + 9*d^2
*Cos[e + f*x] - 3*b^2*d^2*Cos[e + f*x]))/(4*b*(-9 + b^2)^2*(3 + b*Sin[e + f*x]))))/f + ((-2*(144*b*c^3 + 8*b^3
*c^3 - 162*b^2*c^2*d + 135*b*c*d^2 + 21*b^3*c*d^2 - 27*d^3 - 15*b^2*d^3)*EllipticPi[(2*b)/(3 + b), (-e + Pi/2
- f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((3 + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(180*
b*c^2*d + 4*b^3*c^2*d - 108*c*d^2 - 132*b^2*c*d^2 + 72*b*d^3 + 16*b^3*d^3)*Cos[e + f*x]*((b*c - 3*d)*EllipticF
[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + 3*d*EllipticPi[(b*(c + d))/(b*c -
3*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x])/(c
+ d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + 3*d + b*(c + d*Sin[e + f*x])))/(b*(b*c - 3*d)*d^2*Sqrt[
-(c + d)^(-1)]*(3 + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c
+ d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*(-18*b^2*c^2*d + 27*b*c*d^2 + 9*b^3*c*d^2 + 81*d^3 - 27*b^2*d^3)*Cos[e +
f*x]*Cos[2*(e + f*x)]*(2*b*(b*c - 3*d)*(c - d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x
]]], (c + d)/(c - d)] + d*(2*(3 + b)*(b*c - 3*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*
x]]], (c + d)/(c - d)] - (-18 + b^2)*d*EllipticPi[(b*(c + d))/(b*c - 3*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[
c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d
))]*(-(b*c) + 3*d + b*(c + d*Sin[e + f*x])))/(b^2*(b*c - 3*d)*d*Sqrt[-(c + d)^(-1)]*(3 + b*Sin[e + f*x])*Sqrt[
1 - Sin[e + f*x]^2]*(-2*c^2 + d^2 + 4*c*(c + d*Sin[e + f*x]) - 2*(c + d*Sin[e + f*x])^2)*Sqrt[-((c^2 - d^2 - 2
*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]))/(16*(-3 + b)^2*b*(3 + b)^2*f)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1887\) vs. \(2(622)=1244\).

Time = 79.61 (sec) , antiderivative size = 1888, normalized size of antiderivative = 3.69

method	result	size
default	\(\text {Expression too large to display}\)	\(1888\)

[In]

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

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(2*d^3/b^3*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c
+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e
))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-6*d^2/b^4*(a*d-b*c)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+
e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(-c/d+a/b)*Elliptic
Pi(((c+d*sin(f*x+e))/(c-d))^(1/2),(-c/d+1)/(-c/d+a/b),((c-d)/(c+d))^(1/2))+3/b^3*d*(a^2*d^2-2*a*b*c*d+b^2*c^2)
*(-b^2/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(a+b*sin(f*x+e))-a*d/(a^3*d-a^2*b
*c-a*b^2*d+b^3*c)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-
1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/
2))-b*d/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1
/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/
(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))+(3*a^2*d-2*a*
b*c-b^2*d)/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)/b*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/
2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(-c/d+a/b)*EllipticPi(((c+d*sin(f
*x+e))/(c-d))^(1/2),(-c/d+1)/(-c/d+a/b),((c-d)/(c+d))^(1/2)))+1/b^3*(-a^3*d^3+3*a^2*b*c*d^2-3*a*b^2*c^2*d+b^3*
c^3)*(-1/2*b^2/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(a+b*sin(f*x+e))^2-3/4*b^
2*(3*a^2*d-2*a*b*c-b^2*d)/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)^2*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(a+b*sin(f*x
+e))-1/4*d*(7*a^3*d-4*a^2*b*c-a*b^2*d-2*b^3*c)/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d)
)^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/
2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-3/4*b*d*(3*a^2*d-2*a*b*c-b^2*d)/(a^3*d-a^2*b*
c-a*b^2*d+b^3*c)^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)
-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)
/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))+1/4*(15*a^4*d^2-20*a^3*b*c*d+8*a
^2*b^2*c^2-6*a^2*b^2*d^2-4*a*b^3*c*d+4*b^4*c^2+3*b^4*d^2)/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)^2/b*(c/d-1)*((c+d*sin(
f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(
f*x+e)^2)^(1/2)/(-c/d+a/b)*EllipticPi(((c+d*sin(f*x+e))/(c-d))^(1/2),(-c/d+1)/(-c/d+a/b),((c-d)/(c+d))^(1/2)))
)/cos(f*x+e)/(c+d*sin(f*x+e))^(1/2)/f

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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