\(\int (3+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2} \, dx\) [771]

Optimal result

Integrand size = 29, antiderivative size = 1039 \[ \int (3+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2} \, dx=\frac {\sqrt {3+b} (c-d) \sqrt {c+d} \left (513 b c d^2-243 d^3+3 b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c 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))}{192 b^2 (b c-3 d) d f}-\frac {\sqrt {c+d} \left (540 b c d^3-243 d^4-180 b^3 c d \left (c^2+4 d^2\right )-54 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\right )\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(3+b) d},\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right ),\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{64 b^3 \sqrt {3+b} d^2 f}-\frac {\left (513 b c d^2-243 d^3+3 b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{192 b d f \sqrt {3+b \sin (e+f x)}}-\frac {\left (162 b c d-81 d^2+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{96 b f}-\frac {(17 b c-9 d) d \cos (e+f x) (3+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{24 b f}-\frac {d^2 \cos (e+f x) (3+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}+\frac {(3+b)^{3/2} \left (243 d^3-27 b d^2 (17 c+6 d)+9 b^2 d \left (73 c^2+36 c d+28 d^2\right )+b^3 \left (15 c^3+118 c^2 d+284 c d^2+72 d^3\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{192 b^3 d \sqrt {c+d} f} \]

[Out]

-1/64*(20*a^3*b*c*d^3-3*a^4*d^4-60*a*b^3*c*d*(c^2+4*d^2)-6*a^2*b^2*d^2*(15*c^2+4*d^2)+b^4*(5*c^4-120*c^2*d^2-4
8*d^4))*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^3/d^2/f/(a+b)^(1/2)+1/192*(c-d)*(57
*a^2*b*c*d^2-9*a^3*d^3+a*b^2*d*(337*c^2+156*d^2)+b^3*(15*c^3+284*c*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/d/(-a*d+b*c)/f+1/192*(a+b)^(3/2)*(9*a^3*d^3-3*a^2*b*d^2*(17*c+6*d)+3*a*b^2*d*(73*c^
2+36*c*d+28*d^2)+b^3*(15*c^3+118*c^2*d+284*c*d^2+72*d^3))*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/d/f/(c+d)^(1
/2)-1/24*d*(-3*a*d+17*b*c)*cos(f*x+e)*(a+b*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^(1/2)/b/f-1/4*d^2*cos(f*x+e)*(a+
b*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^(1/2)/b/f-1/192*(57*a^2*b*c*d^2-9*a^3*d^3+a*b^2*d*(337*c^2+156*d^2)+b^3*(
15*c^3+284*c*d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/b/d/f/(a+b*sin(f*x+e))^(1/2)-1/96*(54*a*b*c*d-9*a^2*d^2+b
^2*(59*c^2+36*d^2))*cos(f*x+e)*(a+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(1/2)/b/f

Rubi [A] (verified)

Time = 3.36 (sec) , antiderivative size = 1080, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2872, 3128, 3140, 3132, 2890, 3077, 2897, 3075} \[ \int (3+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2} \, dx=-\frac {d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}}{4 b f}-\frac {d (17 b c-3 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)} (a+b \sin (e+f x))^{3/2}}{24 b f}+\frac {\sqrt {a+b} (c-d) \sqrt {c+d} \left (\left (15 c^3+284 d^2 c\right ) b^3+a d \left (337 c^2+156 d^2\right ) b^2+57 a^2 c d^2 b-9 a^3 d^3\right ) E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{192 b^2 d (b c-a d) f}-\frac {\sqrt {c+d} \left (\left (5 c^4-120 d^2 c^2-48 d^4\right ) b^4-60 a c d \left (c^2+4 d^2\right ) b^3-6 a^2 d^2 \left (15 c^2+4 d^2\right ) b^2+20 a^3 c d^3 b-3 a^4 d^4\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{64 b^3 \sqrt {a+b} d^2 f}-\frac {\left (\left (59 c^2+36 d^2\right ) b^2+54 a c d b-9 a^2 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{96 b f}+\frac {(a+b)^{3/2} \left (\left (15 c^3+118 d c^2+284 d^2 c+72 d^3\right ) b^3+3 a d \left (73 c^2+36 d c+28 d^2\right ) b^2-3 a^2 d^2 (17 c+6 d) b+9 a^3 d^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{192 b^3 d \sqrt {c+d} f}-\frac {\left (\left (15 c^3+284 d^2 c\right ) b^3+a d \left (337 c^2+156 d^2\right ) b^2+57 a^2 c d^2 b-9 a^3 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{192 b d f \sqrt {a+b \sin (e+f x)}} \]

[In]

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

[Out]

(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(57*a^2*b*c*d^2 - 9*a^3*d^3 + a*b^2*d*(337*c^2 + 156*d^2) + b^3*(15*c^3 + 284
*c*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]))/(192*b^2*
d*(b*c - a*d)*f) - (Sqrt[c + d]*(20*a^3*b*c*d^3 - 3*a^4*d^4 - 60*a*b^3*c*d*(c^2 + 4*d^2) - 6*a^2*b^2*d^2*(15*c
^2 + 4*d^2) + b^4*(5*c^4 - 120*c^2*d^2 - 48*d^4))*EllipticPi[(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]))/(64*b^3*Sqrt[a + b]*d^2*f) - ((57*a^2*b*c*d^2 - 9*a^
3*d^3 + a*b^2*d*(337*c^2 + 156*d^2) + b^3*(15*c^3 + 284*c*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(192*b*
d*f*Sqrt[a + b*Sin[e + f*x]]) - ((54*a*b*c*d - 9*a^2*d^2 + b^2*(59*c^2 + 36*d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[
e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(96*b*f) - (d*(17*b*c - 3*a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(3/2)*Sqr
t[c + d*Sin[e + f*x]])/(24*b*f) - (d^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(5/2)*Sqrt[c + d*Sin[e + f*x]])/(4*b*
f) + ((a + b)^(3/2)*(9*a^3*d^3 - 3*a^2*b*d^2*(17*c + 6*d) + 3*a*b^2*d*(73*c^2 + 36*c*d + 28*d^2) + b^3*(15*c^3
+ 118*c^2*d + 284*c*d^2 + 72*d^3))*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]))/(192*b^3*d*Sqrt[c + d]*f)

Rule 2872

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

Rule 2890

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[
2*((a + b*Sin[e + f*x])/(d*f*Rt[(a + b)/(c + d), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c -
d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticPi
[b*((c + d)/(d*(a + b))), ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]])],
(a - b)*((c + d)/((a + b)*(c - d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2
, 0] && NeQ[c^2 - d^2, 0] && PosQ[(a + b)/(c + d)]

Rule 2897

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Si
mp[2*((c + d*Sin[e + f*x])/(f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 - Sin[e +
f*x])/((a + b)*(c + d*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 + Sin[e + f*x])/((a - b)*(c + d*Sin[e + f*x]))
)]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], (a + b)*((c -
d)/((a - b)*(c + d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c
^2 - d^2, 0] && PosQ[(c + d)/(a + b)]

Rule 3075

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin
[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A*(c - d)*((a + b*Sin[e + f*x])/(f*(b*c - a*d)^2*Rt[(a + b)/(c +
d), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d)
)*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticE[ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Sin
[e + f*x]]/Sqrt[a + b*Sin[e + f*x]])], (a - b)*((c + d)/((a + b)*(c - d)))], x] /; FreeQ[{a, b, c, d, e, f, A,
B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]

Rule 3077

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*s
in[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[(A - B)/(a - b), Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e
+ f*x]]), x], x] - Dist[(A*b - a*B)/(a - b), Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin
[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2
- d^2, 0] && NeQ[A, B]

Rule 3128

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

Rule 3132

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(((a_.) + (b_.)*sin[(e_.) + (f_.
)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[C/b^2, Int[Sqrt[a + b*Sin[e + f
*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Dist[1/b^2, Int[(A*b^2 - a^2*C + b*(b*B - 2*a*C)*Sin[e + f*x])/((a + b
*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a
*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3140

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) +
(f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(Sqrt[c + d*Sin[
e + f*x]]/(d*f*Sqrt[a + b*Sin[e + f*x]])), x] + Dist[1/(2*d), Int[(1/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Si
n[e + f*x]]))*Simp[2*a*A*d - C*(b*c - a*d) - 2*(a*c*C - d*(A*b + a*B))*Sin[e + f*x] + (2*b*B*d - C*(b*c + a*d)
)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0
] && NeQ[c^2 - d^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}+\frac {\int \frac {(a+b \sin (e+f x))^{3/2} \left (\frac {1}{2} \left (8 b c^3+5 b c d^2+a d^3\right )-d \left (a c d-3 b \left (4 c^2+d^2\right )\right ) \sin (e+f x)+\frac {1}{2} d^2 (17 b c-3 a d) \sin ^2(e+f x)\right )}{\sqrt {c+d \sin (e+f x)}} \, dx}{4 b} \\ & = -\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{24 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}+\frac {\int \frac {\sqrt {a+b \sin (e+f x)} \left (\frac {1}{4} d \left (51 b^2 c^2 d+3 a^2 d^3+a b \left (48 c^3+38 c d^2\right )\right )-\frac {1}{2} d \left (3 a^2 c d^2-5 a b d \left (11 c^2+3 d^2\right )-b^2 \left (24 c^3+49 c d^2\right )\right ) \sin (e+f x)+\frac {1}{4} d^2 \left (54 a b c d-9 a^2 d^2+b^2 \left (59 c^2+36 d^2\right )\right ) \sin ^2(e+f x)\right )}{\sqrt {c+d \sin (e+f x)}} \, dx}{12 b d} \\ & = -\frac {\left (54 a b c d-9 a^2 d^2+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{96 b f}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{24 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}+\frac {\int \frac {\frac {1}{8} d^2 \left (3 a^3 d^3+b^3 c \left (59 c^2+36 d^2\right )+a b^2 d \left (317 c^2+36 d^2\right )+a^2 b c \left (192 c^2+197 d^2\right )\right )-\frac {1}{4} d^2 \left (3 a^3 c d^2-b^3 d \left (161 c^2+36 d^2\right )-a^2 b d \left (166 c^2+57 d^2\right )-a b^2 c \left (133 c^2+290 d^2\right )\right ) \sin (e+f x)+\frac {1}{8} d^2 \left (57 a^2 b c d^2-9 a^3 d^3+a b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c d^2\right )\right ) \sin ^2(e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{24 b d^2} \\ & = -\frac {\left (57 a^2 b c d^2-9 a^3 d^3+a b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{192 b d f \sqrt {a+b \sin (e+f x)}}-\frac {\left (54 a b c d-9 a^2 d^2+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{96 b f}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{24 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}+\frac {\int \frac {-\frac {1}{8} d^2 \left (3 a^4 d^4+4 a b^3 c d \left (51 c^2-50 d^2\right )-2 a^2 b^2 d^2 \left (457 c^2+114 d^2\right )-4 a^3 b c d \left (96 c^2+115 d^2\right )+b^4 \left (15 c^4+284 c^2 d^2\right )\right )+\frac {1}{4} d^2 \left (3 a^4 c d^3+b^4 c d \left (59 c^2+36 d^2\right )+a^3 b d^2 \left (275 c^2+117 d^2\right )+a^2 b^2 c d \left (121 c^2+621 d^2\right )-a b^3 \left (15 c^4-355 c^2 d^2-108 d^4\right )\right ) \sin (e+f x)-\frac {3}{8} d^2 \left (20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c d \left (c^2+4 d^2\right )-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{48 b d^3} \\ & = -\frac {\left (57 a^2 b c d^2-9 a^3 d^3+a b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{192 b d f \sqrt {a+b \sin (e+f x)}}-\frac {\left (54 a b c d-9 a^2 d^2+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{96 b f}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{24 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}+\frac {\int \frac {-\frac {1}{8} b^2 d^2 \left (3 a^4 d^4+4 a b^3 c d \left (51 c^2-50 d^2\right )-2 a^2 b^2 d^2 \left (457 c^2+114 d^2\right )-4 a^3 b c d \left (96 c^2+115 d^2\right )+b^4 \left (15 c^4+284 c^2 d^2\right )\right )+\frac {3}{8} a^2 d^2 \left (20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c d \left (c^2+4 d^2\right )-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\right )\right )+b \left (\frac {1}{4} b d^2 \left (3 a^4 c d^3+b^4 c d \left (59 c^2+36 d^2\right )+a^3 b d^2 \left (275 c^2+117 d^2\right )+a^2 b^2 c d \left (121 c^2+621 d^2\right )-a b^3 \left (15 c^4-355 c^2 d^2-108 d^4\right )\right )+\frac {3}{4} a d^2 \left (20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c d \left (c^2+4 d^2\right )-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\right )\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{48 b^3 d^3}-\frac {\left (20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c d \left (c^2+4 d^2\right )-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{128 b^3 d} \\ & = -\frac {\sqrt {c+d} \left (20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c d \left (c^2+4 d^2\right )-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\right )\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{64 b^3 \sqrt {a+b} d^2 f}-\frac {\left (57 a^2 b c d^2-9 a^3 d^3+a b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{192 b d f \sqrt {a+b \sin (e+f x)}}-\frac {\left (54 a b c d-9 a^2 d^2+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{96 b f}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{24 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}+\frac {\left (-\frac {1}{8} b^2 d^2 \left (3 a^4 d^4+4 a b^3 c d \left (51 c^2-50 d^2\right )-2 a^2 b^2 d^2 \left (457 c^2+114 d^2\right )-4 a^3 b c d \left (96 c^2+115 d^2\right )+b^4 \left (15 c^4+284 c^2 d^2\right )\right )+\frac {3}{8} a^2 d^2 \left (20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c d \left (c^2+4 d^2\right )-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\right )\right )-b \left (\frac {1}{4} b d^2 \left (3 a^4 c d^3+b^4 c d \left (59 c^2+36 d^2\right )+a^3 b d^2 \left (275 c^2+117 d^2\right )+a^2 b^2 c d \left (121 c^2+621 d^2\right )-a b^3 \left (15 c^4-355 c^2 d^2-108 d^4\right )\right )+\frac {3}{4} a d^2 \left (20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c d \left (c^2+4 d^2\right )-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{48 (a-b) b^3 d^3}-\frac {\left (-a b \left (\frac {1}{4} b d^2 \left (3 a^4 c d^3+b^4 c d \left (59 c^2+36 d^2\right )+a^3 b d^2 \left (275 c^2+117 d^2\right )+a^2 b^2 c d \left (121 c^2+621 d^2\right )-a b^3 \left (15 c^4-355 c^2 d^2-108 d^4\right )\right )+\frac {3}{4} a d^2 \left (20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c d \left (c^2+4 d^2\right )-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\right )\right )\right )+b \left (-\frac {1}{8} b^2 d^2 \left (3 a^4 d^4+4 a b^3 c d \left (51 c^2-50 d^2\right )-2 a^2 b^2 d^2 \left (457 c^2+114 d^2\right )-4 a^3 b c d \left (96 c^2+115 d^2\right )+b^4 \left (15 c^4+284 c^2 d^2\right )\right )+\frac {3}{8} a^2 d^2 \left (20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c d \left (c^2+4 d^2\right )-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\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}{48 (a-b) b^3 d^3} \\ & = \frac {\sqrt {a+b} (c-d) \sqrt {c+d} \left (57 a^2 b c d^2-9 a^3 d^3+a b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c 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))}{192 b^2 d (b c-a d) f}-\frac {\sqrt {c+d} \left (20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c d \left (c^2+4 d^2\right )-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\right )\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{64 b^3 \sqrt {a+b} d^2 f}-\frac {\left (57 a^2 b c d^2-9 a^3 d^3+a b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{192 b d f \sqrt {a+b \sin (e+f x)}}-\frac {\left (54 a b c d-9 a^2 d^2+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{96 b f}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{24 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}+\frac {(a+b)^{3/2} \left (9 a^3 d^3-3 a^2 b d^2 (17 c+6 d)+3 a b^2 d \left (73 c^2+36 c d+28 d^2\right )+b^3 \left (15 c^3+118 c^2 d+284 c d^2+72 d^3\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{192 b^3 d \sqrt {c+d} f} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 8.72 (sec) , antiderivative size = 2036, normalized size of antiderivative = 1.96 \[ \int (3+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

-1/384*((-4*(-(b*c) + 3*d)*(-3456*b*c^3 - 133*b^3*c^3 - 2913*b^2*c^2*d - 4059*b*c*d^2 - 356*b^3*c*d^2 + 81*d^3
- 684*b^2*d^3)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e +
Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e
+ f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d
)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*(c + d)*Sqrt[3 +
b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - 4*(-(b*c) + 3*d)*(-1596*b^2*c^3 - 5976*b*c^2*d - 644*b^3*c^2*d + 3
24*c*d^2 - 3480*b^2*c*d^2 - 2052*b*d^3 - 144*b^3*d^3)*((Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*El
lipticF[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 + P
i/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]]*Sq
rt[c + d*Sin[e + f*x]])) + 2*(15*b^3*c^3 + 1011*b^2*c^2*d + 513*b*c*d^2 + 284*b^3*c*d^2 - 243*d^3 + 468*b^2*d^
3)*((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*Sin[e + f*x
])/(3 + b)]], (2*(-(b*c) + 3*d))/((3 - b)*(c + d))]*Sqrt[c + d*Sin[e + f*x]])/(b*d*Sqrt[((3 + b)*Cos[(-e + Pi/
2 - f*x)/2]^2)/(3 + b*Sin[e + f*x])]*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[(3 + b*Sin[e + f*x])/(3 + b)]*Sqrt[((3 + b)
*(c + d*Sin[e + f*x]))/((c + d)*(3 + b*Sin[e + f*x]))]) - (2*(-(b*c) + 3*d)*((((3 + b)*c + 3*d)*Sqrt[((c + d)*
Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e
+ f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2
]^4*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]]) - ((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)))/(b*f) + (Sqrt[3 + b*Sin[
e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*(-1/96*((59*b^2*c^2 + 366*b*c*d + 27*d^2 + 42*b^2*d^2)*Cos[e + f*x])/b + (b
*d^2*Cos[3*(e + f*x)])/16 - (d*(17*b*c + 27*d)*Sin[2*(e + f*x)])/48))/f

Maple [B] (warning: unable to verify)

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

Time = 26.86 (sec) , antiderivative size = 514068, normalized size of antiderivative = 494.77

method	result	size
default	\(\text {Expression too large to display}\)	\(514068\)

[In]

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

[Out]

result too large to display

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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