\(\int (3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2} \, dx\) [778]

Optimal result

Integrand size = 29, antiderivative size = 1030 \[ \int (3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2} \, dx=\frac {\sqrt {3+b} (c-d) \sqrt {c+d} \left (3033 b c d^2+405 d^3+3 b^2 d \left (57 c^2+284 d^2\right )-b^3 \left (9 c^3-156 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 (b c-3 d) d^2 f}+\frac {\sqrt {c+d} \left (1620 b c d^3-405 d^4-60 b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+270 b^2 d^2 \left (3 c^2+4 d^2\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^2 \sqrt {3+b} d^3 f}-\frac {\left (3033 b c d^2+405 d^3+3 b^2 d \left (57 c^2+284 d^2\right )-b^3 \left (9 c^3-156 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{192 d^2 f \sqrt {3+b \sin (e+f x)}}-\frac {\left (162 b c d+531 d^2-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{96 d f}-\frac {(3+b)^{3/2} \left (405 d^3-135 b d^2 (11 c+2 d)-3 b^2 d \left (51 c^2+172 c d+212 d^2\right )+b^3 \left (9 c^3-6 c^2 d-156 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^2 d^2 \sqrt {c+d} f}+\frac {b (3 b c-51 d) \cos (e+f x) \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{24 d f}-\frac {b^2 \cos (e+f x) \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f} \]

[Out]

1/64*(60*a^3*b*c*d^3-5*a^4*d^4-20*a*b^3*c*d*(c^2-12*d^2)+3*b^4*(c^2+4*d^2)^2+30*a^2*b^2*d^2*(3*c^2+4*d^2))*Ell
ipticPi((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^2/d^3/f/(a+b)^(1/2)+1/192*(c-d)*(337*a^2*b*c*d
^2+15*a^3*d^3+a*b^2*d*(57*c^2+284*d^2)-b^3*(9*c^3-156*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/d^2/(-a*d+b*c)/f+1/24*b*(-17*a*d+3*b*c)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)*(a+b*sin(f*x+e))^(1/2)
/d/f-1/4*b^2*cos(f*x+e)*(c+d*sin(f*x+e))^(5/2)*(a+b*sin(f*x+e))^(1/2)/d/f-1/192*(a+b)^(3/2)*(15*a^3*d^3-15*a^2
*b*d^2*(11*c+2*d)-a*b^2*d*(51*c^2+172*c*d+212*d^2)+b^3*(9*c^3-6*c^2*d-156*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*si
n(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^2/d^2/f/(c+d)^(1/2)-1/192*(337*a^2*b*c*d^2+15*a^3*d^3+a*b^2*d*(57*c^2+284*d^2)-b^3*(9*c^3-156
*c*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)-1/96*(54*a*b*c*d+59*a^2*d^2-9*b^2*(c^2
-4*d^2))*cos(f*x+e)*(a+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(1/2)/d/f

Rubi [A] (verified)

Time = 3.27 (sec) , antiderivative size = 1071, 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))^{5/2} (c+d \sin (e+f x))^{3/2} \, dx=-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}+\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{24 d f}-\frac {(a+b)^{3/2} \left (\left (9 c^3-6 d c^2-156 d^2 c-72 d^3\right ) b^3-a d \left (51 c^2+172 d c+212 d^2\right ) b^2-15 a^2 d^2 (11 c+2 d) b+15 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^2 d^2 \sqrt {c+d} f}-\frac {\left (-9 \left (c^2-4 d^2\right ) b^2+54 a c d b+59 a^2 d^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{96 d f}-\frac {\left (-\left (\left (9 c^3-156 c d^2\right ) b^3\right )+a d \left (57 c^2+284 d^2\right ) b^2+337 a^2 c d^2 b+15 a^3 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{192 d^2 f \sqrt {a+b \sin (e+f x)}}+\frac {\sqrt {a+b} (c-d) \sqrt {c+d} \left (-\left (\left (9 c^3-156 c d^2\right ) b^3\right )+a d \left (57 c^2+284 d^2\right ) b^2+337 a^2 c d^2 b+15 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 d^2 (b c-a d) f}+\frac {\sqrt {c+d} \left (3 \left (c^2+4 d^2\right )^2 b^4-20 a c d \left (c^2-12 d^2\right ) b^3+30 a^2 d^2 \left (3 c^2+4 d^2\right ) b^2+60 a^3 c d^3 b-5 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^2 \sqrt {a+b} d^3 f} \]

[In]

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

[Out]

(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(337*a^2*b*c*d^2 + 15*a^3*d^3 + a*b^2*d*(57*c^2 + 284*d^2) - b^3*(9*c^3 - 156
*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*d^
2*(b*c - a*d)*f) + (Sqrt[c + d]*(60*a^3*b*c*d^3 - 5*a^4*d^4 - 20*a*b^3*c*d*(c^2 - 12*d^2) + 3*b^4*(c^2 + 4*d^2
)^2 + 30*a^2*b^2*d^2*(3*c^2 + 4*d^2))*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^2*Sqrt[a + b]*d^3*f) - ((337*a^2*b*c*d^2 + 15*a^3*d^3 + a*
b^2*d*(57*c^2 + 284*d^2) - b^3*(9*c^3 - 156*c*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(192*d^2*f*Sqrt[a +
b*Sin[e + f*x]]) - ((54*a*b*c*d + 59*a^2*d^2 - 9*b^2*(c^2 - 4*d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqr
t[c + d*Sin[e + f*x]])/(96*d*f) - ((a + b)^(3/2)*(15*a^3*d^3 - 15*a^2*b*d^2*(11*c + 2*d) - a*b^2*d*(51*c^2 + 1
72*c*d + 212*d^2) + b^3*(9*c^3 - 6*c^2*d - 156*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^2*d^2*Sqrt[c + d]*f) + (b*(3*b*c - 17*a*d)*Cos[e + f*x]*S
qrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2))/(24*d*f) - (b^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c +
d*Sin[e + f*x])^(5/2))/(4*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 {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}+\frac {\int \frac {(c+d \sin (e+f x))^{3/2} \left (\frac {1}{2} \left (b^3 c+8 a^3 d+5 a b^2 d\right )-b \left (a b c-12 a^2 d-3 b^2 d\right ) \sin (e+f x)-\frac {1}{2} b^2 (3 b c-17 a d) \sin ^2(e+f x)\right )}{\sqrt {a+b \sin (e+f x)}} \, dx}{4 d} \\ & = \frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{24 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}+\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (\frac {1}{4} b \left (3 b^3 c^2+48 a^3 c d+38 a b^2 c d+51 a^2 b d^2\right )+\frac {1}{2} b \left (55 a^2 b c d+15 b^3 c d+24 a^3 d^2-a b^2 \left (3 c^2-49 d^2\right )\right ) \sin (e+f x)+\frac {1}{4} b^2 \left (54 a b c d+59 a^2 d^2-9 b^2 \left (c^2-4 d^2\right )\right ) \sin ^2(e+f x)\right )}{\sqrt {a+b \sin (e+f x)}} \, dx}{12 b d} \\ & = -\frac {\left (54 a b c d+59 a^2 d^2-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{96 d f}+\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{24 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}+\frac {\int \frac {\frac {1}{8} b^2 \left (317 a^2 b c d^2+a b^2 d \left (197 c^2+36 d^2\right )+a^3 d \left (192 c^2+59 d^2\right )+3 b^3 \left (c^3+12 c d^2\right )\right )+\frac {1}{4} b^2 \left (133 a^3 c d^2-a b^2 c \left (3 c^2-290 d^2\right )+3 b^3 d \left (19 c^2+12 d^2\right )+a^2 b d \left (166 c^2+161 d^2\right )\right ) \sin (e+f x)+\frac {1}{8} b^2 \left (337 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (57 c^2+284 d^2\right )-b^3 \left (9 c^3-156 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^2 d} \\ & = -\frac {\left (337 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (57 c^2+284 d^2\right )-b^3 \left (9 c^3-156 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{192 d^2 f \sqrt {a+b \sin (e+f x)}}-\frac {\left (54 a b c d+59 a^2 d^2-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{96 d f}+\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{24 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}+\frac {\int \frac {\frac {1}{8} b^2 \left (956 a^3 b c d^3-4 a b^3 c d \left (15 c^2+14 d^2\right )+a^4 d^2 \left (384 c^2+133 d^2\right )+2 a^2 b^2 d^2 \left (57 c^2+178 d^2\right )+3 b^4 \left (3 c^4-52 c^2 d^2\right )\right )+\frac {1}{4} b^2 \left (251 a^4 c d^3-a^2 b^2 c d \left (63 c^2-613 d^2\right )+3 b^4 c d \left (c^2+12 d^2\right )+a^3 b d^2 \left (187 c^2+381 d^2\right )+a b^3 \left (9 c^4+155 c^2 d^2+108 d^4\right )\right ) \sin (e+f x)+\frac {3}{8} b^2 \left (60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+30 a^2 b^2 d^2 \left (3 c^2+4 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}{48 b^2 d^2} \\ & = -\frac {\left (337 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (57 c^2+284 d^2\right )-b^3 \left (9 c^3-156 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{192 d^2 f \sqrt {a+b \sin (e+f x)}}-\frac {\left (54 a b c d+59 a^2 d^2-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{96 d f}+\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{24 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}+\frac {\int \frac {-\frac {3}{8} a^2 b^2 \left (60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\right )\right )+\frac {1}{8} b^4 \left (956 a^3 b c d^3-4 a b^3 c d \left (15 c^2+14 d^2\right )+a^4 d^2 \left (384 c^2+133 d^2\right )+2 a^2 b^2 d^2 \left (57 c^2+178 d^2\right )+3 b^4 \left (3 c^4-52 c^2 d^2\right )\right )+b \left (-\frac {3}{4} a b^2 \left (60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\right )\right )+\frac {1}{4} b^3 \left (251 a^4 c d^3-a^2 b^2 c d \left (63 c^2-613 d^2\right )+3 b^4 c d \left (c^2+12 d^2\right )+a^3 b d^2 \left (187 c^2+381 d^2\right )+a b^3 \left (9 c^4+155 c^2 d^2+108 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^4 d^2}+\frac {\left (60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{128 b^2 d^2} \\ & = \frac {\sqrt {c+d} \left (60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\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^2 \sqrt {a+b} d^3 f}-\frac {\left (337 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (57 c^2+284 d^2\right )-b^3 \left (9 c^3-156 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{192 d^2 f \sqrt {a+b \sin (e+f x)}}-\frac {\left (54 a b c d+59 a^2 d^2-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{96 d f}+\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{24 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}+\frac {\left (-\frac {3}{8} a^2 b^2 \left (60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\right )\right )+\frac {1}{8} b^4 \left (956 a^3 b c d^3-4 a b^3 c d \left (15 c^2+14 d^2\right )+a^4 d^2 \left (384 c^2+133 d^2\right )+2 a^2 b^2 d^2 \left (57 c^2+178 d^2\right )+3 b^4 \left (3 c^4-52 c^2 d^2\right )\right )-b \left (-\frac {3}{4} a b^2 \left (60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\right )\right )+\frac {1}{4} b^3 \left (251 a^4 c d^3-a^2 b^2 c d \left (63 c^2-613 d^2\right )+3 b^4 c d \left (c^2+12 d^2\right )+a^3 b d^2 \left (187 c^2+381 d^2\right )+a b^3 \left (9 c^4+155 c^2 d^2+108 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^4 d^2}-\frac {\left (b \left (-\frac {3}{8} a^2 b^2 \left (60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\right )\right )+\frac {1}{8} b^4 \left (956 a^3 b c d^3-4 a b^3 c d \left (15 c^2+14 d^2\right )+a^4 d^2 \left (384 c^2+133 d^2\right )+2 a^2 b^2 d^2 \left (57 c^2+178 d^2\right )+3 b^4 \left (3 c^4-52 c^2 d^2\right )\right )\right )-a b \left (-\frac {3}{4} a b^2 \left (60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\right )\right )+\frac {1}{4} b^3 \left (251 a^4 c d^3-a^2 b^2 c d \left (63 c^2-613 d^2\right )+3 b^4 c d \left (c^2+12 d^2\right )+a^3 b d^2 \left (187 c^2+381 d^2\right )+a b^3 \left (9 c^4+155 c^2 d^2+108 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^4 d^2} \\ & = \frac {\sqrt {a+b} (c-d) \sqrt {c+d} \left (337 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (57 c^2+284 d^2\right )-b^3 \left (9 c^3-156 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 d^2 (b c-a d) f}+\frac {\sqrt {c+d} \left (60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\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^2 \sqrt {a+b} d^3 f}-\frac {\left (337 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (57 c^2+284 d^2\right )-b^3 \left (9 c^3-156 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{192 d^2 f \sqrt {a+b \sin (e+f x)}}-\frac {\left (54 a b c d+59 a^2 d^2-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{96 d f}-\frac {(a+b)^{3/2} \left (15 a^3 d^3-15 a^2 b d^2 (11 c+2 d)-a b^2 d \left (51 c^2+172 c d+212 d^2\right )+b^3 \left (9 c^3-6 c^2 d-156 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^2 d^2 \sqrt {c+d} f}+\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{24 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 9.86 (sec) , antiderivative size = 2035, normalized size of antiderivative = 1.98 \[ \int (3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

((-4*(-(b*c) + 3*d)*(-(b^3*c^3) + 3456*c^2*d + 451*b^2*c^2*d + 2913*b*c*d^2 + 76*b^3*c*d^2 + 1197*d^3 + 356*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]*Si
n[(-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)*(-12*b^2*c^3 + 1992*b*c^2*d + 76*b^3*c^2*d + 4788*c*d^2 +
1160*b^2*c*d^2 + 1932*b*d^3 + 48*b^3*d^3)*((Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcS
in[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^3 - 57*b^2*c^2*d - 1011*b*c*d^2 - 52*b^3*c*d^2 - 135*d^3 - 284*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)))/(128*d*f) + (Sqrt[3 + b*Sin[e + f*x]]*S
qrt[c + d*Sin[e + f*x]]*(-1/32*((b^2*c^2 + 122*b*c*d + 177*d^2 + 14*b^2*d^2)*Cos[e + f*x])/d + (b^2*d*Cos[3*(e
+ f*x)])/16 - (b*(3*b*c + 17*d)*Sin[2*(e + f*x)])/16))/f

Maple [B] (warning: unable to verify)

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

Time = 23.97 (sec) , antiderivative size = 514069, normalized size of antiderivative = 499.10

method	result	size
default	\(\text {Expression too large to display}\)	\(514069\)

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

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[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]

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

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

[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)