\(\int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx\) [738]

Optimal result

Integrand size = 27, antiderivative size = 470 \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=-\frac {2 \left (1701 b c d^2+2835 d^3-27 b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d^2 f}+\frac {2 b \left (162 b c d-1701 d^2-b^2 \left (8 c^2+49 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac {8 b^2 (b c-15 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac {2 b^2 \cos (e+f x) (3+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac {2 \left (11340 c d^3+1701 b d^2 \left (c^2+3 d^2\right )-3 b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{315 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (c^2-d^2\right ) \left (1701 b c d^2+2835 d^3-27 b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{315 d^3 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

2/315*b*(54*a*b*c*d-189*a^2*d^2-b^2*(8*c^2+49*d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/d^2/f+8/63*b^2*(-5*a*d+b
*c)*cos(f*x+e)*(c+d*sin(f*x+e))^(5/2)/d^2/f-2/9*b^2*cos(f*x+e)*(a+b*sin(f*x+e))*(c+d*sin(f*x+e))^(5/2)/d/f-2/3
15*(189*a^2*b*c*d^2+105*a^3*d^3-9*a*b^2*d*(6*c^2-25*d^2)+b^3*(8*c^3+39*c*d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^(1/
2)/d^2/f-2/315*(420*a^3*c*d^3+189*a^2*b*d^2*(c^2+3*d^2)-a*b^2*(54*c^3*d-738*c*d^3)+b^3*(8*c^4+33*c^2*d^2+147*d
^4))*(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)/d^3/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+2/315*(c^2-d^2)*(189*a^2*b*c*d^2
+105*a^3*d^3-9*a*b^2*d*(6*c^2-25*d^2)+b^3*(8*c^3+39*c*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)/d^3/f/
(c+d*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2872, 3102, 2832, 2831, 2742, 2740, 2734, 2732} \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=\frac {2 b \left (-189 a^2 d^2+54 a b c d-\left (b^2 \left (8 c^2+49 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}-\frac {2 \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d^2 f}-\frac {2 \left (c^2-d^2\right ) \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{315 d^3 f \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (420 a^3 c d^3+189 a^2 b d^2 \left (c^2+3 d^2\right )-a b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{315 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f} \]

[In]

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

[Out]

(-2*(189*a^2*b*c*d^2 + 105*a^3*d^3 - 9*a*b^2*d*(6*c^2 - 25*d^2) + b^3*(8*c^3 + 39*c*d^2))*Cos[e + f*x]*Sqrt[c
+ d*Sin[e + f*x]])/(315*d^2*f) + (2*b*(54*a*b*c*d - 189*a^2*d^2 - b^2*(8*c^2 + 49*d^2))*Cos[e + f*x]*(c + d*Si
n[e + f*x])^(3/2))/(315*d^2*f) + (8*b^2*(b*c - 5*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(63*d^2*f) - (2
*b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2))/(9*d*f) + (2*(420*a^3*c*d^3 + 189*a^2*b*d^2
*(c^2 + 3*d^2) - a*b^2*(54*c^3*d - 738*c*d^3) + b^3*(8*c^4 + 33*c^2*d^2 + 147*d^4))*EllipticE[(e - Pi/2 + f*x)
/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(315*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(c^2 - d^2)*(
189*a^2*b*c*d^2 + 105*a^3*d^3 - 9*a*b^2*d*(6*c^2 - 25*d^2) + b^3*(8*c^3 + 39*c*d^2))*EllipticF[(e - Pi/2 + f*x
)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(315*d^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 2831

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*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]

Rule 2832

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d
)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(f*(m + 1))), x] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Sim
p[b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

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 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((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 + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac {2 \int (c+d \sin (e+f x))^{3/2} \left (\frac {1}{2} \left (2 b^3 c+9 a^3 d+5 a b^2 d\right )-\frac {1}{2} b \left (2 a b c-27 a^2 d-7 b^2 d\right ) \sin (e+f x)-2 b^2 (b c-5 a d) \sin ^2(e+f x)\right ) \, dx}{9 d} \\ & = \frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac {4 \int (c+d \sin (e+f x))^{3/2} \left (-\frac {3}{4} d \left (2 b^3 c-21 a^3 d-45 a b^2 d\right )-\frac {1}{4} b \left (54 a b c d-189 a^2 d^2-b^2 \left (8 c^2+49 d^2\right )\right ) \sin (e+f x)\right ) \, dx}{63 d^2} \\ & = \frac {2 b \left (54 a b c d-189 a^2 d^2-b^2 \left (8 c^2+49 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac {8 \int \sqrt {c+d \sin (e+f x)} \left (\frac {3}{8} d \left (105 a^3 c d+171 a b^2 c d+189 a^2 b d^2-b^3 \left (2 c^2-49 d^2\right )\right )+\frac {3}{8} \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \sin (e+f x)\right ) \, dx}{315 d^2} \\ & = -\frac {2 \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d^2 f}+\frac {2 b \left (54 a b c d-189 a^2 d^2-b^2 \left (8 c^2+49 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac {16 \int \frac {\frac {3}{16} d \left (756 a^2 b c d^2+105 a^3 d \left (3 c^2+d^2\right )+9 a b^2 d \left (51 c^2+25 d^2\right )+2 b^3 \left (c^3+93 c d^2\right )\right )+\frac {3}{16} \left (420 a^3 c d^3+189 a^2 b d^2 \left (c^2+3 d^2\right )-a b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{945 d^2} \\ & = -\frac {2 \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d^2 f}+\frac {2 b \left (54 a b c d-189 a^2 d^2-b^2 \left (8 c^2+49 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}-\frac {\left (\left (c^2-d^2\right ) \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{315 d^3}+\frac {\left (420 a^3 c d^3+189 a^2 b d^2 \left (c^2+3 d^2\right )-a b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{315 d^3} \\ & = -\frac {2 \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d^2 f}+\frac {2 b \left (54 a b c d-189 a^2 d^2-b^2 \left (8 c^2+49 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac {\left (\left (420 a^3 c d^3+189 a^2 b d^2 \left (c^2+3 d^2\right )-a b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{315 d^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (\left (c^2-d^2\right ) \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c 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}{315 d^3 \sqrt {c+d \sin (e+f x)}} \\ & = -\frac {2 \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d^2 f}+\frac {2 b \left (54 a b c d-189 a^2 d^2-b^2 \left (8 c^2+49 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac {2 \left (420 a^3 c d^3+189 a^2 b d^2 \left (c^2+3 d^2\right )-a b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\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)}}{315 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (c^2-d^2\right ) \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c 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}}}{315 d^3 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.43 (sec) , antiderivative size = 385, normalized size of antiderivative = 0.82 \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=\frac {-8 \left (d^2 \left (6804 b c d^2+2 b^3 \left (c^3+93 c d^2\right )+2835 \left (3 c^2 d+d^3\right )+27 b^2 \left (51 c^2 d+25 d^3\right )\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (11340 c d^3+1701 b d^2 \left (c^2+3 d^2\right )-54 b^2 \left (3 c^3 d-41 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+d (c+d \sin (e+f x)) \left (\left (-27216 b c d^2-22680 d^3+4 b^3 \left (8 c^3-201 c d^2\right )-54 b^2 \left (12 c^2 d+115 d^3\right )\right ) \cos (e+f x)+b d \left (10 b d (10 b c+81 d) \cos (3 (e+f x))-2 \left (1296 b c d+3402 d^2+b^2 \left (6 c^2+133 d^2\right )-35 b^2 d^2 \cos (2 (e+f x))\right ) \sin (2 (e+f x))\right )\right )}{1260 d^3 f \sqrt {c+d \sin (e+f x)}} \]

[In]

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

[Out]

(-8*(d^2*(6804*b*c*d^2 + 2*b^3*(c^3 + 93*c*d^2) + 2835*(3*c^2*d + d^3) + 27*b^2*(51*c^2*d + 25*d^3))*EllipticF
[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (11340*c*d^3 + 1701*b*d^2*(c^2 + 3*d^2) - 54*b^2*(3*c^3*d - 41*c*d^3)
+ b^3*(8*c^4 + 33*c^2*d^2 + 147*d^4))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[
(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + d*(c + d*Sin[e + f*x])*((-27216*b
*c*d^2 - 22680*d^3 + 4*b^3*(8*c^3 - 201*c*d^2) - 54*b^2*(12*c^2*d + 115*d^3))*Cos[e + f*x] + b*d*(10*b*d*(10*b
*c + 81*d)*Cos[3*(e + f*x)] - 2*(1296*b*c*d + 3402*d^2 + b^2*(6*c^2 + 133*d^2) - 35*b^2*d^2*Cos[2*(e + f*x)])*
Sin[2*(e + f*x)])))/(1260*d^3*f*Sqrt[c + d*Sin[e + f*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2111\) vs. \(2(534)=1068\).

Time = 18.22 (sec) , antiderivative size = 2112, normalized size of antiderivative = 4.49

method	result	size
default	\(\text {Expression too large to display}\)	\(2112\)
parts	\(\text {Expression too large to display}\)	\(3591\)

[In]

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

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(2*a^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))+b^3*d^2*(-2/9/d*sin(f*x+e)^3*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+16/6
3*c/d^2*sin(f*x+e)^2*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)-2/5*(7/9+16/21*c^2/d^2)/d*sin(f*x+e)*(-(-d*sin(f*
x+e)-c)*cos(f*x+e)^2)^(1/2)-2/315*(-64*c^3-62*c*d^2)/d^4*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2/315*(32*c^3
+36*c*d^2)/d^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))
+2/315*(128*c^4+108*c^2*d^2+147*d^4)/d^4*(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*b^2*
d^2+2*b^3*c*d)*(-2/7/d*sin(f*x+e)^2*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+12/35*c/d^2*sin(f*x+e)*(-(-d*sin(f
*x+e)-c)*cos(f*x+e)^2)^(1/2)-2/3*(5/7+24/35*c^2/d^2)/d*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2*(-4/35*c^2/d^
2+5/21)*(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))+2/105*
(-48*c^3-44*c*d^2)/d^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)*((-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))))+2*(2*a^3*c*d+3*a^2*b*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*s
in(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))+Ellip
ticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))+(3*a^2*b*d^2+6*a*b^2*c*d+b^3*c^2)*(-2/5/d*sin(f*x+e)
*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+8/15*c/d^2*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+4/15*c/d*(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))+2*(3/5+8/15*c^2/d^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*si
n(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))+Ellipt
icF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+(a^3*d^2+6*a^2*b*c*d+3*a*b^2*c^2)*(-2/3/d*(-(-d*sin(
f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2/3*(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))-4/3*c/d*(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)*(-s
in(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)))))/cos(f*x+e)/(c+d*sin(f
*x+e))^(1/2)/f

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.22 (sec) , antiderivative size = 881, normalized size of antiderivative = 1.87 \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/945*(sqrt(2)*(16*b^3*c^5 - 108*a*b^2*c^4*d + 6*(63*a^2*b + 10*b^3)*c^3*d^2 - 3*(35*a^3 - 33*a*b^2)*c^2*d^3
- 6*(189*a^2*b + 44*b^3)*c*d^4 - 45*(7*a^3 + 15*a*b^2)*d^5)*sqrt(I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)
/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d) + sqrt(2)*(16*b^
3*c^5 - 108*a*b^2*c^4*d + 6*(63*a^2*b + 10*b^3)*c^3*d^2 - 3*(35*a^3 - 33*a*b^2)*c^2*d^3 - 6*(189*a^2*b + 44*b^
3)*c*d^4 - 45*(7*a^3 + 15*a*b^2)*d^5)*sqrt(-I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3
+ 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) + 3*sqrt(2)*(8*I*b^3*c^4*d - 54*I*a*
b^2*c^3*d^2 + 3*I*(63*a^2*b + 11*b^3)*c^2*d^3 + 6*I*(70*a^3 + 123*a*b^2)*c*d^4 + 21*I*(27*a^2*b + 7*b^3)*d^5)*
sqrt(I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*
(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d))
+ 3*sqrt(2)*(-8*I*b^3*c^4*d + 54*I*a*b^2*c^3*d^2 - 3*I*(63*a^2*b + 11*b^3)*c^2*d^3 - 6*I*(70*a^3 + 123*a*b^2)*
c*d^4 - 21*I*(27*a^2*b + 7*b^3)*d^5)*sqrt(-I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*
I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x
+ e) + 3*I*d*sin(f*x + e) + 2*I*c)/d)) - 6*(5*(10*b^3*c*d^4 + 27*a*b^2*d^5)*cos(f*x + e)^3 + (4*b^3*c^3*d^2 -
27*a*b^2*c^2*d^3 - 6*(63*a^2*b + 23*b^3)*c*d^4 - 15*(7*a^3 + 24*a*b^2)*d^5)*cos(f*x + e) + (35*b^3*d^5*cos(f*
x + e)^3 - 3*(b^3*c^2*d^3 + 72*a*b^2*c*d^4 + 7*(9*a^2*b + 4*b^3)*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(d*sin(f
*x + e) + c))/(d^4*f)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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