\(\int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx\) [737]

Optimal result

Integrand size = 27, antiderivative size = 609 \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=-\frac {2 \left (49896 c d^3+4455 b d^2 \left (3 c^2+5 d^2\right )-198 b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3465 d^2 f}-\frac {2 \left (13365 b c d^2+18711 d^3-99 b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}+\frac {2 b \left (198 b c d-2673 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-18 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (3+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {2 \left (6237 d^3 \left (23 c^2+9 d^2\right )+4455 b c d^2 \left (3 c^2+29 d^2\right )-99 b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3465 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (c^2-d^2\right ) \left (49896 c d^3+4455 b d^2 \left (3 c^2+5 d^2\right )-198 b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3465 d^3 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

-2/3465*(1485*a^2*b*c*d^2+693*a^3*d^3-33*a*b^2*d*(10*c^2-49*d^2)+5*b^3*(8*c^3+67*c*d^2))*cos(f*x+e)*(c+d*sin(f
*x+e))^(3/2)/d^2/f+2/693*b*(66*a*b*c*d-297*a^2*d^2-b^2*(8*c^2+81*d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^(5/2)/d^2/f
+8/99*b^2*(-6*a*d+b*c)*cos(f*x+e)*(c+d*sin(f*x+e))^(7/2)/d^2/f-2/11*b^2*cos(f*x+e)*(a+b*sin(f*x+e))*(c+d*sin(f
*x+e))^(7/2)/d/f-2/3465*(1848*a^3*c*d^3+495*a^2*b*d^2*(3*c^2+5*d^2)-66*a*b^2*d*(5*c^3-57*c*d^2)+5*b^3*(8*c^4+5
7*c^2*d^2+135*d^4))*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/d^2/f-2/3465*(231*a^3*d^3*(23*c^2+9*d^2)+495*a^2*b*c*d^2
*(3*c^2+29*d^2)-33*a*b^2*d*(10*c^4-279*c^2*d^2-147*d^4)+5*b^3*(8*c^5+51*c^3*d^2+741*c*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)+2/3465*(c^2-d^2)*(1848*a^3*c*d^3+495*a^2*b*d^2*(3*c^2+5
*d^2)-66*a*b^2*d*(5*c^3-57*c*d^2)+5*b^3*(8*c^4+57*c^2*d^2+135*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)*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 = 0.88 (sec) , antiderivative size = 642, normalized size of antiderivative = 1.05, number of steps used = 10, 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} \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\frac {2 b \left (-297 a^2 d^2+66 a b c d-\left (b^2 \left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3465 d^2 f}-\frac {2 \left (c^2-d^2\right ) \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{3465 d^3 f \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (693 a^3 d^3+1485 a^2 b c d^2-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}+\frac {2 \left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3465 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f} \]

[In]

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

[Out]

(-2*(1848*a^3*c*d^3 - 66*a*b^2*c*d*(5*c^2 - 57*d^2) + 495*a^2*b*d^2*(3*c^2 + 5*d^2) + 5*b^3*(8*c^4 + 57*c^2*d^
2 + 135*d^4))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3465*d^2*f) - (2*(1485*a^2*b*c*d^2 + 693*a^3*d^3 - 33*a*
b^2*d*(10*c^2 - 49*d^2) + 5*b^3*(8*c^3 + 67*c*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(3465*d^2*f) + (2
*b*(66*a*b*c*d - 297*a^2*d^2 - b^2*(8*c^2 + 81*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(693*d^2*f) + (8
*b^2*(b*c - 6*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/2))/(99*d^2*f) - (2*b^2*Cos[e + f*x]*(a + b*Sin[e + f*
x])*(c + d*Sin[e + f*x])^(7/2))/(11*d*f) + (2*(231*a^3*d^3*(23*c^2 + 9*d^2) + 495*a^2*b*c*d^2*(3*c^2 + 29*d^2)
- 33*a*b^2*d*(10*c^4 - 279*c^2*d^2 - 147*d^4) + 5*b^3*(8*c^5 + 51*c^3*d^2 + 741*c*d^4))*EllipticE[(e - Pi/2 +
f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3465*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(c^2 -
d^2)*(1848*a^3*c*d^3 - 66*a*b^2*c*d*(5*c^2 - 57*d^2) + 495*a^2*b*d^2*(3*c^2 + 5*d^2) + 5*b^3*(8*c^4 + 57*c^2*d
^2 + 135*d^4))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3465*d^3*f*Sq
rt[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))^{7/2}}{11 d f}+\frac {2 \int (c+d \sin (e+f x))^{5/2} \left (\frac {1}{2} \left (2 b^3 c+11 a^3 d+7 a b^2 d\right )-\frac {1}{2} b \left (2 a b c-33 a^2 d-9 b^2 d\right ) \sin (e+f x)-2 b^2 (b c-6 a d) \sin ^2(e+f x)\right ) \, dx}{11 d} \\ & = \frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {4 \int (c+d \sin (e+f x))^{5/2} \left (-\frac {1}{4} d \left (10 b^3 c-99 a^3 d-231 a b^2 d\right )-\frac {1}{4} b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \sin (e+f x)\right ) \, dx}{99 d^2} \\ & = \frac {2 b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {8 \int (c+d \sin (e+f x))^{3/2} \left (\frac {3}{8} d \left (231 a^3 c d+429 a b^2 c d+495 a^2 b d^2-5 b^3 \left (2 c^2-27 d^2\right )\right )+\frac {1}{8} \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \sin (e+f x)\right ) \, dx}{693 d^2} \\ & = -\frac {2 \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}+\frac {2 b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {16 \int \sqrt {c+d \sin (e+f x)} \left (\frac {3}{16} d \left (3960 a^2 b c d^2+231 a^3 d \left (5 c^2+3 d^2\right )+33 a b^2 d \left (55 c^2+49 d^2\right )-10 b^3 \left (c^3-101 c d^2\right )\right )+\frac {3}{16} \left (1848 a^3 c d^3-66 a b^2 c d \left (5 c^2-57 d^2\right )+495 a^2 b d^2 \left (3 c^2+5 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \sin (e+f x)\right ) \, dx}{3465 d^2} \\ & = -\frac {2 \left (1848 a^3 c d^3-66 a b^2 c d \left (5 c^2-57 d^2\right )+495 a^2 b d^2 \left (3 c^2+5 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3465 d^2 f}-\frac {2 \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}+\frac {2 b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {32 \int \frac {\frac {3}{32} d \left (495 a^2 b d^2 \left (27 c^2+5 d^2\right )+231 a^3 c d \left (15 c^2+17 d^2\right )+33 a b^2 c d \left (155 c^2+261 d^2\right )+5 b^3 \left (2 c^4+663 c^2 d^2+135 d^4\right )\right )+\frac {3}{32} \left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{10395 d^2} \\ & = -\frac {2 \left (1848 a^3 c d^3-66 a b^2 c d \left (5 c^2-57 d^2\right )+495 a^2 b d^2 \left (3 c^2+5 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3465 d^2 f}-\frac {2 \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}+\frac {2 b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}-\frac {\left (\left (c^2-d^2\right ) \left (1848 a^3 c d^3-66 a b^2 c d \left (5 c^2-57 d^2\right )+495 a^2 b d^2 \left (3 c^2+5 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{3465 d^3}+\frac {\left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{3465 d^3} \\ & = -\frac {2 \left (1848 a^3 c d^3-66 a b^2 c d \left (5 c^2-57 d^2\right )+495 a^2 b d^2 \left (3 c^2+5 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3465 d^2 f}-\frac {2 \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}+\frac {2 b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {\left (\left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c 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}{3465 d^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (\left (c^2-d^2\right ) \left (1848 a^3 c d^3-66 a b^2 c d \left (5 c^2-57 d^2\right )+495 a^2 b d^2 \left (3 c^2+5 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\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}{3465 d^3 \sqrt {c+d \sin (e+f x)}} \\ & = -\frac {2 \left (1848 a^3 c d^3-66 a b^2 c d \left (5 c^2-57 d^2\right )+495 a^2 b d^2 \left (3 c^2+5 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3465 d^2 f}-\frac {2 \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}+\frac {2 b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {2 \left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c 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)}}{3465 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (c^2-d^2\right ) \left (1848 a^3 c d^3-66 a b^2 c d \left (5 c^2-57 d^2\right )+495 a^2 b d^2 \left (3 c^2+5 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\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}}}{3465 d^3 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.72 (sec) , antiderivative size = 512, normalized size of antiderivative = 0.84 \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\frac {-16 \left (d^2 \left (6237 \left (15 c^3 d+17 c d^3\right )+99 b^2 \left (155 c^3 d+261 c d^3\right )+4455 b \left (27 c^2 d^2+5 d^4\right )+5 b^3 \left (2 c^4+663 c^2 d^2+135 d^4\right )\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (4455 b \left (3 c^3 d^2+29 c d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )-99 b^2 \left (10 c^4 d-279 c^2 d^3-147 d^5\right )+6237 \left (23 c^2 d^3+9 d^5\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+d (c+d \sin (e+f x)) \left (2 \left (-548856 c d^3-198 b^2 \left (20 c^3 d+747 c d^3\right )+5 b^3 \left (32 c^4-1866 c^2 d^2-1305 d^4\right )-8910 b \left (36 c^2 d^2+23 d^4\right )\right ) \cos (e+f x)+5 b d^2 \left (7524 b c d+10692 d^2+b^2 \left (452 c^2+513 d^2\right )\right ) \cos (3 (e+f x))-315 b^3 d^4 \cos (5 (e+f x))-4 d \left (80190 b c d^2+37422 d^3+5 b^3 \left (6 c^3+619 c d^2\right )+99 b^2 \left (150 c^2 d+133 d^3\right )\right ) \sin (2 (e+f x))+70 b^2 d^3 (23 b c+99 d) \sin (4 (e+f x))\right )}{27720 d^3 f \sqrt {c+d \sin (e+f x)}} \]

[In]

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

[Out]

(-16*(d^2*(6237*(15*c^3*d + 17*c*d^3) + 99*b^2*(155*c^3*d + 261*c*d^3) + 4455*b*(27*c^2*d^2 + 5*d^4) + 5*b^3*(
2*c^4 + 663*c^2*d^2 + 135*d^4))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (4455*b*(3*c^3*d^2 + 29*c*d^
4) + 5*b^3*(8*c^5 + 51*c^3*d^2 + 741*c*d^4) - 99*b^2*(10*c^4*d - 279*c^2*d^3 - 147*d^5) + 6237*(23*c^2*d^3 + 9
*d^5))*((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])*(2*(-548856*c*d^3 - 198*b^2*(20*c^3*d + 74
7*c*d^3) + 5*b^3*(32*c^4 - 1866*c^2*d^2 - 1305*d^4) - 8910*b*(36*c^2*d^2 + 23*d^4))*Cos[e + f*x] + 5*b*d^2*(75
24*b*c*d + 10692*d^2 + b^2*(452*c^2 + 513*d^2))*Cos[3*(e + f*x)] - 315*b^3*d^4*Cos[5*(e + f*x)] - 4*d*(80190*b
*c*d^2 + 37422*d^3 + 5*b^3*(6*c^3 + 619*c*d^2) + 99*b^2*(150*c^2*d + 133*d^3))*Sin[2*(e + f*x)] + 70*b^2*d^3*(
23*b*c + 99*d)*Sin[4*(e + f*x)]))/(27720*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. \(2727\) vs. \(2(676)=1352\).

Time = 58.37 (sec) , antiderivative size = 2728, normalized size of antiderivative = 4.48

method	result	size
default	\(\text {Expression too large to display}\)	\(2728\)
parts	\(\text {Expression too large to display}\)	\(4390\)

[In]

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

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(2*a^3*c^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))+b^3*d^3*(-2/11/d*sin(f*x+e)^4*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+20/
99*c/d^2*sin(f*x+e)^3*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)-2/7*(9/11+80/99*c^2/d^2)/d*sin(f*x+e)^2*(-(-d*si
n(f*x+e)-c)*cos(f*x+e)^2)^(1/2)-2/3465*(-480*c^3-472*c*d^2)/d^4*sin(f*x+e)*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(
1/2)-2/3465*(640*c^4+596*c^2*d^2+675*d^4)/d^5*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2/3465*(-320*c^4-348*c^2
*d^2+675*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)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/
2))+2/3465*(-1280*c^5-1032*c^3*d^2-1146*c*d^4)/d^5*(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^3+3*b^3*c*d^2)*(-2/9/d*sin(f*x+e)^3*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+16/63*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+10
8*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))))+2*(3*a^3*c^2*d+3*a^2*b*c^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))+Ell
ipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))+(3*a^2*b*d^3+9*a*b^2*c*d^2+3*b^3*c^2*d)*(-2/7/d*si
n(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*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))))+(3*a^3*c*d^2+9*a^2*b*c^2*d+3*a*b^2*c^3)*(-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)*(-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))))+(a^3*d^3+9*a^2*b
*c*d^2+9*a*b^2*c^2*d+b^3*c^3)*(-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*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 = 0.18 (sec) , antiderivative size = 1051, normalized size of antiderivative = 1.73 \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/10395*(sqrt(2)*(80*b^3*c^6 - 660*a*b^2*c^5*d + 30*(99*a^2*b + 16*b^3)*c^4*d^2 + 33*(7*a^3 + 93*a*b^2)*c^3*d
^3 - 15*(759*a^2*b + 169*b^3)*c^2*d^4 - 99*(77*a^3 + 163*a*b^2)*c*d^5 - 675*(11*a^2*b + 3*b^3)*d^6)*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*s
in(f*x + e) - 2*I*c)/d) + sqrt(2)*(80*b^3*c^6 - 660*a*b^2*c^5*d + 30*(99*a^2*b + 16*b^3)*c^4*d^2 + 33*(7*a^3 +
93*a*b^2)*c^3*d^3 - 15*(759*a^2*b + 169*b^3)*c^2*d^4 - 99*(77*a^3 + 163*a*b^2)*c*d^5 - 675*(11*a^2*b + 3*b^3)
*d^6)*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)*(40*I*b^3*c^5*d - 330*I*a*b^2*c^4*d^2 + 15*I*(99*a^2*b +
17*b^3)*c^3*d^3 + 33*I*(161*a^3 + 279*a*b^2)*c^2*d^4 + 15*I*(957*a^2*b + 247*b^3)*c*d^5 + 693*I*(3*a^3 + 7*a*
b^2)*d^6)*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, weierstrassPInv
erse(-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)*(-40*I*b^3*c^5*d + 330*I*a*b^2*c^4*d^2 - 15*I*(99*a^2*b + 17*b^3)*c^3*d^3 - 33*I*(161*a^
3 + 279*a*b^2)*c^2*d^4 - 15*I*(957*a^2*b + 247*b^3)*c*d^5 - 693*I*(3*a^3 + 7*a*b^2)*d^6)*sqrt(-I*d)*weierstras
sZeta(-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*(315*b^3*d^6*c
os(f*x + e)^5 - 5*(113*b^3*c^2*d^4 + 627*a*b^2*c*d^5 + 9*(33*a^2*b + 23*b^3)*d^6)*cos(f*x + e)^3 - (20*b^3*c^4
*d^2 - 165*a*b^2*c^3*d^3 - 15*(297*a^2*b + 106*b^3)*c^2*d^4 - 33*(77*a^3 + 258*a*b^2)*c*d^5 - 45*(88*a^2*b + 3
1*b^3)*d^6)*cos(f*x + e) - (35*(23*b^3*c*d^5 + 33*a*b^2*d^6)*cos(f*x + e)^3 - 3*(5*b^3*c^3*d^3 + 825*a*b^2*c^2
*d^4 + 5*(297*a^2*b + 130*b^3)*c*d^5 + 231*(a^3 + 4*a*b^2)*d^6)*cos(f*x + e))*sin(f*x + e))*sqrt(d*sin(f*x + e
) + c))/(d^4*f)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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