\(\int (3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx\) [730]

Optimal result

Integrand size = 27, antiderivative size = 432 \[ \int (3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=-\frac {4 \left (756 c d^2+45 b d \left (3 c^2+5 d^2\right )-b^2 \left (5 c^3-57 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d f}-\frac {2 \left (-10 b c (b c-27 d)+7 \left (81+7 b^2\right ) d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac {4 b (b c-27 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {2 \left (189 d^2 \left (23 c^2+9 d^2\right )+90 b d \left (3 c^3+29 c d^2\right )-b^2 \left (10 c^4-279 c^2 d^2-147 d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{315 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 \left (c^2-d^2\right ) \left (5 b^2 c^3-135 b c^2 d-756 c d^2-57 b^2 c d^2-225 b d^3\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{315 d^2 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

-2/315*(7*(9*a^2+7*b^2)*d^2-10*b*c*(-9*a*d+b*c))*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/d/f+4/63*b*(-9*a*d+b*c)*cos
(f*x+e)*(c+d*sin(f*x+e))^(5/2)/d/f-2/9*b^2*cos(f*x+e)*(c+d*sin(f*x+e))^(7/2)/d/f-4/315*(84*a^2*c*d^2+15*a*b*d*
(3*c^2+5*d^2)-b^2*(5*c^3-57*c*d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/d/f-2/315*(21*a^2*d^2*(23*c^2+9*d^2)+30*
a*b*d*(3*c^3+29*c*d^2)-b^2*(10*c^4-279*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^2/f/((c+d*sin(f
*x+e))/(c+d))^(1/2)-4/315*(c^2-d^2)*(-84*a^2*c*d^2-45*a*b*c^2*d-75*a*b*d^3+5*b^2*c^3-57*b^2*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^2/f/(c+d*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2870, 2832, 2831, 2742, 2740, 2734, 2732} \[ \int (3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\frac {2 \left (21 a^2 d^2 \left (23 c^2+9 d^2\right )+30 a b c d \left (3 c^2+29 d^2\right )-\left (b^2 \left (10 c^4-279 c^2 d^2-147 d^4\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{315 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {4 \left (84 a^2 c d^2+15 a b d \left (3 c^2+5 d^2\right )-\left (b^2 \left (5 c^3-57 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d f}+\frac {4 \left (c^2-d^2\right ) \left (-84 a^2 c d^2-45 a b c^2 d-75 a b d^3+5 b^2 c^3-57 b^2 c d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{315 d^2 f \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (7 d^2 \left (9 a^2+7 b^2\right )-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f} \]

[In]

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

[Out]

(-4*(84*a^2*c*d^2 + 15*a*b*d*(3*c^2 + 5*d^2) - b^2*(5*c^3 - 57*c*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/
(315*d*f) - (2*(7*(9*a^2 + 7*b^2)*d^2 - 10*b*c*(b*c - 9*a*d))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(315*d*
f) + (4*b*(b*c - 9*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(63*d*f) - (2*b^2*Cos[e + f*x]*(c + d*Sin[e +
f*x])^(7/2))/(9*d*f) + (2*(21*a^2*d^2*(23*c^2 + 9*d^2) + 30*a*b*c*d*(3*c^2 + 29*d^2) - b^2*(10*c^4 - 279*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^2*f*Sqrt[(c + d*
Sin[e + f*x])/(c + d)]) + (4*(c^2 - d^2)*(5*b^2*c^3 - 45*a*b*c^2*d - 84*a^2*c*d^2 - 57*b^2*c*d^2 - 75*a*b*d^3)
*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(315*d^2*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 2870

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

Rubi steps \begin{align*} \text {integral}& = -\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {2 \int (c+d \sin (e+f x))^{5/2} \left (\frac {1}{2} \left (9 a^2+7 b^2\right ) d-b (b c-9 a d) \sin (e+f x)\right ) \, dx}{9 d} \\ & = \frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {4 \int (c+d \sin (e+f x))^{3/2} \left (\frac {3}{4} d \left (21 a^2 c+13 b^2 c+30 a b d\right )+\frac {1}{4} \left (7 \left (9 a^2+7 b^2\right ) d^2-10 b c (b c-9 a d)\right ) \sin (e+f x)\right ) \, dx}{63 d} \\ & = -\frac {2 \left (7 \left (9 a^2+7 b^2\right ) d^2-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {8 \int \sqrt {c+d \sin (e+f x)} \left (\frac {3}{8} d \left (240 a b c d+21 a^2 \left (5 c^2+3 d^2\right )+b^2 \left (55 c^2+49 d^2\right )\right )+\frac {3}{4} \left (84 a^2 c d^2+15 a b d \left (3 c^2+5 d^2\right )-b^2 \left (5 c^3-57 c d^2\right )\right ) \sin (e+f x)\right ) \, dx}{315 d} \\ & = -\frac {4 \left (84 a^2 c d^2+15 a b d \left (3 c^2+5 d^2\right )-b^2 \left (5 c^3-57 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d f}-\frac {2 \left (7 \left (9 a^2+7 b^2\right ) d^2-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {16 \int \frac {\frac {3}{16} d \left (30 a b d \left (27 c^2+5 d^2\right )+b^2 c \left (155 c^2+261 d^2\right )+21 a^2 \left (15 c^3+17 c d^2\right )\right )+\frac {3}{16} \left (21 a^2 d^2 \left (23 c^2+9 d^2\right )+30 a b c d \left (3 c^2+29 d^2\right )-b^2 \left (10 c^4-279 c^2 d^2-147 d^4\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{945 d} \\ & = -\frac {4 \left (84 a^2 c d^2+15 a b d \left (3 c^2+5 d^2\right )-b^2 \left (5 c^3-57 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d f}-\frac {2 \left (7 \left (9 a^2+7 b^2\right ) d^2-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {\left (2 \left (c^2-d^2\right ) \left (5 b^2 c^3-45 a b c^2 d-84 a^2 c d^2-57 b^2 c d^2-75 a b d^3\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{315 d^2}+\frac {\left (21 a^2 d^2 \left (23 c^2+9 d^2\right )+30 a b c d \left (3 c^2+29 d^2\right )-b^2 \left (10 c^4-279 c^2 d^2-147 d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{315 d^2} \\ & = -\frac {4 \left (84 a^2 c d^2+15 a b d \left (3 c^2+5 d^2\right )-b^2 \left (5 c^3-57 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d f}-\frac {2 \left (7 \left (9 a^2+7 b^2\right ) d^2-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {\left (\left (21 a^2 d^2 \left (23 c^2+9 d^2\right )+30 a b c d \left (3 c^2+29 d^2\right )-b^2 \left (10 c^4-279 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^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (2 \left (c^2-d^2\right ) \left (5 b^2 c^3-45 a b c^2 d-84 a^2 c d^2-57 b^2 c d^2-75 a b d^3\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^2 \sqrt {c+d \sin (e+f x)}} \\ & = -\frac {4 \left (84 a^2 c d^2+15 a b d \left (3 c^2+5 d^2\right )-b^2 \left (5 c^3-57 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d f}-\frac {2 \left (7 \left (9 a^2+7 b^2\right ) d^2-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {2 \left (21 a^2 d^2 \left (23 c^2+9 d^2\right )+30 a b c d \left (3 c^2+29 d^2\right )-b^2 \left (10 c^4-279 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^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 \left (c^2-d^2\right ) \left (5 b^2 c^3-45 a b c^2 d-84 a^2 c d^2-57 b^2 c d^2-75 a b d^3\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^2 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.79 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.83 \[ \int (3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\frac {8 \left (-d^2 \left (5 \left (567+31 b^2\right ) c^3+2430 b c^2 d+9 \left (357+29 b^2\right ) c d^2+450 b d^3\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (-90 b \left (3 c^3 d+29 c d^3\right )+b^2 \left (10 c^4-279 c^2 d^2-147 d^4\right )-189 \left (23 c^2 d^2+9 d^4\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-d (c+d \sin (e+f x)) \left (2 \left (8316 c d^2+b^2 \left (20 c^3+747 c d^2\right )+90 b \left (36 c^2 d+23 d^3\right )\right ) \cos (e+f x)-10 b d^2 (19 b c+54 d) \cos (3 (e+f x))+2 d \left (1620 b c d+1134 d^2+b^2 \left (150 c^2+133 d^2\right )\right ) \sin (2 (e+f x))-35 b^2 d^3 \sin (4 (e+f x))\right )}{1260 d^2 f \sqrt {c+d \sin (e+f x)}} \]

[In]

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

[Out]

(8*(-(d^2*(5*(567 + 31*b^2)*c^3 + 2430*b*c^2*d + 9*(357 + 29*b^2)*c*d^2 + 450*b*d^3)*EllipticF[(-2*e + Pi - 2*
f*x)/4, (2*d)/(c + d)]) + (-90*b*(3*c^3*d + 29*c*d^3) + b^2*(10*c^4 - 279*c^2*d^2 - 147*d^4) - 189*(23*c^2*d^2
+ 9*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])*(2*(8316*c*d^2 + b^2*(20*c^3 + 747*c*d
^2) + 90*b*(36*c^2*d + 23*d^3))*Cos[e + f*x] - 10*b*d^2*(19*b*c + 54*d)*Cos[3*(e + f*x)] + 2*d*(1620*b*c*d + 1
134*d^2 + b^2*(150*c^2 + 133*d^2))*Sin[2*(e + f*x)] - 35*b^2*d^3*Sin[4*(e + f*x)]))/(1260*d^2*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. \(2(489)=978\).

Time = 21.70 (sec) , antiderivative size = 2112, normalized size of antiderivative = 4.89

method	result	size
default	\(\text {Expression too large to display}\)	\(2112\)
parts	\(\text {Expression too large to display}\)	\(3030\)

[In]

int((a+b*sin(f*x+e))^2*(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^2*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^2*d^3*(-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))))+(2*a*b*d^
3+3*b^2*c*d^2)*(-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*(3*a^2*c^2*d+2*a*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*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)))+(a^2*d^3+6*a*b*c*d^2+3*b^2*c^2*d)*(-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))))+(3*a^2*c*d^2+6*a*b*c^2*d+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)*(-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 = 0.16 (sec) , antiderivative size = 789, normalized size of antiderivative = 1.83 \[ \int (3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\frac {\sqrt {2} {\left (20 \, b^{2} c^{5} - 180 \, a b c^{4} d + 690 \, a b c^{2} d^{3} + 450 \, a b d^{5} - 3 \, {\left (7 \, a^{2} + 31 \, b^{2}\right )} c^{3} d^{2} + 3 \, {\left (231 \, a^{2} + 163 \, b^{2}\right )} c d^{4}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) + \sqrt {2} {\left (20 \, b^{2} c^{5} - 180 \, a b c^{4} d + 690 \, a b c^{2} d^{3} + 450 \, a b d^{5} - 3 \, {\left (7 \, a^{2} + 31 \, b^{2}\right )} c^{3} d^{2} + 3 \, {\left (231 \, a^{2} + 163 \, b^{2}\right )} c d^{4}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) - 3 \, \sqrt {2} {\left (-10 i \, b^{2} c^{4} d + 90 i \, a b c^{3} d^{2} + 870 i \, a b c d^{4} + 3 i \, {\left (161 \, a^{2} + 93 \, b^{2}\right )} c^{2} d^{3} + 21 i \, {\left (9 \, a^{2} + 7 \, b^{2}\right )} d^{5}\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) - 3 \, \sqrt {2} {\left (10 i \, b^{2} c^{4} d - 90 i \, a b c^{3} d^{2} - 870 i \, a b c d^{4} - 3 i \, {\left (161 \, a^{2} + 93 \, b^{2}\right )} c^{2} d^{3} - 21 i \, {\left (9 \, a^{2} + 7 \, b^{2}\right )} d^{5}\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right ) + 6 \, {\left (5 \, {\left (19 \, b^{2} c d^{4} + 18 \, a b d^{5}\right )} \cos \left (f x + e\right )^{3} - {\left (5 \, b^{2} c^{3} d^{2} + 270 \, a b c^{2} d^{3} + 240 \, a b d^{5} + 3 \, {\left (77 \, a^{2} + 86 \, b^{2}\right )} c d^{4}\right )} \cos \left (f x + e\right ) + {\left (35 \, b^{2} d^{5} \cos \left (f x + e\right )^{3} - 3 \, {\left (25 \, b^{2} c^{2} d^{3} + 90 \, a b c d^{4} + 7 \, {\left (3 \, a^{2} + 4 \, b^{2}\right )} d^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{945 \, d^{3} f} \]

[In]

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

[Out]

1/945*(sqrt(2)*(20*b^2*c^5 - 180*a*b*c^4*d + 690*a*b*c^2*d^3 + 450*a*b*d^5 - 3*(7*a^2 + 31*b^2)*c^3*d^2 + 3*(2
31*a^2 + 163*b^2)*c*d^4)*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)*(20*b^2*c^5 - 180*a*b*c^4*d + 690*a*b*c^2
*d^3 + 450*a*b*d^5 - 3*(7*a^2 + 31*b^2)*c^3*d^2 + 3*(231*a^2 + 163*b^2)*c*d^4)*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)*(-10*I*b^2*c^4*d + 90*I*a*b*c^3*d^2 + 870*I*a*b*c*d^4 + 3*I*(161*a^2 + 93*b^2)*c^2*d^3 + 21*I
*(9*a^2 + 7*b^2)*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, wei
erstrassPInverse(-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)*(10*I*b^2*c^4*d - 90*I*a*b*c^3*d^2 - 870*I*a*b*c*d^4 - 3*I*(161*a^2 + 93*b^2
)*c^2*d^3 - 21*I*(9*a^2 + 7*b^2)*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*(19*b^2*c*d^4 + 18*a*b*d^5)*cos(f*x + e)^3 - (5*b^2*c^3*d^2 +
270*a*b*c^2*d^3 + 240*a*b*d^5 + 3*(77*a^2 + 86*b^2)*c*d^4)*cos(f*x + e) + (35*b^2*d^5*cos(f*x + e)^3 - 3*(25*b
^2*c^2*d^3 + 90*a*b*c*d^4 + 7*(3*a^2 + 4*b^2)*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(d*sin(f*x + e) + c))/(d^3*
f)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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