3.730 \(\int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx\)

Optimal. Leaf size=451 \[ -\frac{4 \left (84 a^2 c d^2+15 a b d \left (3 c^2+5 d^2\right )+b^2 \left (-\left (5 c^3-57 c d^2\right )\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d f}+\frac{4 \left (c^2-d^2\right ) \left (-84 a^2 c d^2-45 a b c^2 d-75 a b d^3+5 b^2 c^3-57 b^2 c d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{315 d^2 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (21 a^2 d^2 \left (23 c^2+9 d^2\right )+30 a b d \left (3 c^3+29 c d^2\right )+b^2 \left (-\left (-279 c^2 d^2+10 c^4-147 d^4\right )\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{315 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 \left (7 d^2 \left (9 a^2+7 b^2\right )-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac{4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac{2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f} \]

[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*d*(3*c^3 + 29*c*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]])

________________________________________________________________________________________

Rubi [A]  time = 0.94531, antiderivative size = 451, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2791, 2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac{4 \left (84 a^2 c d^2+15 a b d \left (3 c^2+5 d^2\right )+b^2 \left (-\left (5 c^3-57 c d^2\right )\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d f}+\frac{4 \left (c^2-d^2\right ) \left (-84 a^2 c d^2-45 a b c^2 d-75 a b d^3+5 b^2 c^3-57 b^2 c d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{315 d^2 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (21 a^2 d^2 \left (23 c^2+9 d^2\right )+30 a b d \left (3 c^3+29 c d^2\right )+b^2 \left (-\left (-279 c^2 d^2+10 c^4-147 d^4\right )\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{315 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 \left (7 d^2 \left (9 a^2+7 b^2\right )-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac{4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac{2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f} \]

Antiderivative was successfully verified.

[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*d*(3*c^3 + 29*c*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 2791

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]

Rule 2753

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)*Simp[
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 2752

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 2663

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*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2655

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*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

\begin{align*} \int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx &=-\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 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 ) \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 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 ) \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 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 ) \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 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^3-45 a b c^2 d-84 a^2 c d^2-57 b^2 c d^2-75 a b d^3\right ) F\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]  time = 1.74823, size = 382, normalized size = 0.85 \[ \frac{8 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left (\left (-21 a^2 d^2 \left (23 c^2+9 d^2\right )-30 a b d \left (3 c^3+29 c d^2\right )+b^2 \left (-279 c^2 d^2+10 c^4-147 d^4\right )\right ) \left ((c+d) E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-c F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )-d^2 \left (21 a^2 \left (15 c^3+17 c d^2\right )+30 a b d \left (27 c^2+5 d^2\right )+b^2 c \left (155 c^2+261 d^2\right )\right ) F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )-d (c+d \sin (e+f x)) \left (2 d \left (126 a^2 d^2+540 a b c d+b^2 \left (150 c^2+133 d^2\right )\right ) \sin (2 (e+f x))+2 \left (924 a^2 c d^2+30 a b d \left (36 c^2+23 d^2\right )+b^2 \left (20 c^3+747 c d^2\right )\right ) \cos (e+f x)-10 b d^2 (18 a d+19 b c) \cos (3 (e+f x))-35 b^2 d^3 \sin (4 (e+f x))\right )}{1260 d^2 f \sqrt{c+d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*(-(d^2*(30*a*b*d*(27*c^2 + 5*d^2) + b^2*c*(155*c^2 + 261*d^2) + 21*a^2*(15*c^3 + 17*c*d^2))*EllipticF[(-2*e
+ Pi - 2*f*x)/4, (2*d)/(c + d)]) + (-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))*((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*(924*a^2*c*d^2 + 30*
a*b*d*(36*c^2 + 23*d^2) + b^2*(20*c^3 + 747*c*d^2))*Cos[e + f*x] - 10*b*d^2*(19*b*c + 18*a*d)*Cos[3*(e + f*x)]
+ 2*d*(540*a*b*c*d + 126*a^2*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]  time = 5.909, size = 2112, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(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)+
16/63*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*si
n(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)*((-sin(f*x+e)-1)*d/(c
-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)*((-sin(f*x+e)-1)*d/(c-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)*((-sin(f*x+e)-1)*d/(c-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)*((-sin(f*x+e)-1)*d/
(c-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^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)*((-sin(f*x+e)-1)*d/(c-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)*((-sin(f*x+e)-1)*d/(c-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^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)*((-sin(f*x+e)-1)*d/(c-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)*((-sin(f*x+e)-1)*d/(c-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)*((-sin(f*x+e)-1)*d/(c-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^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)*((-sin(f*x+e)-1)*d/(c-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)))/cos(f*x+e)/(c+d*sin(f*x+e))^(1/2)/f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{2} d^{2} \cos \left (f x + e\right )^{4} + 4 \, a b c d +{\left (a^{2} + b^{2}\right )} c^{2} +{\left (a^{2} + b^{2}\right )} d^{2} -{\left (b^{2} c^{2} + 4 \, a b c d +{\left (a^{2} + 2 \, b^{2}\right )} d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \,{\left (a b c^{2} + a b d^{2} +{\left (a^{2} + b^{2}\right )} c d -{\left (b^{2} c d + a b d^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt{d \sin \left (f x + e\right ) + c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*d^2*cos(f*x + e)^4 + 4*a*b*c*d + (a^2 + b^2)*c^2 + (a^2 + b^2)*d^2 - (b^2*c^2 + 4*a*b*c*d + (a^2
+ 2*b^2)*d^2)*cos(f*x + e)^2 + 2*(a*b*c^2 + a*b*d^2 + (a^2 + b^2)*c*d - (b^2*c*d + a*b*d^2)*cos(f*x + e)^2)*s
in(f*x + e))*sqrt(d*sin(f*x + e) + c), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out