3.536 \(\int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx\)

Optimal. Leaf size=328 \[ -\frac{2 a^3 \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{8 a^2 (5 c-d) (c+d) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3465 d f}-\frac{4 a^3 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x)}{3465 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f}-\frac{4 a (c+d) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{1155 f} \]

[Out]

(-4*a^3*(c + d)*(15*c^2 + 10*c*d + 7*d^2)*(3*c^2 - 38*c*d + 355*d^2)*Cos[e + f*x])/(3465*d^2*f*Sqrt[a + a*Sin[
e + f*x]]) - (8*a^2*(5*c - d)*(c + d)*(3*c^2 - 38*c*d + 355*d^2)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3465*
d*f) - (4*a*(c + d)*(3*c^2 - 38*c*d + 355*d^2)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(1155*f) - (2*a^3*(3*c
^2 - 38*c*d + 355*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(693*d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (2*a^3*(3*c
- 23*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(99*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*Cos[e + f*x]*Sqrt[a
+ a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^4)/(11*d*f)

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Rubi [A]  time = 0.656315, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2763, 2981, 2770, 2761, 2751, 2646} \[ -\frac{2 a^3 \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{8 a^2 (5 c-d) (c+d) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3465 d f}-\frac{4 a^3 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x)}{3465 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f}-\frac{4 a (c+d) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{1155 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a^3*(c + d)*(15*c^2 + 10*c*d + 7*d^2)*(3*c^2 - 38*c*d + 355*d^2)*Cos[e + f*x])/(3465*d^2*f*Sqrt[a + a*Sin[
e + f*x]]) - (8*a^2*(5*c - d)*(c + d)*(3*c^2 - 38*c*d + 355*d^2)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3465*
d*f) - (4*a*(c + d)*(3*c^2 - 38*c*d + 355*d^2)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(1155*f) - (2*a^3*(3*c
^2 - 38*c*d + 355*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(693*d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (2*a^3*(3*c
- 23*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(99*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*Cos[e + f*x]*Sqrt[a
+ a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^4)/(11*d*f)

Rule 2763

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 - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c*(m - 2) + b^2*d*(n + 1) + a^2*d*(
m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] &&  !LtQ[n, -1] && (IntegersQ[2*m, 2*
n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2981

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2*b*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(2*n + 3)*Sqr
t[a + b*Sin[e + f*x]]), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b*d*(2*n + 3)), Int[Sqrt[a + b*
Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] &&
EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[n, -1]

Rule 2770

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

Rule 2761

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] && EqQ[a^2 - b^2, 0] &&  !LtQ[m, -1]

Rule 2751

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[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*Sin
[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m,
-2^(-1)]

Rule 2646

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

Rubi steps

\begin{align*} \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx &=-\frac{2 a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}+\frac{2 \int \sqrt{a+a \sin (e+f x)} \left (\frac{1}{2} a^2 (c+19 d)-\frac{1}{2} a^2 (3 c-23 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^3 \, dx}{11 d}\\ &=\frac{2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}+\frac{\left (a^2 \left (3 c^2-38 c d+355 d^2\right )\right ) \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^3 \, dx}{99 d^2}\\ &=-\frac{2 a^3 \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt{a+a \sin (e+f x)}}+\frac{2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}+\frac{\left (2 a^2 (c+d) \left (3 c^2-38 c d+355 d^2\right )\right ) \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \, dx}{231 d^2}\\ &=-\frac{4 a (c+d) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 f}-\frac{2 a^3 \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt{a+a \sin (e+f x)}}+\frac{2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}+\frac{\left (4 a (c+d) \left (3 c^2-38 c d+355 d^2\right )\right ) \int \sqrt{a+a \sin (e+f x)} \left (\frac{1}{2} a \left (5 c^2+3 d^2\right )+a (5 c-d) d \sin (e+f x)\right ) \, dx}{1155 d^2}\\ &=-\frac{8 a^2 (5 c-d) (c+d) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{3465 d f}-\frac{4 a (c+d) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 f}-\frac{2 a^3 \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt{a+a \sin (e+f x)}}+\frac{2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}+\frac{\left (2 a^2 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (3 c^2-38 c d+355 d^2\right )\right ) \int \sqrt{a+a \sin (e+f x)} \, dx}{3465 d^2}\\ &=-\frac{4 a^3 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x)}{3465 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{8 a^2 (5 c-d) (c+d) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{3465 d f}-\frac{4 a (c+d) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 f}-\frac{2 a^3 \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt{a+a \sin (e+f x)}}+\frac{2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}\\ \end{align*}

Mathematica [A]  time = 6.35429, size = 246, normalized size = 0.75 \[ -\frac{a^2 \sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (-8 \left (5940 c^2 d+693 c^3+8382 c d^2+3250 d^3\right ) \cos (2 (e+f x))+199980 c^2 d \sin (e+f x)-5940 c^2 d \sin (3 (e+f x))+411840 c^2 d+51744 c^3 \sin (e+f x)+164472 c^3+205656 c d^2 \sin (e+f x)-17160 c d^2 \sin (3 (e+f x))+70 d^2 (33 c+32 d) \cos (4 (e+f x))+373098 c d^2+69890 d^3 \sin (e+f x)-8675 d^3 \sin (3 (e+f x))+315 d^3 \sin (5 (e+f x))+114640 d^3\right )}{27720 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(164472*c^3 + 411840*c^2*d + 373098*c*d
^2 + 114640*d^3 - 8*(693*c^3 + 5940*c^2*d + 8382*c*d^2 + 3250*d^3)*Cos[2*(e + f*x)] + 70*d^2*(33*c + 32*d)*Cos
[4*(e + f*x)] + 51744*c^3*Sin[e + f*x] + 199980*c^2*d*Sin[e + f*x] + 205656*c*d^2*Sin[e + f*x] + 69890*d^3*Sin
[e + f*x] - 5940*c^2*d*Sin[3*(e + f*x)] - 17160*c*d^2*Sin[3*(e + f*x)] - 8675*d^3*Sin[3*(e + f*x)] + 315*d^3*S
in[5*(e + f*x)]))/(27720*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))

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Maple [A]  time = 0.668, size = 249, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( fx+e \right ) \right ){a}^{3} \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( 315\,{d}^{3} \left ( \sin \left ( fx+e \right ) \right ) ^{5}+1155\,c{d}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{4}+1120\,{d}^{3} \left ( \sin \left ( fx+e \right ) \right ) ^{4}+1485\,{c}^{2}d \left ( \sin \left ( fx+e \right ) \right ) ^{3}+4290\,c{d}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}+1775\,{d}^{3} \left ( \sin \left ( fx+e \right ) \right ) ^{3}+693\,{c}^{3} \left ( \sin \left ( fx+e \right ) \right ) ^{2}+5940\,{c}^{2}d \left ( \sin \left ( fx+e \right ) \right ) ^{2}+7227\,c{d}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{2}+2130\,{d}^{3} \left ( \sin \left ( fx+e \right ) \right ) ^{2}+3234\,{c}^{3}\sin \left ( fx+e \right ) +11385\,{c}^{2}d\sin \left ( fx+e \right ) +9636\,\sin \left ( fx+e \right ){d}^{2}c+2840\,{d}^{3}\sin \left ( fx+e \right ) +9933\,{c}^{3}+22770\,{c}^{2}d+19272\,c{d}^{2}+5680\,{d}^{3} \right ) }{3465\,f\cos \left ( fx+e \right ) }{\frac{1}{\sqrt{a+a\sin \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(1+sin(f*x+e))*a^3*(-1+sin(f*x+e))*(315*d^3*sin(f*x+e)^5+1155*c*d^2*sin(f*x+e)^4+1120*d^3*sin(f*x+e)^4+
1485*c^2*d*sin(f*x+e)^3+4290*c*d^2*sin(f*x+e)^3+1775*d^3*sin(f*x+e)^3+693*c^3*sin(f*x+e)^2+5940*c^2*d*sin(f*x+
e)^2+7227*c*d^2*sin(f*x+e)^2+2130*d^3*sin(f*x+e)^2+3234*c^3*sin(f*x+e)+11385*c^2*d*sin(f*x+e)+9636*sin(f*x+e)*
d^2*c+2840*d^3*sin(f*x+e)+9933*c^3+22770*c^2*d+19272*c*d^2+5680*d^3)/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}}{\left (d \sin \left (f x + e\right ) + c\right )}^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.81451, size = 1227, normalized size = 3.74 \begin{align*} -\frac{2 \,{\left (315 \, a^{2} d^{3} \cos \left (f x + e\right )^{6} + 35 \,{\left (33 \, a^{2} c d^{2} + 32 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{5} + 7392 \, a^{2} c^{3} + 15840 \, a^{2} c^{2} d + 13728 \, a^{2} c d^{2} + 4000 \, a^{2} d^{3} - 5 \,{\left (297 \, a^{2} c^{2} d + 627 \, a^{2} c d^{2} + 320 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{4} -{\left (693 \, a^{2} c^{3} + 5940 \, a^{2} c^{2} d + 9537 \, a^{2} c d^{2} + 4370 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} +{\left (2541 \, a^{2} c^{3} + 8415 \, a^{2} c^{2} d + 8679 \, a^{2} c d^{2} + 2965 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \,{\left (5313 \, a^{2} c^{3} + 14355 \, a^{2} c^{2} d + 13827 \, a^{2} c d^{2} + 4465 \, a^{2} d^{3}\right )} \cos \left (f x + e\right ) +{\left (315 \, a^{2} d^{3} \cos \left (f x + e\right )^{5} - 7392 \, a^{2} c^{3} - 15840 \, a^{2} c^{2} d - 13728 \, a^{2} c d^{2} - 4000 \, a^{2} d^{3} - 35 \,{\left (33 \, a^{2} c d^{2} + 23 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{4} - 5 \,{\left (297 \, a^{2} c^{2} d + 858 \, a^{2} c d^{2} + 481 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \,{\left (231 \, a^{2} c^{3} + 1485 \, a^{2} c^{2} d + 1749 \, a^{2} c d^{2} + 655 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \,{\left (1617 \, a^{2} c^{3} + 6435 \, a^{2} c^{2} d + 6963 \, a^{2} c d^{2} + 2465 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt{a \sin \left (f x + e\right ) + a}}{3465 \,{\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3465*(315*a^2*d^3*cos(f*x + e)^6 + 35*(33*a^2*c*d^2 + 32*a^2*d^3)*cos(f*x + e)^5 + 7392*a^2*c^3 + 15840*a^2
*c^2*d + 13728*a^2*c*d^2 + 4000*a^2*d^3 - 5*(297*a^2*c^2*d + 627*a^2*c*d^2 + 320*a^2*d^3)*cos(f*x + e)^4 - (69
3*a^2*c^3 + 5940*a^2*c^2*d + 9537*a^2*c*d^2 + 4370*a^2*d^3)*cos(f*x + e)^3 + (2541*a^2*c^3 + 8415*a^2*c^2*d +
8679*a^2*c*d^2 + 2965*a^2*d^3)*cos(f*x + e)^2 + 2*(5313*a^2*c^3 + 14355*a^2*c^2*d + 13827*a^2*c*d^2 + 4465*a^2
*d^3)*cos(f*x + e) + (315*a^2*d^3*cos(f*x + e)^5 - 7392*a^2*c^3 - 15840*a^2*c^2*d - 13728*a^2*c*d^2 - 4000*a^2
*d^3 - 35*(33*a^2*c*d^2 + 23*a^2*d^3)*cos(f*x + e)^4 - 5*(297*a^2*c^2*d + 858*a^2*c*d^2 + 481*a^2*d^3)*cos(f*x
+ e)^3 + 3*(231*a^2*c^3 + 1485*a^2*c^2*d + 1749*a^2*c*d^2 + 655*a^2*d^3)*cos(f*x + e)^2 + 2*(1617*a^2*c^3 + 6
435*a^2*c^2*d + 6963*a^2*c*d^2 + 2465*a^2*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x
+ e) + f*sin(f*x + e) + f)

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

[Out]

Timed out