3.6.36 \(\int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx\) [536]

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

[Out]

-4/1155*a*(c+d)*(3*c^2-38*c*d+355*d^2)*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/f-4/3465*a^3*(c+d)*(15*c^2+10*c*d+7*d
^2)*(3*c^2-38*c*d+355*d^2)*cos(f*x+e)/d^2/f/(a+a*sin(f*x+e))^(1/2)-2/693*a^3*(3*c^2-38*c*d+355*d^2)*cos(f*x+e)
*(c+d*sin(f*x+e))^3/d^2/f/(a+a*sin(f*x+e))^(1/2)+2/99*a^3*(3*c-23*d)*cos(f*x+e)*(c+d*sin(f*x+e))^4/d^2/f/(a+a*
sin(f*x+e))^(1/2)-8/3465*a^2*(5*c-d)*(c+d)*(3*c^2-38*c*d+355*d^2)*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/d/f-2/11*a
^2*cos(f*x+e)*(c+d*sin(f*x+e))^4*(a+a*sin(f*x+e))^(1/2)/d/f

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Rubi [A]
time = 0.44, antiderivative size = 328, normalized size of antiderivative = 1.00, 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 = {2842, 3060, 2849, 2840, 2830, 2725} \begin {gather*} -\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 {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 {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 {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} \end {gather*}

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 2725

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]

Rule 2830

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*S
in[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 2840

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 2842

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-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 2849

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 3060

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)*Sqrt
[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]

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*}

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Mathematica [A]
time = 3.99, size = 246, normalized size = 0.75 \begin {gather*} -\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \left (164472 c^3+411840 c^2 d+373098 c d^2+114640 d^3-8 \left (693 c^3+5940 c^2 d+8382 c d^2+3250 d^3\right ) \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 \sin (5 (e+f x))\right )}{27720 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/27720*(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 + 37
3098*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 + 3
2*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)] + 3
15*d^3*Sin[5*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))

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Maple [A]
time = 2.58, size = 249, normalized size = 0.76

method	result	size
default	\(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (315 d^{3} \left (\sin ^{5}\left (f x +e \right )\right )+1155 c \,d^{2} \left (\sin ^{4}\left (f x +e \right )\right )+1120 d^{3} \left (\sin ^{4}\left (f x +e \right )\right )+1485 c^{2} d \left (\sin ^{3}\left (f x +e \right )\right )+4290 c \,d^{2} \left (\sin ^{3}\left (f x +e \right )\right )+1775 d^{3} \left (\sin ^{3}\left (f x +e \right )\right )+693 c^{3} \left (\sin ^{2}\left (f x +e \right )\right )+5940 c^{2} d \left (\sin ^{2}\left (f x +e \right )\right )+7227 c \,d^{2} \left (\sin ^{2}\left (f x +e \right )\right )+2130 d^{3} \left (\sin ^{2}\left (f x +e \right )\right )+3234 c^{3} \sin \left (f x +e \right )+11385 c^{2} d \sin \left (f x +e \right )+9636 c \,d^{2} \sin \left (f x +e \right )+2840 d^{3} \sin \left (f x +e \right )+9933 c^{3}+22770 c^{2} d +19272 c \,d^{2}+5680 d^{3}\right )}{3465 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(249\)

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,method=_RETURNVERBOSE)

[Out]

2/3465*(1+sin(f*x+e))*a^3*(sin(f*x+e)-1)*(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+1
485*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*c*d^2*sin(f*
x+e)+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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.35, size = 508, normalized size = 1.55 \begin {gather*} -\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 {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}} \left (c + d \sin {\left (e + f x \right )}\right )^{3}\, dx \end {gather*}

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]

Integral((a*(sin(e + f*x) + 1))**(5/2)*(c + d*sin(e + f*x))**3, x)

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Giac [A]
time = 0.69, size = 510, normalized size = 1.55 \begin {gather*} \frac {\sqrt {2} {\left (315 \, a^{2} d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {11}{4} \, \pi + \frac {11}{2} \, f x + \frac {11}{2} \, e\right ) + 6930 \, {\left (40 \, a^{2} c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 90 \, a^{2} c^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 78 \, a^{2} c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 23 \, a^{2} d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2310 \, {\left (20 \, a^{2} c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 66 \, a^{2} c^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 60 \, a^{2} c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 19 \, a^{2} d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) + 693 \, {\left (8 \, a^{2} c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 60 \, a^{2} c^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 72 \, a^{2} c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 25 \, a^{2} d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) + 495 \, {\left (12 \, a^{2} c^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 30 \, a^{2} c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 13 \, a^{2} d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) + 385 \, {\left (6 \, a^{2} c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 5 \, a^{2} d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right )\right )} \sqrt {a}}{55440 \, f} \end {gather*}

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]

1/55440*sqrt(2)*(315*a^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-11/4*pi + 11/2*f*x + 11/2*e) + 6930*(40*
a^2*c^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 90*a^2*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 78*a^2*c*d^2*
sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 23*a^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-1/4*pi + 1/2*f*x +
1/2*e) + 2310*(20*a^2*c^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 66*a^2*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e
)) + 60*a^2*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 19*a^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-3
/4*pi + 3/2*f*x + 3/2*e) + 693*(8*a^2*c^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 60*a^2*c^2*d*sgn(cos(-1/4*pi +
1/2*f*x + 1/2*e)) + 72*a^2*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 25*a^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x +
1/2*e)))*sin(-5/4*pi + 5/2*f*x + 5/2*e) + 495*(12*a^2*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 30*a^2*c*d^
2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 13*a^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-7/4*pi + 7/2*f*x
+ 7/2*e) + 385*(6*a^2*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 5*a^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)
))*sin(-9/4*pi + 9/2*f*x + 9/2*e))*sqrt(a)/f

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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