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

Optimal. Leaf size=429 \[ \frac{2 a^3 \left (11 A d (3 c-19 d)-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a^3 \left (15 c^2+10 c d+7 d^2\right ) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (-40 c^2 d+5 c^3+169 c d^2-710 d^3\right )\right ) \cos (e+f x)}{3465 d^3 f \sqrt{a \sin (e+f x)+a}}-\frac{4 a^2 (5 c-d) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (-40 c^2 d+5 c^3+169 c d^2-710 d^3\right )\right ) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3465 d^2 f}+\frac{2 a^2 (-11 A d+5 B c-14 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac{2 a \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (-40 c^2 d+5 c^3+169 c d^2-710 d^3\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{1155 d f}-\frac{2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}{11 d f} \]

[Out]

(-2*a^3*(15*c^2 + 10*c*d + 7*d^2)*(11*A*d*(c^2 - 10*c*d + 73*d^2) - B*(5*c^3 - 40*c^2*d + 169*c*d^2 - 710*d^3)
)*Cos[e + f*x])/(3465*d^3*f*Sqrt[a + a*Sin[e + f*x]]) - (4*a^2*(5*c - d)*(11*A*d*(c^2 - 10*c*d + 73*d^2) - B*(
5*c^3 - 40*c^2*d + 169*c*d^2 - 710*d^3))*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3465*d^2*f) - (2*a*(11*A*d*(c
^2 - 10*c*d + 73*d^2) - B*(5*c^3 - 40*c^2*d + 169*c*d^2 - 710*d^3))*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(
1155*d*f) + (2*a^3*(11*A*(3*c - 19*d)*d - B*(15*c^2 - 65*c*d + 194*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/
(693*d^3*f*Sqrt[a + a*Sin[e + f*x]]) + (2*a^2*(5*B*c - 11*A*d - 14*B*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*
(c + d*Sin[e + f*x])^3)/(99*d^2*f) - (2*a*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^3)/(1
1*d*f)

________________________________________________________________________________________

Rubi [A]  time = 1.06576, antiderivative size = 429, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {2976, 2981, 2761, 2751, 2646} \[ \frac{2 a^3 \left (11 A d (3 c-19 d)-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a^3 \left (15 c^2+10 c d+7 d^2\right ) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (-40 c^2 d+5 c^3+169 c d^2-710 d^3\right )\right ) \cos (e+f x)}{3465 d^3 f \sqrt{a \sin (e+f x)+a}}-\frac{4 a^2 (5 c-d) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (-40 c^2 d+5 c^3+169 c d^2-710 d^3\right )\right ) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3465 d^2 f}+\frac{2 a^2 (-11 A d+5 B c-14 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac{2 a \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (-40 c^2 d+5 c^3+169 c d^2-710 d^3\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{1155 d f}-\frac{2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}{11 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^3*(15*c^2 + 10*c*d + 7*d^2)*(11*A*d*(c^2 - 10*c*d + 73*d^2) - B*(5*c^3 - 40*c^2*d + 169*c*d^2 - 710*d^3)
)*Cos[e + f*x])/(3465*d^3*f*Sqrt[a + a*Sin[e + f*x]]) - (4*a^2*(5*c - d)*(11*A*d*(c^2 - 10*c*d + 73*d^2) - B*(
5*c^3 - 40*c^2*d + 169*c*d^2 - 710*d^3))*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3465*d^2*f) - (2*a*(11*A*d*(c
^2 - 10*c*d + 73*d^2) - B*(5*c^3 - 40*c^2*d + 169*c*d^2 - 710*d^3))*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(
1155*d*f) + (2*a^3*(11*A*(3*c - 19*d)*d - B*(15*c^2 - 65*c*d + 194*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/
(693*d^3*f*Sqrt[a + a*Sin[e + f*x]]) + (2*a^2*(5*B*c - 11*A*d - 14*B*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*
(c + d*Sin[e + f*x])^3)/(99*d^2*f) - (2*a*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^3)/(1
1*d*f)

Rule 2976

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])
^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x]
)^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*S
in[e + f*x], x], 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] && GtQ[m, 1/2] &&  !LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || 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 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} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx &=-\frac{2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3}{11 d f}+\frac{2 \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2 \left (\frac{1}{2} a (11 A d+3 B (c+2 d))-\frac{1}{2} a (5 B c-11 A d-14 B d) \sin (e+f x)\right ) \, dx}{11 d}\\ &=\frac{2 a^2 (5 B c-11 A d-14 B d) \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac{2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3}{11 d f}+\frac{4 \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \left (\frac{1}{4} a^2 \left (11 A d (c+15 d)-B \left (5 c^2-11 c d-138 d^2\right )\right )-\frac{1}{4} a^2 \left (11 A (3 c-19 d) d-B \left (15 c^2-65 c d+194 d^2\right )\right ) \sin (e+f x)\right ) \, dx}{99 d^2}\\ &=\frac{2 a^3 \left (11 A (3 c-19 d) d-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^3 f \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 (5 B c-11 A d-14 B d) \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac{2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3}{11 d f}+\frac{\left (a^2 \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right )\right ) \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \, dx}{231 d^3}\\ &=-\frac{2 a \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 d f}+\frac{2 a^3 \left (11 A (3 c-19 d) d-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^3 f \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 (5 B c-11 A d-14 B d) \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac{2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3}{11 d f}+\frac{\left (2 a \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\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^3}\\ &=-\frac{4 a^2 (5 c-d) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{3465 d^2 f}-\frac{2 a \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 d f}+\frac{2 a^3 \left (11 A (3 c-19 d) d-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^3 f \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 (5 B c-11 A d-14 B d) \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac{2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3}{11 d f}+\frac{\left (a^2 \left (15 c^2+10 c d+7 d^2\right ) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right )\right ) \int \sqrt{a+a \sin (e+f x)} \, dx}{3465 d^3}\\ &=-\frac{2 a^3 \left (15 c^2+10 c d+7 d^2\right ) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x)}{3465 d^3 f \sqrt{a+a \sin (e+f x)}}-\frac{4 a^2 (5 c-d) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{3465 d^2 f}-\frac{2 a \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 d f}+\frac{2 a^3 \left (11 A (3 c-19 d) d-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^3 f \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 (5 B c-11 A d-14 B d) \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac{2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3}{11 d f}\\ \end{align*}

Mathematica [B]  time = 6.61678, size = 891, normalized size = 2.08 \[ \frac{(a (\sin (e+f x)+1))^{5/2} \left (-277200 A \cos \left (\frac{1}{2} (e+f x)\right ) c^2-207900 B \cos \left (\frac{1}{2} (e+f x)\right ) c^2-46200 A \cos \left (\frac{3}{2} (e+f x)\right ) c^2-50820 B \cos \left (\frac{3}{2} (e+f x)\right ) c^2+5544 A \cos \left (\frac{5}{2} (e+f x)\right ) c^2+13860 B \cos \left (\frac{5}{2} (e+f x)\right ) c^2+1980 B \cos \left (\frac{7}{2} (e+f x)\right ) c^2+277200 A \sin \left (\frac{1}{2} (e+f x)\right ) c^2+207900 B \sin \left (\frac{1}{2} (e+f x)\right ) c^2-46200 A \sin \left (\frac{3}{2} (e+f x)\right ) c^2-50820 B \sin \left (\frac{3}{2} (e+f x)\right ) c^2-5544 A \sin \left (\frac{5}{2} (e+f x)\right ) c^2-13860 B \sin \left (\frac{5}{2} (e+f x)\right ) c^2+1980 B \sin \left (\frac{7}{2} (e+f x)\right ) c^2-415800 A d \cos \left (\frac{1}{2} (e+f x)\right ) c-360360 B d \cos \left (\frac{1}{2} (e+f x)\right ) c-101640 A d \cos \left (\frac{3}{2} (e+f x)\right ) c-92400 B d \cos \left (\frac{3}{2} (e+f x)\right ) c+27720 A d \cos \left (\frac{5}{2} (e+f x)\right ) c+33264 B d \cos \left (\frac{5}{2} (e+f x)\right ) c+3960 A d \cos \left (\frac{7}{2} (e+f x)\right ) c+9900 B d \cos \left (\frac{7}{2} (e+f x)\right ) c-1540 B d \cos \left (\frac{9}{2} (e+f x)\right ) c+415800 A d \sin \left (\frac{1}{2} (e+f x)\right ) c+360360 B d \sin \left (\frac{1}{2} (e+f x)\right ) c-101640 A d \sin \left (\frac{3}{2} (e+f x)\right ) c-92400 B d \sin \left (\frac{3}{2} (e+f x)\right ) c-27720 A d \sin \left (\frac{5}{2} (e+f x)\right ) c-33264 B d \sin \left (\frac{5}{2} (e+f x)\right ) c+3960 A d \sin \left (\frac{7}{2} (e+f x)\right ) c+9900 B d \sin \left (\frac{7}{2} (e+f x)\right ) c+1540 B d \sin \left (\frac{9}{2} (e+f x)\right ) c-180180 A d^2 \cos \left (\frac{1}{2} (e+f x)\right )-159390 B d^2 \cos \left (\frac{1}{2} (e+f x)\right )-46200 A d^2 \cos \left (\frac{3}{2} (e+f x)\right )-43890 B d^2 \cos \left (\frac{3}{2} (e+f x)\right )+16632 A d^2 \cos \left (\frac{5}{2} (e+f x)\right )+17325 B d^2 \cos \left (\frac{5}{2} (e+f x)\right )+4950 A d^2 \cos \left (\frac{7}{2} (e+f x)\right )+6435 B d^2 \cos \left (\frac{7}{2} (e+f x)\right )-770 A d^2 \cos \left (\frac{9}{2} (e+f x)\right )-1925 B d^2 \cos \left (\frac{9}{2} (e+f x)\right )-315 B d^2 \cos \left (\frac{11}{2} (e+f x)\right )+180180 A d^2 \sin \left (\frac{1}{2} (e+f x)\right )+159390 B d^2 \sin \left (\frac{1}{2} (e+f x)\right )-46200 A d^2 \sin \left (\frac{3}{2} (e+f x)\right )-43890 B d^2 \sin \left (\frac{3}{2} (e+f x)\right )-16632 A d^2 \sin \left (\frac{5}{2} (e+f x)\right )-17325 B d^2 \sin \left (\frac{5}{2} (e+f x)\right )+4950 A d^2 \sin \left (\frac{7}{2} (e+f x)\right )+6435 B d^2 \sin \left (\frac{7}{2} (e+f x)\right )+770 A d^2 \sin \left (\frac{9}{2} (e+f x)\right )+1925 B d^2 \sin \left (\frac{9}{2} (e+f x)\right )-315 B d^2 \sin \left (\frac{11}{2} (e+f x)\right )\right )}{55440 f \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*(1 + Sin[e + f*x]))^(5/2)*(-277200*A*c^2*Cos[(e + f*x)/2] - 207900*B*c^2*Cos[(e + f*x)/2] - 415800*A*c*d*C
os[(e + f*x)/2] - 360360*B*c*d*Cos[(e + f*x)/2] - 180180*A*d^2*Cos[(e + f*x)/2] - 159390*B*d^2*Cos[(e + f*x)/2
] - 46200*A*c^2*Cos[(3*(e + f*x))/2] - 50820*B*c^2*Cos[(3*(e + f*x))/2] - 101640*A*c*d*Cos[(3*(e + f*x))/2] -
92400*B*c*d*Cos[(3*(e + f*x))/2] - 46200*A*d^2*Cos[(3*(e + f*x))/2] - 43890*B*d^2*Cos[(3*(e + f*x))/2] + 5544*
A*c^2*Cos[(5*(e + f*x))/2] + 13860*B*c^2*Cos[(5*(e + f*x))/2] + 27720*A*c*d*Cos[(5*(e + f*x))/2] + 33264*B*c*d
*Cos[(5*(e + f*x))/2] + 16632*A*d^2*Cos[(5*(e + f*x))/2] + 17325*B*d^2*Cos[(5*(e + f*x))/2] + 1980*B*c^2*Cos[(
7*(e + f*x))/2] + 3960*A*c*d*Cos[(7*(e + f*x))/2] + 9900*B*c*d*Cos[(7*(e + f*x))/2] + 4950*A*d^2*Cos[(7*(e + f
*x))/2] + 6435*B*d^2*Cos[(7*(e + f*x))/2] - 1540*B*c*d*Cos[(9*(e + f*x))/2] - 770*A*d^2*Cos[(9*(e + f*x))/2] -
1925*B*d^2*Cos[(9*(e + f*x))/2] - 315*B*d^2*Cos[(11*(e + f*x))/2] + 277200*A*c^2*Sin[(e + f*x)/2] + 207900*B*
c^2*Sin[(e + f*x)/2] + 415800*A*c*d*Sin[(e + f*x)/2] + 360360*B*c*d*Sin[(e + f*x)/2] + 180180*A*d^2*Sin[(e + f
*x)/2] + 159390*B*d^2*Sin[(e + f*x)/2] - 46200*A*c^2*Sin[(3*(e + f*x))/2] - 50820*B*c^2*Sin[(3*(e + f*x))/2] -
101640*A*c*d*Sin[(3*(e + f*x))/2] - 92400*B*c*d*Sin[(3*(e + f*x))/2] - 46200*A*d^2*Sin[(3*(e + f*x))/2] - 438
90*B*d^2*Sin[(3*(e + f*x))/2] - 5544*A*c^2*Sin[(5*(e + f*x))/2] - 13860*B*c^2*Sin[(5*(e + f*x))/2] - 27720*A*c
*d*Sin[(5*(e + f*x))/2] - 33264*B*c*d*Sin[(5*(e + f*x))/2] - 16632*A*d^2*Sin[(5*(e + f*x))/2] - 17325*B*d^2*Si
n[(5*(e + f*x))/2] + 1980*B*c^2*Sin[(7*(e + f*x))/2] + 3960*A*c*d*Sin[(7*(e + f*x))/2] + 9900*B*c*d*Sin[(7*(e
+ f*x))/2] + 4950*A*d^2*Sin[(7*(e + f*x))/2] + 6435*B*d^2*Sin[(7*(e + f*x))/2] + 1540*B*c*d*Sin[(9*(e + f*x))/
2] + 770*A*d^2*Sin[(9*(e + f*x))/2] + 1925*B*d^2*Sin[(9*(e + f*x))/2] - 315*B*d^2*Sin[(11*(e + f*x))/2]))/(554
40*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)

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Maple [A]  time = 0.928, size = 257, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( fx+e \right ) \right ){a}^{3} \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( 315\,B{d}^{2}\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( -990\,Acd-1430\,A{d}^{2}-495\,B{c}^{2}-2860\,Bcd-2405\,B{d}^{2} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) + \left ( 3234\,A{c}^{2}+8580\,Acd+4642\,A{d}^{2}+4290\,B{c}^{2}+9284\,Bcd+4930\,B{d}^{2} \right ) \sin \left ( fx+e \right ) + \left ( 385\,A{d}^{2}+770\,Bcd+1120\,B{d}^{2} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( -693\,A{c}^{2}-3960\,Acd-3179\,A{d}^{2}-1980\,B{c}^{2}-6358\,Bcd-4370\,B{d}^{2} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+10626\,A{c}^{2}+19140\,Acd+9218\,A{d}^{2}+9570\,B{c}^{2}+18436\,Bcd+8930\,B{d}^{2} \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)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x)

[Out]

2/3465*(1+sin(f*x+e))*a^3*(-1+sin(f*x+e))*(315*B*d^2*sin(f*x+e)*cos(f*x+e)^4+(-990*A*c*d-1430*A*d^2-495*B*c^2-
2860*B*c*d-2405*B*d^2)*cos(f*x+e)^2*sin(f*x+e)+(3234*A*c^2+8580*A*c*d+4642*A*d^2+4290*B*c^2+9284*B*c*d+4930*B*
d^2)*sin(f*x+e)+(385*A*d^2+770*B*c*d+1120*B*d^2)*cos(f*x+e)^4+(-693*A*c^2-3960*A*c*d-3179*A*d^2-1980*B*c^2-635
8*B*c*d-4370*B*d^2)*cos(f*x+e)^2+10626*A*c^2+19140*A*c*d+9218*A*d^2+9570*B*c^2+18436*B*c*d+8930*B*d^2)/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 (B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}}{\left (d \sin \left (f x + e\right ) + c\right )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.96476, size = 1500, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3465*(315*B*a^2*d^2*cos(f*x + e)^6 + 35*(22*B*a^2*c*d + (11*A + 32*B)*a^2*d^2)*cos(f*x + e)^5 + 1056*(7*A +
5*B)*a^2*c^2 + 704*(15*A + 13*B)*a^2*c*d + 32*(143*A + 125*B)*a^2*d^2 - 5*(99*B*a^2*c^2 + 22*(9*A + 19*B)*a^2
*c*d + (209*A + 320*B)*a^2*d^2)*cos(f*x + e)^4 - (99*(7*A + 20*B)*a^2*c^2 + 22*(180*A + 289*B)*a^2*c*d + (3179
*A + 4370*B)*a^2*d^2)*cos(f*x + e)^3 + (33*(77*A + 85*B)*a^2*c^2 + 22*(255*A + 263*B)*a^2*c*d + (2893*A + 2965
*B)*a^2*d^2)*cos(f*x + e)^2 + 2*(33*(161*A + 145*B)*a^2*c^2 + 22*(435*A + 419*B)*a^2*c*d + (4609*A + 4465*B)*a
^2*d^2)*cos(f*x + e) + (315*B*a^2*d^2*cos(f*x + e)^5 - 1056*(7*A + 5*B)*a^2*c^2 - 704*(15*A + 13*B)*a^2*c*d -
32*(143*A + 125*B)*a^2*d^2 - 35*(22*B*a^2*c*d + (11*A + 23*B)*a^2*d^2)*cos(f*x + e)^4 - 5*(99*B*a^2*c^2 + 22*(
9*A + 26*B)*a^2*c*d + 13*(22*A + 37*B)*a^2*d^2)*cos(f*x + e)^3 + 3*(33*(7*A + 15*B)*a^2*c^2 + 22*(45*A + 53*B)
*a^2*c*d + (583*A + 655*B)*a^2*d^2)*cos(f*x + e)^2 + 2*(33*(49*A + 65*B)*a^2*c^2 + 22*(195*A + 211*B)*a^2*c*d
+ (2321*A + 2465*B)*a^2*d^2)*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)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))**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+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x, algorithm="giac")

[Out]

Timed out