\(\int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx\) [352]

Optimal result

Integrand size = 33, antiderivative size = 264 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {2 \left (9 a^2 A+7 A b^2+14 a b B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {10 \left (9 b^2 B+11 a (2 A b+a B)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {10 \left (9 b^2 B+11 a (2 A b+a B)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (9 a^2 A+7 A b^2+14 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 \left (9 b^2 B+11 a (2 A b+a B)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 b (11 A b+13 a B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 b B \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d} \]

[Out]

2/15*(9*A*a^2+7*A*b^2+14*B*a*b)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2
^(1/2))/d+10/231*(9*b^2*B+11*a*(2*A*b+B*a))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*
d*x+1/2*c),2^(1/2))/d+2/45*(9*A*a^2+7*A*b^2+14*B*a*b)*cos(d*x+c)^(3/2)*sin(d*x+c)/d+2/77*(9*b^2*B+11*a*(2*A*b+
B*a))*cos(d*x+c)^(5/2)*sin(d*x+c)/d+2/99*b*(11*A*b+13*B*a)*cos(d*x+c)^(7/2)*sin(d*x+c)/d+2/11*b*B*cos(d*x+c)^(
7/2)*(a+b*cos(d*x+c))*sin(d*x+c)/d+10/231*(9*b^2*B+11*a*(2*A*b+B*a))*sin(d*x+c)*cos(d*x+c)^(1/2)/d

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3069, 3102, 2827, 2715, 2719, 2720} \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {2 \left (9 a^2 A+14 a b B+7 A b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (9 a^2 A+14 a b B+7 A b^2\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 d}+\frac {10 \left (11 a (a B+2 A b)+9 b^2 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {2 \left (11 a (a B+2 A b)+9 b^2 B\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{77 d}+\frac {10 \left (11 a (a B+2 A b)+9 b^2 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{231 d}+\frac {2 b (13 a B+11 A b) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{99 d}+\frac {2 b B \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d} \]

[In]

Int[Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]

[Out]

(2*(9*a^2*A + 7*A*b^2 + 14*a*b*B)*EllipticE[(c + d*x)/2, 2])/(15*d) + (10*(9*b^2*B + 11*a*(2*A*b + a*B))*Ellip
ticF[(c + d*x)/2, 2])/(231*d) + (10*(9*b^2*B + 11*a*(2*A*b + a*B))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) +
(2*(9*a^2*A + 7*A*b^2 + 14*a*b*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(9*b^2*B + 11*a*(2*A*b + a*B))*
Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*b*(11*A*b + 13*a*B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*
b*B*Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(11*d)

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 2827

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 3069

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

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*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[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {2 b B \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}+\frac {2}{11} \int \cos ^{\frac {5}{2}}(c+d x) \left (\frac {1}{2} a (11 a A+7 b B)+\frac {1}{2} \left (9 b^2 B+11 a (2 A b+a B)\right ) \cos (c+d x)+\frac {1}{2} b (11 A b+13 a B) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 b (11 A b+13 a B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 b B \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}+\frac {4}{99} \int \cos ^{\frac {5}{2}}(c+d x) \left (\frac {11}{4} \left (9 a^2 A+7 A b^2+14 a b B\right )+\frac {9}{4} \left (9 b^2 B+11 a (2 A b+a B)\right ) \cos (c+d x)\right ) \, dx \\ & = \frac {2 b (11 A b+13 a B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 b B \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}+\frac {1}{9} \left (9 a^2 A+7 A b^2+14 a b B\right ) \int \cos ^{\frac {5}{2}}(c+d x) \, dx+\frac {1}{11} \left (9 b^2 B+11 a (2 A b+a B)\right ) \int \cos ^{\frac {7}{2}}(c+d x) \, dx \\ & = \frac {2 \left (9 a^2 A+7 A b^2+14 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 \left (9 b^2 B+11 a (2 A b+a B)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 b (11 A b+13 a B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 b B \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}+\frac {1}{15} \left (9 a^2 A+7 A b^2+14 a b B\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{77} \left (5 \left (9 b^2 B+11 a (2 A b+a B)\right )\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx \\ & = \frac {2 \left (9 a^2 A+7 A b^2+14 a b B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {10 \left (9 b^2 B+11 a (2 A b+a B)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (9 a^2 A+7 A b^2+14 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 \left (9 b^2 B+11 a (2 A b+a B)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 b (11 A b+13 a B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 b B \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}+\frac {1}{231} \left (5 \left (9 b^2 B+11 a (2 A b+a B)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 \left (9 a^2 A+7 A b^2+14 a b B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {10 \left (9 b^2 B+11 a (2 A b+a B)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {10 \left (9 b^2 B+11 a (2 A b+a B)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (9 a^2 A+7 A b^2+14 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 \left (9 b^2 B+11 a (2 A b+a B)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 b (11 A b+13 a B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 b B \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.33 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.74 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {3696 \left (9 a^2 A+7 A b^2+14 a b B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+1200 \left (22 a A b+11 a^2 B+9 b^2 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 \sqrt {\cos (c+d x)} \left (154 \left (36 a^2 A+43 A b^2+86 a b B\right ) \cos (c+d x)+180 \left (22 a A b+11 a^2 B+16 b^2 B\right ) \cos (2 (c+d x))+770 b (A b+2 a B) \cos (3 (c+d x))+15 \left (1144 a A b+572 a^2 B+531 b^2 B+21 b^2 B \cos (4 (c+d x))\right )\right ) \sin (c+d x)}{27720 d} \]

[In]

Integrate[Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]

[Out]

(3696*(9*a^2*A + 7*A*b^2 + 14*a*b*B)*EllipticE[(c + d*x)/2, 2] + 1200*(22*a*A*b + 11*a^2*B + 9*b^2*B)*Elliptic
F[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(154*(36*a^2*A + 43*A*b^2 + 86*a*b*B)*Cos[c + d*x] + 180*(22*a*A*b +
11*a^2*B + 16*b^2*B)*Cos[2*(c + d*x)] + 770*b*(A*b + 2*a*B)*Cos[3*(c + d*x)] + 15*(1144*a*A*b + 572*a^2*B + 53
1*b^2*B + 21*b^2*B*Cos[4*(c + d*x)]))*Sin[c + d*x])/(27720*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(665\) vs. \(2(292)=584\).

Time = 42.91 (sec) , antiderivative size = 666, normalized size of antiderivative = 2.52

method	result	size
default	\(-\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (20160 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}+\left (-12320 A \,b^{2}-24640 B a b -50400 B \,b^{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (15840 A a b +24640 A \,b^{2}+7920 B \,a^{2}+49280 B a b +56880 B \,b^{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-5544 A \,a^{2}-23760 A a b -22792 A \,b^{2}-11880 B \,a^{2}-45584 B a b -34920 B \,b^{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (5544 A \,a^{2}+18480 A a b +10472 A \,b^{2}+9240 B \,a^{2}+20944 B a b +13860 B \,b^{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-1386 A \,a^{2}-5280 A a b -1848 A \,b^{2}-2640 B \,a^{2}-3696 B a b -2790 B \,b^{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+1650 A a b \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-2079 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2}-1617 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{2}+825 B \,a^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+675 B \,b^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3234 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a b \right )}{3465 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\)	\(666\)
parts	\(\text {Expression too large to display}\)	\(866\)

[In]

int(cos(d*x+c)^(5/2)*(a+cos(d*x+c)*b)^2*(A+B*cos(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-2/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(20160*B*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)
^12*b^2+(-12320*A*b^2-24640*B*a*b-50400*B*b^2)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(15840*A*a*b+24640*A*b
^2+7920*B*a^2+49280*B*a*b+56880*B*b^2)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-5544*A*a^2-23760*A*a*b-22792*
A*b^2-11880*B*a^2-45584*B*a*b-34920*B*b^2)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(5544*A*a^2+18480*A*a*b+104
72*A*b^2+9240*B*a^2+20944*B*a*b+13860*B*b^2)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-1386*A*a^2-5280*A*a*b-1
848*A*b^2-2640*B*a^2-3696*B*a*b-2790*B*b^2)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+1650*A*a*b*(sin(1/2*d*x+1/
2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-2079*A*(sin(1/2*d*x+1/2*c
)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^2-1617*A*(sin(1/2*d*x+1/2*
c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*b^2+825*B*a^2*(sin(1/2*d*x+
1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+675*B*b^2*(sin(1/2*d*x+
1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-3234*B*(sin(1/2*d*x+1/2
*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b)/(-2*sin(1/2*d*x+1/2*c
)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.13 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.13 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {2 \, {\left (315 \, B b^{2} \cos \left (d x + c\right )^{4} + 385 \, {\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{3} + 825 \, B a^{2} + 1650 \, A a b + 675 \, B b^{2} + 45 \, {\left (11 \, B a^{2} + 22 \, A a b + 9 \, B b^{2}\right )} \cos \left (d x + c\right )^{2} + 77 \, {\left (9 \, A a^{2} + 14 \, B a b + 7 \, A b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 75 \, \sqrt {2} {\left (11 i \, B a^{2} + 22 i \, A a b + 9 i \, B b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 75 \, \sqrt {2} {\left (-11 i \, B a^{2} - 22 i \, A a b - 9 i \, B b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 \, \sqrt {2} {\left (-9 i \, A a^{2} - 14 i \, B a b - 7 i \, A b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 \, \sqrt {2} {\left (9 i \, A a^{2} + 14 i \, B a b + 7 i \, A b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{3465 \, d} \]

[In]

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

[Out]

1/3465*(2*(315*B*b^2*cos(d*x + c)^4 + 385*(2*B*a*b + A*b^2)*cos(d*x + c)^3 + 825*B*a^2 + 1650*A*a*b + 675*B*b^
2 + 45*(11*B*a^2 + 22*A*a*b + 9*B*b^2)*cos(d*x + c)^2 + 77*(9*A*a^2 + 14*B*a*b + 7*A*b^2)*cos(d*x + c))*sqrt(c
os(d*x + c))*sin(d*x + c) - 75*sqrt(2)*(11*I*B*a^2 + 22*I*A*a*b + 9*I*B*b^2)*weierstrassPInverse(-4, 0, cos(d*
x + c) + I*sin(d*x + c)) - 75*sqrt(2)*(-11*I*B*a^2 - 22*I*A*a*b - 9*I*B*b^2)*weierstrassPInverse(-4, 0, cos(d*
x + c) - I*sin(d*x + c)) - 231*sqrt(2)*(-9*I*A*a^2 - 14*I*B*a*b - 7*I*A*b^2)*weierstrassZeta(-4, 0, weierstras
sPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 231*sqrt(2)*(9*I*A*a^2 + 14*I*B*a*b + 7*I*A*b^2)*weierstras
sZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))))/d

Sympy [F(-1)]

Timed out. \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**(5/2)*(a+b*cos(d*x+c))**2*(A+B*cos(d*x+c)),x)

[Out]

Timed out

Maxima [F]

\[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {5}{2}} \,d x } \]

[In]

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

[Out]

integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^2*cos(d*x + c)^(5/2), x)

Giac [F]

\[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {5}{2}} \,d x } \]

[In]

integrate(cos(d*x+c)^(5/2)*(a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)),x, algorithm="giac")

[Out]

integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^2*cos(d*x + c)^(5/2), x)

Mupad [B] (verification not implemented)

Time = 1.70 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.04 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=-\frac {2\,A\,a^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,b^2\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,b^2\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{13\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,A\,a\,b\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,B\,a\,b\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]

[In]

int(cos(c + d*x)^(5/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^2,x)

[Out]

- (2*A*a^2*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^
(1/2)) - (2*B*a^2*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d
*x)^2)^(1/2)) - (2*A*b^2*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 15/4, cos(c + d*x)^2))/(11*d*
(sin(c + d*x)^2)^(1/2)) - (2*B*b^2*cos(c + d*x)^(13/2)*sin(c + d*x)*hypergeom([1/2, 13/4], 17/4, cos(c + d*x)^
2))/(13*d*(sin(c + d*x)^2)^(1/2)) - (4*A*a*b*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c
+ d*x)^2))/(9*d*(sin(c + d*x)^2)^(1/2)) - (4*B*a*b*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 15
/4, cos(c + d*x)^2))/(11*d*(sin(c + d*x)^2)^(1/2))