\(\int \frac {\cos ^{\frac {5}{2}}(c+d x) (A+B \cos (c+d x)+C \cos ^2(c+d x))}{a+b \cos (c+d x)} \, dx\) [1095]

Optimal result

Integrand size = 43, antiderivative size = 285 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\frac {2 \left (5 a^2 b B+3 b^3 B-5 a^3 C-a b^2 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^4 d}-\frac {2 \left (21 a^3 b B+7 a b^3 B-21 a^4 C-7 a^2 b^2 (3 A+C)-b^4 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 b^5 d}-\frac {2 a^3 \left (A b^2-a (b B-a C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b^5 (a+b) d}+\frac {2 \left (7 A b^2-7 a b B+7 a^2 C+5 b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 b^3 d}+\frac {2 (b B-a C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 b^2 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 b d} \] Output:

2/5*(5*B*a^2*b+3*B*b^3-5*a^3*C-a*b^2*(5*A+3*C))*EllipticE(sin(1/2*d*x+1/2* 
c),2^(1/2))/b^4/d-2/21*(21*B*a^3*b+7*B*a*b^3-21*a^4*C-7*a^2*b^2*(3*A+C)-b^ 
4*(7*A+5*C))*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/b^5/d-2*a^3*(A*b^2-a*( 
B*b-C*a))*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(a+b),2^(1/2))/b^5/(a+b)/d+2/2 
1*(7*A*b^2-7*B*a*b+7*C*a^2+5*C*b^2)*cos(d*x+c)^(1/2)*sin(d*x+c)/b^3/d+2/5* 
(B*b-C*a)*cos(d*x+c)^(3/2)*sin(d*x+c)/b^2/d+2/7*C*cos(d*x+c)^(5/2)*sin(d*x 
+c)/b/d
 

Mathematica [A] (warning: unable to verify)

Time = 4.86 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.18 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\frac {-\frac {2 \left (-35 a^2 b B-63 b^3 B+35 a^3 C+a b^2 (35 A+13 C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {4 \left (35 A b^2+28 a b B-28 a^2 C+25 b^2 C\right ) \left ((a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-a \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )\right )}{a+b}+2 \sqrt {\cos (c+d x)} \left (70 A b^2-70 a b B+70 a^2 C+65 b^2 C+42 b (b B-a C) \cos (c+d x)+15 b^2 C \cos (2 (c+d x))\right ) \sin (c+d x)-\frac {42 \left (-5 a^2 b B-3 b^3 B+5 a^3 C+a b^2 (5 A+3 C)\right ) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (-2 a^2+b^2\right ) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a b^2 \sqrt {\sin ^2(c+d x)}}}{210 b^3 d} \] Input:

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

Output:

((-2*(-35*a^2*b*B - 63*b^3*B + 35*a^3*C + a*b^2*(35*A + 13*C))*EllipticPi[ 
(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (4*(35*A*b^2 + 28*a*b*B - 28*a^2 
*C + 25*b^2*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a 
+ b), (c + d*x)/2, 2]))/(a + b) + 2*Sqrt[Cos[c + d*x]]*(70*A*b^2 - 70*a*b* 
B + 70*a^2*C + 65*b^2*C + 42*b*(b*B - a*C)*Cos[c + d*x] + 15*b^2*C*Cos[2*( 
c + d*x)])*Sin[c + d*x] - (42*(-5*a^2*b*B - 3*b^3*B + 5*a^3*C + a*b^2*(5*A 
 + 3*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*E 
llipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a 
), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x] 
^2]))/(210*b^3*d)
 

Rubi [A] (verified)

Time = 2.35 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.09, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {3042, 3528, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3538, 27, 3042, 3119, 3481, 3042, 3120, 3284}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {2 \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (7 (b B-a C) \cos ^2(c+d x)+b (7 A+5 C) \cos (c+d x)+5 a C\right )}{2 (a+b \cos (c+d x))}dx}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (7 (b B-a C) \cos ^2(c+d x)+b (7 A+5 C) \cos (c+d x)+5 a C\right )}{a+b \cos (c+d x)}dx}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (7 (b B-a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (7 A+5 C) \sin \left (c+d x+\frac {\pi }{2}\right )+5 a C\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {\frac {2 \int \frac {\sqrt {\cos (c+d x)} \left (5 \left (7 C a^2-7 b B a+7 A b^2+5 b^2 C\right ) \cos ^2(c+d x)+b (21 b B+4 a C) \cos (c+d x)+21 a (b B-a C)\right )}{2 (a+b \cos (c+d x))}dx}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\sqrt {\cos (c+d x)} \left (5 \left (7 C a^2-7 b B a+7 A b^2+5 b^2 C\right ) \cos ^2(c+d x)+b (21 b B+4 a C) \cos (c+d x)+21 a (b B-a C)\right )}{a+b \cos (c+d x)}dx}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (5 \left (7 C a^2-7 b B a+7 A b^2+5 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (21 b B+4 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+21 a (b B-a C)\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {\frac {\frac {2 \int \frac {21 \left (-5 C a^3+5 b B a^2-b^2 (5 A+3 C) a+3 b^3 B\right ) \cos ^2(c+d x)+b \left (-28 C a^2+28 b B a+35 A b^2+25 b^2 C\right ) \cos (c+d x)+5 a \left (7 C a^2-7 b B a+7 A b^2+5 b^2 C\right )}{2 \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{3 b}+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {21 \left (-5 C a^3+5 b B a^2-b^2 (5 A+3 C) a+3 b^3 B\right ) \cos ^2(c+d x)+b \left (-28 C a^2+28 b B a+35 A b^2+25 b^2 C\right ) \cos (c+d x)+5 a \left (7 C a^2-7 b B a+7 A b^2+5 b^2 C\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{3 b}+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \frac {21 \left (-5 C a^3+5 b B a^2-b^2 (5 A+3 C) a+3 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b \left (-28 C a^2+28 b B a+35 A b^2+25 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+5 a \left (7 C a^2-7 b B a+7 A b^2+5 b^2 C\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 b}+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3538

\(\displaystyle \frac {\frac {\frac {\frac {21 \left (-5 a^3 C+5 a^2 b B-a b^2 (5 A+3 C)+3 b^3 B\right ) \int \sqrt {\cos (c+d x)}dx}{b}-\frac {\int -\frac {5 \left (a b \left (7 C a^2-7 b B a+7 A b^2+5 b^2 C\right )-\left (-21 C a^4+21 b B a^3-7 b^2 (3 A+C) a^2+7 b^3 B a-b^4 (7 A+5 C)\right ) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{3 b}+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {21 \left (-5 a^3 C+5 a^2 b B-a b^2 (5 A+3 C)+3 b^3 B\right ) \int \sqrt {\cos (c+d x)}dx}{b}+\frac {5 \int \frac {a b \left (7 C a^2-7 b B a+7 A b^2+5 b^2 C\right )-\left (-21 C a^4+21 b B a^3-7 b^2 (3 A+C) a^2+7 b^3 B a-b^4 (7 A+5 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{3 b}+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {21 \left (-5 a^3 C+5 a^2 b B-a b^2 (5 A+3 C)+3 b^3 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}+\frac {5 \int \frac {a b \left (7 C a^2-7 b B a+7 A b^2+5 b^2 C\right )+\left (21 C a^4-21 b B a^3+7 b^2 (3 A+C) a^2-7 b^3 B a+b^4 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}}{3 b}+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\frac {\frac {\frac {5 \int \frac {a b \left (7 C a^2-7 b B a+7 A b^2+5 b^2 C\right )+\left (21 C a^4-21 b B a^3+7 b^2 (3 A+C) a^2-7 b^3 B a+b^4 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-5 a^3 C+5 a^2 b B-a b^2 (5 A+3 C)+3 b^3 B\right )}{b d}}{3 b}+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3481

\(\displaystyle \frac {\frac {\frac {\frac {5 \left (-\frac {21 a^3 \left (A b^2-a (b B-a C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}-\frac {\left (-21 a^4 C+21 a^3 b B-7 a^2 b^2 (3 A+C)+7 a b^3 B-b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{b}\right )}{b}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-5 a^3 C+5 a^2 b B-a b^2 (5 A+3 C)+3 b^3 B\right )}{b d}}{3 b}+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {5 \left (-\frac {21 a^3 \left (A b^2-a (b B-a C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {\left (-21 a^4 C+21 a^3 b B-7 a^2 b^2 (3 A+C)+7 a b^3 B-b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )}{b}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-5 a^3 C+5 a^2 b B-a b^2 (5 A+3 C)+3 b^3 B\right )}{b d}}{3 b}+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {\frac {\frac {\frac {5 \left (-\frac {21 a^3 \left (A b^2-a (b B-a C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (-21 a^4 C+21 a^3 b B-7 a^2 b^2 (3 A+C)+7 a b^3 B-b^4 (7 A+5 C)\right )}{b d}\right )}{b}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-5 a^3 C+5 a^2 b B-a b^2 (5 A+3 C)+3 b^3 B\right )}{b d}}{3 b}+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {\frac {\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}+\frac {\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-5 a^3 C+5 a^2 b B-a b^2 (5 A+3 C)+3 b^3 B\right )}{b d}+\frac {5 \left (-\frac {42 a^3 \left (A b^2-a (b B-a C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b d (a+b)}-\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (-21 a^4 C+21 a^3 b B-7 a^2 b^2 (3 A+C)+7 a b^3 B-b^4 (7 A+5 C)\right )}{b d}\right )}{b}}{3 b}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

Input:

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

Output:

(2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*b*d) + ((14*(b*B - a*C)*Cos[c + d 
*x]^(3/2)*Sin[c + d*x])/(5*b*d) + (((42*(5*a^2*b*B + 3*b^3*B - 5*a^3*C - a 
*b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(b*d) + (5*((-2*(21*a^3*b*B + 
 7*a*b^3*B - 21*a^4*C - 7*a^2*b^2*(3*A + C) - b^4*(7*A + 5*C))*EllipticF[( 
c + d*x)/2, 2])/(b*d) - (42*a^3*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*b)/( 
a + b), (c + d*x)/2, 2])/(b*(a + b)*d)))/b)/(3*b) + (10*(7*A*b^2 - 7*a*b*B 
 + 7*a^2*C + 5*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*d))/(5*b))/(7* 
b)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3119

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 3120

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 3284

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

rule 3481

Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ 
B/d   Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d   Int[(a + b* 
Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, 
 B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
 

rule 3528

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) 
+ (f_.)*(x_)])^(n_.)*((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*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + 
n + 2))   Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* 
d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a 
*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + 
 n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} 
, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ 
m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
 

rule 3538

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 
2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d)   Int[Sqrt[a + b*Sin[e + f*x]], x] 
, x] - Simp[1/(b*d)   Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ 
e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre 
eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 
] && NeQ[c^2 - d^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1096\) vs. \(2(274)=548\).

Time = 17.56 (sec) , antiderivative size = 1097, normalized size of antiderivative = 3.85

method	result	size
default	\(\text {Expression too large to display}\)	\(1097\)

Input:

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

Output:

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*(A*a^2*b^2+A 
*a*b^3+A*b^4-B*a^3*b-B*a^2*b^2-B*a*b^3-B*b^4+C*a^4+C*a^3*b+C*a^2*b^2+C*a*b 
^3+C*b^4)/b^5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/ 
2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2* 
d*x+1/2*c),2^(1/2))+8/105*C/b*(60*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)- 
258*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6+448*cos(1/2*d*x+1/2*c)*sin(1/2 
*d*x+1/2*c)^4-167*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+85*(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))-168*(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)))/(-2*sin(1/2*d*x+1/2*c)^4+sin(1 
/2*d*x+1/2*c)^2)^(1/2)-2/b^4*(A*a*b^2+2*A*b^3-B*a^2*b-2*B*a*b^2-3*B*b^3+C* 
a^3+2*C*a^2*b+3*C*a*b^2+4*C*b^3)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2* 
d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2) 
*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/ 
2)))+4/3/b^3*(A*b^2-B*a*b-3*B*b^2+C*a^2+3*C*a*b+6*C*b^2)*(2*cos(1/2*d*x+1/ 
2*c)*sin(1/2*d*x+1/2*c)^4-cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+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))-3*(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)))/(-2*sin(1/2*d*x+1/2*c)^4+ 
sin(1/2*d*x+1/2*c)^2)^(1/2)-4/5/b^2*(B*b-C*a-4*C*b)/(-2*sin(1/2*d*x+1/2...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{a+b\,\cos \left (c+d\,x\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}}{\cos \left (d x +c \right ) b +a}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\cos \left (d x +c \right ) b +a}d x \right ) b +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{\cos \left (d x +c \right ) b +a}d x \right ) a \] Input:

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

Output:

int((sqrt(cos(c + d*x))*cos(c + d*x)**4)/(cos(c + d*x)*b + a),x)*c + int(( 
sqrt(cos(c + d*x))*cos(c + d*x)**3)/(cos(c + d*x)*b + a),x)*b + int((sqrt( 
cos(c + d*x))*cos(c + d*x)**2)/(cos(c + d*x)*b + a),x)*a