\(\int \frac {\cos ^{\frac {3}{2}}(c+d x) (B \cos (c+d x)+C \cos ^2(c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx\) [933]

Optimal result

Integrand size = 44, antiderivative size = 674 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\frac {\left (6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{3 a (a-b) b^3 (a+b)^{3/2} d}-\frac {\left (a^2 b (6 B-5 C)-3 b^3 (4 B-C)-15 a^3 C+a b^2 (2 B+21 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{3 b^3 \sqrt {a+b} \left (a^2-b^2\right ) d}-\frac {\sqrt {a+b} (2 b B-5 a C) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{b^4 d}+\frac {2 a (b B-a C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 a \left (2 a^2 b B-6 b^3 B-5 a^3 C+9 a b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {\left (6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}} \] Output:

1/3*(6*B*a^3*b-14*B*a*b^3-15*C*a^4+26*C*a^2*b^2-3*C*b^4)*cot(d*x+c)*Ellipt 
icE((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),(-(a+b)/(a-b))^(1/ 
2))*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a/(a-b)/ 
b^3/(a+b)^(3/2)/d-1/3*(a^2*b*(6*B-5*C)-3*b^3*(4*B-C)-15*a^3*C+a*b^2*(2*B+2 
1*C))*cot(d*x+c)*EllipticF((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^( 
1/2),(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c) 
)/(a-b))^(1/2)/b^3/(a+b)^(1/2)/(a^2-b^2)/d-(a+b)^(1/2)*(2*B*b-5*C*a)*cot(d 
*x+c)*EllipticPi((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),(a+b) 
/b,(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/ 
(a-b))^(1/2)/b^4/d+2/3*a*(B*b-C*a)*cos(d*x+c)^(3/2)*sin(d*x+c)/b/(a^2-b^2) 
/d/(a+b*cos(d*x+c))^(3/2)+2/3*a*(2*B*a^2*b-6*B*b^3-5*C*a^3+9*C*a*b^2)*cos( 
d*x+c)^(1/2)*sin(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^(1/2)-1/3*(6*B* 
a^3*b-14*B*a*b^3-15*C*a^4+26*C*a^2*b^2-3*C*b^4)*(a+b*cos(d*x+c))^(1/2)*sin 
(d*x+c)/b^3/(a^2-b^2)^2/d/cos(d*x+c)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 7.68 (sec) , antiderivative size = 1396, normalized size of antiderivative = 2.07 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx =\text {Too large to display} \] Input:

Integrate[(Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b* 
Cos[c + d*x])^(5/2),x]
 

Output:

(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((-2*(-(a^2*b*B*Sin[c + d*x]) 
 + a^3*C*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) - (2*( 
-3*a^3*b*B*Sin[c + d*x] + 7*a*b^3*B*Sin[c + d*x] + 6*a^4*C*Sin[c + d*x] - 
10*a^2*b^2*C*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/ 
d + ((-4*a*(-2*a^3*b*B + 2*a*b^3*B + 5*a^4*C - 8*a^2*b^2*C + 3*b^4*C)*Sqrt 
[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[( 
c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c 
 + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a] 
/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x] 
]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(2*a^2*b^2*B + 6*b^4*B + 4*a^3*b*C - 12* 
a*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos 
[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x) 
/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c 
+ d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sq 
rt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/ 
2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[ 
((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b 
), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2* 
a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + 
d*x]])) + 2*(-6*a^3*b*B + 14*a*b^3*B + 15*a^4*C - 26*a^2*b^2*C + 3*b^4*...
 

Rubi [A] (verified)

Time = 3.53 (sec) , antiderivative size = 695, normalized size of antiderivative = 1.03, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {3042, 3508, 3042, 3468, 27, 3042, 3526, 27, 3042, 3540, 3042, 3532, 3042, 3288, 3477, 3042, 3295, 3473}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3508

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3468

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3526

\(\displaystyle \frac {\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {2 \int -\frac {-\left (\left (-15 C a^4+6 b B a^3+26 b^2 C a^2-14 b^3 B a-3 b^4 C\right ) \cos ^2(c+d x)\right )+b \left (2 C a^3+b B a^2-6 b^2 C a+3 b^3 B\right ) \cos (c+d x)+a \left (-5 C a^3+2 b B a^2+9 b^2 C a-6 b^3 B\right )}{2 \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx}{b \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {-\left (\left (-15 C a^4+6 b B a^3+26 b^2 C a^2-14 b^3 B a-3 b^4 C\right ) \cos ^2(c+d x)\right )+b \left (2 C a^3+b B a^2-6 b^2 C a+3 b^3 B\right ) \cos (c+d x)+a \left (-5 C a^3+2 b B a^2+9 b^2 C a-6 b^3 B\right )}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx}{b \left (a^2-b^2\right )}+\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {\left (15 C a^4-6 b B a^3-26 b^2 C a^2+14 b^3 B a+3 b^4 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b \left (2 C a^3+b B a^2-6 b^2 C a+3 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (-5 C a^3+2 b B a^2+9 b^2 C a-6 b^3 B\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b \left (a^2-b^2\right )}+\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3540

\(\displaystyle \frac {\frac {\frac {\int \frac {3 \left (a^2-b^2\right )^2 (2 b B-5 a C) \cos ^2(c+d x)+2 a b \left (-5 C a^3+2 b B a^2+9 b^2 C a-6 b^3 B\right ) \cos (c+d x)+a \left (-15 C a^4+6 b B a^3+26 b^2 C a^2-14 b^3 B a-3 b^4 C\right )}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{2 b}-\frac {\left (-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B-3 b^4 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \frac {3 \left (a^2-b^2\right )^2 (2 b B-5 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a b \left (-5 C a^3+2 b B a^2+9 b^2 C a-6 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (-15 C a^4+6 b B a^3+26 b^2 C a^2-14 b^3 B a-3 b^4 C\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}-\frac {\left (-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B-3 b^4 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3532

\(\displaystyle \frac {\frac {\frac {3 \left (a^2-b^2\right )^2 (2 b B-5 a C) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}}dx+\int \frac {a \left (-15 C a^4+6 b B a^3+26 b^2 C a^2-14 b^3 B a-3 b^4 C\right )+2 a b \left (-5 C a^3+2 b B a^2+9 b^2 C a-6 b^3 B\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{2 b}-\frac {\left (-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B-3 b^4 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {3 \left (a^2-b^2\right )^2 (2 b B-5 a C) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\int \frac {a \left (-15 C a^4+6 b B a^3+26 b^2 C a^2-14 b^3 B a-3 b^4 C\right )+2 a b \left (-5 C a^3+2 b B a^2+9 b^2 C a-6 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}-\frac {\left (-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B-3 b^4 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3288

\(\displaystyle \frac {\frac {\frac {\int \frac {a \left (-15 C a^4+6 b B a^3+26 b^2 C a^2-14 b^3 B a-3 b^4 C\right )+2 a b \left (-5 C a^3+2 b B a^2+9 b^2 C a-6 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \left (a^2-b^2\right )^2 (2 b B-5 a C) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}-\frac {\left (-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B-3 b^4 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3477

\(\displaystyle \frac {\frac {\frac {-a (a-b) \left (-15 a^3 C+a^2 b (6 B-5 C)+a b^2 (2 B+21 C)-3 b^3 (4 B-C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+a \left (-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B-3 b^4 C\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx-\frac {6 \sqrt {a+b} \left (a^2-b^2\right )^2 (2 b B-5 a C) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}-\frac {\left (-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B-3 b^4 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {-a (a-b) \left (-15 a^3 C+a^2 b (6 B-5 C)+a b^2 (2 B+21 C)-3 b^3 (4 B-C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+a \left (-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B-3 b^4 C\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \left (a^2-b^2\right )^2 (2 b B-5 a C) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}-\frac {\left (-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B-3 b^4 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3295

\(\displaystyle \frac {\frac {\frac {a \left (-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B-3 b^4 C\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \left (a^2-b^2\right )^2 (2 b B-5 a C) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}-\frac {2 (a-b) \sqrt {a+b} \left (-15 a^3 C+a^2 b (6 B-5 C)+a b^2 (2 B+21 C)-3 b^3 (4 B-C)\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{2 b}-\frac {\left (-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B-3 b^4 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3473

\(\displaystyle \frac {2 a (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac {\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {\frac {-\frac {6 \sqrt {a+b} \left (a^2-b^2\right )^2 (2 b B-5 a C) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}-\frac {2 (a-b) \sqrt {a+b} \left (-15 a^3 C+a^2 b (6 B-5 C)+a b^2 (2 B+21 C)-3 b^3 (4 B-C)\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}+\frac {2 (a-b) \sqrt {a+b} \left (-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B-3 b^4 C\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a d}}{2 b}-\frac {\left (-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B-3 b^4 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}}{b \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}\)

Input:

Int[(Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c 
+ d*x])^(5/2),x]
 

Output:

(2*a*(b*B - a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + 
b*Cos[c + d*x])^(3/2)) + ((2*a*(2*a^2*b*B - 6*b^3*B - 5*a^3*C + 9*a*b^2*C) 
*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x] 
]) + (((2*(a - b)*Sqrt[a + b]*(6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2* 
b^2*C - 3*b^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(S 
qrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + 
d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (2*(a - b)*S 
qrt[a + b]*(a^2*b*(6*B - 5*C) - 3*b^3*(4*B - C) - 15*a^3*C + a*b^2*(2*B + 
21*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b] 
*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a 
+ b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d - (6*Sqrt[a + b]*(a^2 - b^2) 
^2*(2*b*B - 5*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Co 
s[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a 
*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d)) 
/(2*b) - ((6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Sqr 
t[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]))/(b*(a^2 - b^ 
2)))/(3*b*(a^2 - b^2))
 

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 3288

Int[Sqrt[(b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.) 
*(x_)]], x_Symbol] :> Simp[2*b*(Tan[e + f*x]/(d*f))*Rt[(c + d)/b, 2]*Sqrt[c 
*((1 + Csc[e + f*x])/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*Ellipti 
cPi[(c + d)/d, ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + 
 d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - 
 d^2, 0] && PosQ[(c + d)/b]
 

rule 3295

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f 
_.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr 
t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli 
pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] 
], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] 
&& PosQ[(a + b)/d]
 

rule 3468

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

rule 3473

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) 
^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A* 
(c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x] 
)/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + 
d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], 
x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && 
PosQ[(c + d)/b]
 

rule 3477

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

rule 3508

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

rule 3526

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^2*C - B*c*d + A*d^2))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - 
d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2))   Int[(a + b*Sin[e + f*x])^(m - 
 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* 
d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 
1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x 
] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f 
*x]^2, x], x], x] /; FreeQ[{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] && GtQ[m, 0] && LtQ[n, -1]
 

rule 3532

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 
2)/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e 
_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[C/b^2   Int[Sqrt[a + b*Sin[e + f*x]] 
/Sqrt[c + d*Sin[e + f*x]], x], x] + Simp[1/b^2   Int[(A*b^2 - a^2*C + b*(b* 
B - 2*a*C)*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x 
]]), x], x] /; FreeQ[{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]
 

rule 3540

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 
2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) 
 + (f_.)*(x_)]]), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(Sqrt[c + d*Sin[e + f 
*x]]/(d*f*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[1/(2*d)   Int[(1/((a + b*Si 
n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]))*Simp[2*a*A*d - C*(b*c - a*d) - 
 2*(a*c*C - d*(A*b + a*B))*Sin[e + f*x] + (2*b*B*d - C*(b*c + a*d))*Sin[e + 
 f*x]^2, x], x], x] /; FreeQ[{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] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(4189\) vs. \(2(619)=1238\).

Time = 27.02 (sec) , antiderivative size = 4190, normalized size of antiderivative = 6.22

method	result	size
default	\(\text {Expression too large to display}\)	\(4190\)
parts	\(\text {Expression too large to display}\)	\(4229\)

Input:

int(cos(d*x+c)^(3/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2), 
x,method=_RETURNVERBOSE)
 

Output:

-1/3/d*(-3*C*b^6*cos(d*x+c)^3*sin(d*x+c)+B*(cos(d*x+c)/(1+cos(d*x+c)))^(1/ 
2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*b^6*EllipticPi(-csc(d*x 
+c)+cot(d*x+c),-1,(-(a-b)/(a+b))^(1/2))*(12*cos(d*x+c)^3+24*cos(d*x+c)^2+1 
2*cos(d*x+c))+6*sin(d*x+c)*cos(d*x+c)*B*a^5*b+14*B*a*b^5*cos(d*x+c)^2*sin( 
d*x+c)-6*C*a*b^5*cos(d*x+c)^2*sin(d*x+c)+(-2*cos(d*x+c)^3-16*cos(d*x+c)^2- 
26*cos(d*x+c)-12)*B*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d* 
x+c))/(1+cos(d*x+c)))^(1/2)*a^2*b^4*EllipticF(-csc(d*x+c)+cot(d*x+c),(-(a- 
b)/(a+b))^(1/2))+(-12*cos(d*x+c)^3-30*cos(d*x+c)^2-24*cos(d*x+c)-6)*B*(cos 
(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1 
/2)*a*b^5*EllipticF(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))+(-10*cos( 
d*x+c)^2-20*cos(d*x+c)-10)*C*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a 
+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^5*b*EllipticF(-csc(d*x+c)+cot(d*x+c 
),(-(a-b)/(a+b))^(1/2))+(-10*cos(d*x+c)^3-24*cos(d*x+c)^2-18*cos(d*x+c)-4) 
*C*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+ 
c)))^(1/2)*a^4*b^2*EllipticF(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))+ 
(-4*cos(d*x+c)^3+10*cos(d*x+c)^2+32*cos(d*x+c)+18)*C*(cos(d*x+c)/(1+cos(d* 
x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^3*b^3*Ellip 
ticF(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))+(18*cos(d*x+c)^3+48*cos( 
d*x+c)^2+42*cos(d*x+c)+12)*C*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a 
+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^2*b^4*EllipticF(-csc(d*x+c)+cot(...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)^(3/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^ 
(5/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**(3/2)*(B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+b*cos(d*x+c) 
)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(cos(d*x+c)^(3/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^ 
(5/2),x, algorithm="maxima")
 

Output:

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c))*cos(d*x + c)^(3/2)/(b*cos(d* 
x + c) + a)^(5/2), x)
 

Giac [F]

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

integrate(cos(d*x+c)^(3/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^ 
(5/2),x, algorithm="giac")
 

Output:

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c))*cos(d*x + c)^(3/2)/(b*cos(d* 
x + c) + a)^(5/2), x)
 

Mupad [F(-1)]

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

int((cos(c + d*x)^(3/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c 
+ d*x))^(5/2),x)
 

Output:

int((cos(c + d*x)^(3/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c 
+ d*x))^(5/2), x)
 

Reduce [F]

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

int(cos(d*x+c)^(3/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2), 
x)
 

Output:

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