Optimal. Leaf size=629 \[ \frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{1287 b^3 d}+\frac {2 \sin (c+d x) \left (-48 a^3 C+104 a^2 b B-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{9009 b^3 d}+\frac {2 \sin (c+d x) \left (-240 a^4 C+520 a^3 b B-10 a^2 b^2 (143 A+124 C)+4355 a b^3 B+539 b^4 (13 A+11 C)\right ) (a+b \cos (c+d x))^{3/2}}{45045 b^3 d}+\frac {2 \sin (c+d x) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{45045 b^3 d}-\frac {2 \left (a^2-b^2\right ) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (-240 a^6 C+520 a^5 b B-10 a^4 b^2 (143 A+76 C)+3315 a^3 b^3 B+3 a^2 b^4 (13299 A+10223 C)+48165 a b^5 B+1617 b^6 (13 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{143 b^2 d}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d} \]
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Rubi [A] time = 1.51, antiderivative size = 629, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{1287 b^3 d}+\frac {2 \sin (c+d x) \left (104 a^2 b B-48 a^3 C-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{9009 b^3 d}+\frac {2 \sin (c+d x) \left (-10 a^2 b^2 (143 A+124 C)+520 a^3 b B-240 a^4 C+4355 a b^3 B+539 b^4 (13 A+11 C)\right ) (a+b \cos (c+d x))^{3/2}}{45045 b^3 d}+\frac {2 \sin (c+d x) \left (-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+520 a^4 b B-240 a^5 C+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{45045 b^3 d}-\frac {2 \left (a^2-b^2\right ) \left (-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+520 a^4 b B-240 a^5 C+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (-10 a^4 b^2 (143 A+76 C)+3 a^2 b^4 (13299 A+10223 C)+3315 a^3 b^3 B+520 a^5 b B-240 a^6 C+48165 a b^5 B+1617 b^6 (13 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{143 b^2 d}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rule 3023
Rule 3049
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {2 \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (2 a C+\frac {1}{2} b (13 A+11 C) \cos (c+d x)+\frac {1}{2} (13 b B-6 a C) \cos ^2(c+d x)\right ) \, dx}{13 b}\\ &=\frac {2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {4 \int (a+b \cos (c+d x))^{5/2} \left (\frac {1}{2} a (13 b B-6 a C)+\frac {1}{4} b (117 b B-10 a C) \cos (c+d x)+\frac {1}{4} \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx}{143 b^2}\\ &=\frac {2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac {2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {8 \int (a+b \cos (c+d x))^{5/2} \left (\frac {1}{8} b \left (1001 A b^2-130 a b B+60 a^2 C+847 b^2 C\right )+\frac {1}{8} \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) \cos (c+d x)\right ) \, dx}{1287 b^3}\\ &=\frac {2 \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac {2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {16 \int (a+b \cos (c+d x))^{3/2} \left (-\frac {3}{16} b \left (130 a^2 b B-1755 b^3 B-60 a^3 C-a b^2 (1859 A+1423 C)\right )+\frac {1}{16} \left (520 a^3 b B+4355 a b^3 B-240 a^4 C+539 b^4 (13 A+11 C)-10 a^2 b^2 (143 A+124 C)\right ) \cos (c+d x)\right ) \, dx}{9009 b^3}\\ &=\frac {2 \left (520 a^3 b B+4355 a b^3 B-240 a^4 C+539 b^4 (13 A+11 C)-10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac {2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {32 \int \sqrt {a+b \cos (c+d x)} \left (-\frac {3}{32} b \left (130 a^3 b B-13130 a b^3 B-60 a^4 C-539 b^4 (13 A+11 C)-5 a^2 b^2 (1573 A+1175 C)\right )+\frac {3}{32} \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \cos (c+d x)\right ) \, dx}{45045 b^3}\\ &=\frac {2 \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (520 a^3 b B+4355 a b^3 B-240 a^4 C+539 b^4 (13 A+11 C)-10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac {2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {64 \int \frac {\frac {3}{64} b \left (130 a^4 b B+43095 a^2 b^3 B+8775 b^5 B-60 a^5 C+5 a^3 b^2 (4433 A+3337 C)+3 a b^4 (12441 A+10277 C)\right )+\frac {3}{64} \left (520 a^5 b B+3315 a^3 b^3 B+48165 a b^5 B-240 a^6 C+1617 b^6 (13 A+11 C)-10 a^4 b^2 (143 A+76 C)+3 a^2 b^4 (13299 A+10223 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{135135 b^3}\\ &=\frac {2 \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (520 a^3 b B+4355 a b^3 B-240 a^4 C+539 b^4 (13 A+11 C)-10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac {2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}-\frac {\left (\left (a^2-b^2\right ) \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right )\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{45045 b^4}+\frac {\left (520 a^5 b B+3315 a^3 b^3 B+48165 a b^5 B-240 a^6 C+1617 b^6 (13 A+11 C)-10 a^4 b^2 (143 A+76 C)+3 a^2 b^4 (13299 A+10223 C)\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{45045 b^4}\\ &=\frac {2 \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (520 a^3 b B+4355 a b^3 B-240 a^4 C+539 b^4 (13 A+11 C)-10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac {2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {\left (\left (520 a^5 b B+3315 a^3 b^3 B+48165 a b^5 B-240 a^6 C+1617 b^6 (13 A+11 C)-10 a^4 b^2 (143 A+76 C)+3 a^2 b^4 (13299 A+10223 C)\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{45045 b^4 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (\left (a^2-b^2\right ) \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{45045 b^4 \sqrt {a+b \cos (c+d x)}}\\ &=\frac {2 \left (520 a^5 b B+3315 a^3 b^3 B+48165 a b^5 B-240 a^6 C+1617 b^6 (13 A+11 C)-10 a^4 b^2 (143 A+76 C)+3 a^2 b^4 (13299 A+10223 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (520 a^3 b B+4355 a b^3 B-240 a^4 C+539 b^4 (13 A+11 C)-10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac {2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}\\ \end {align*}
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Mathematica [A] time = 4.04, size = 501, normalized size = 0.80 \[ \frac {b (a+b \cos (c+d x)) \left (b \left (5 b \left (7 b \left (4 \sin (4 (c+d x)) \left (159 a^2 C+299 a b B+143 A b^2+220 b^2 C\right )+9 b ((54 a C+26 b B) \sin (5 (c+d x))+11 b C \sin (6 (c+d x)))\right )+2 \sin (3 (c+d x)) \left (60 a^3 C+5876 a^2 b B+a b^2 (10868 A+13939 C)+6669 b^3 B\right )\right )+\sin (2 (c+d x)) \left (-1440 a^4 C+3120 a^3 b B+120 a^2 b^2 (1430 A+1457 C)+321880 a b^3 B+77 b^4 (1976 A+1897 C)\right )\right )+4 \sin (c+d x) \left (960 a^5 C-2080 a^4 b B+10 a^3 b^2 (572 A+331 C)+121290 a^2 b^3 B+3 a b^4 (71214 A+60793 C)+84825 b^5 B\right )\right )+32 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (-60 a^5 C+130 a^4 b B+5 a^3 b^2 (4433 A+3337 C)+43095 a^2 b^3 B+3 a b^4 (12441 A+10277 C)+8775 b^5 B\right ) F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-\left (240 a^6 C-520 a^5 b B+10 a^4 b^2 (143 A+76 C)-3315 a^3 b^3 B-3 a^2 b^4 (13299 A+10223 C)-48165 a b^5 B-1617 b^6 (13 A+11 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )\right )\right )}{720720 b^4 d \sqrt {a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{2} \cos \left (d x + c\right )^{6} + {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{5} + A a^{2} \cos \left (d x + c\right )^{2} + {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{4} + {\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.62, size = 3165, normalized size = 5.03 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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