Optimal. Leaf size=834 \[ \frac {C \sqrt {\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d}+\frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d}+\frac {\left (-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d}+\frac {\left (-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right ) \sqrt {\cos (c+d x)} \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{320 b d}+\frac {\left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{1920 b^2 d \sqrt {\cos (c+d x)}}-\frac {(a-b) \sqrt {a+b} \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}}}{1920 a b^2 d}-\frac {\sqrt {a+b} \left (45 C a^4-30 b (5 B+C) a^3-4 b^2 (660 A+295 B+423 C) a^2-8 b^3 (260 A+355 B+193 C) a-16 b^4 (80 A+45 B+64 C)\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}}}{1920 b^2 d}+\frac {\sqrt {a+b} \left (-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right ) \cot (c+d x) \Pi \left (\frac {a+b}{b};\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}}}{128 b^3 d} \]
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Rubi [A] time = 3.79, antiderivative size = 834, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {3049, 3061, 3053, 2809, 2998, 2816, 2994} \[ \frac {C \sqrt {\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d}+\frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d}+\frac {\left (-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d}+\frac {\left (-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right ) \sqrt {\cos (c+d x)} \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{320 b d}+\frac {\left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{1920 b^2 d \sqrt {\cos (c+d x)}}-\frac {(a-b) \sqrt {a+b} \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}}}{1920 a b^2 d}-\frac {\sqrt {a+b} \left (45 C a^4-30 b (5 B+C) a^3-4 b^2 (660 A+295 B+423 C) a^2-8 b^3 (260 A+355 B+193 C) a-16 b^4 (80 A+45 B+64 C)\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}}}{1920 b^2 d}+\frac {\sqrt {a+b} \left (-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right ) \cot (c+d x) \Pi \left (\frac {a+b}{b};\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}}}{128 b^3 d} \]
Antiderivative was successfully verified.
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Rule 2809
Rule 2816
Rule 2994
Rule 2998
Rule 3049
Rule 3053
Rule 3061
Rubi steps
\begin {align*} \int \sqrt {\cos (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 {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac {\int \frac {(a+b \cos (c+d x))^{5/2} \left (\frac {a C}{2}+b (5 A+4 C) \cos (c+d x)+\frac {1}{2} (10 b B-3 a C) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{5 b}\\ &=\frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac {\int \frac {(a+b \cos (c+d x))^{3/2} \left (\frac {5}{4} a (2 b B+a C)+\frac {1}{2} b (40 a A+30 b B+27 a C) \cos (c+d x)+\frac {1}{4} \left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{20 b}\\ &=\frac {\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac {\int \frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{8} a \left (80 A b^2+110 a b B+15 a^2 C+64 b^2 C\right )+\frac {1}{4} b \left (310 a b B+32 b^2 (5 A+4 C)+3 a^2 (80 A+49 C)\right ) \cos (c+d x)+\frac {3}{8} \left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{60 b}\\ &=\frac {\left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{320 b d}+\frac {\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac {\int \frac {\frac {1}{16} a \left (590 a^2 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\right )+\frac {1}{8} b \left (1610 a^2 b B+360 b^3 B+4 a b^2 (380 A+289 C)+a^3 (960 A+573 C)\right ) \cos (c+d x)+\frac {1}{16} \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{120 b}\\ &=\frac {\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{320 b d}+\frac {\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac {\int \frac {-\frac {1}{16} a \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right )+\frac {1}{8} a b \left (590 a^2 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\right ) \cos (c+d x)-\frac {15}{16} \left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 (2 A+C)-80 a b^4 (4 A+3 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{240 b^2}\\ &=\frac {\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{320 b d}+\frac {\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac {\int \frac {-\frac {1}{16} a \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right )+\frac {1}{8} a b \left (590 a^2 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{240 b^2}-\frac {\left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 (2 A+C)-80 a b^4 (4 A+3 C)\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}} \, dx}{256 b^2}\\ &=\frac {\sqrt {a+b} \left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 (2 A+C)-80 a b^4 (4 A+3 C)\right ) \cot (c+d x) \Pi \left (\frac {a+b}{b};\sin ^{-1}\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}}}{128 b^3 d}+\frac {\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{320 b d}+\frac {\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}-\frac {\left (a \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right )\right ) \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{3840 b^2}-\frac {\left (a \left (45 a^4 C-30 a^3 b (5 B+C)-16 b^4 (80 A+45 B+64 C)-8 a b^3 (260 A+355 B+193 C)-4 a^2 b^2 (660 A+295 B+423 C)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{3840 b^2}\\ &=-\frac {(a-b) \sqrt {a+b} \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \cot (c+d x) E\left (\sin ^{-1}\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}}}{1920 a b^2 d}-\frac {\sqrt {a+b} \left (45 a^4 C-30 a^3 b (5 B+C)-16 b^4 (80 A+45 B+64 C)-8 a b^3 (260 A+355 B+193 C)-4 a^2 b^2 (660 A+295 B+423 C)\right ) \cot (c+d x) F\left (\sin ^{-1}\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}}}{1920 b^2 d}+\frac {\sqrt {a+b} \left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 (2 A+C)-80 a b^4 (4 A+3 C)\right ) \cot (c+d x) \Pi \left (\frac {a+b}{b};\sin ^{-1}\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}}}{128 b^3 d}+\frac {\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{320 b d}+\frac {\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}\\ \end {align*}
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Mathematica [C] time = 6.73, size = 1410, normalized size = 1.69 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 6.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{2} \cos \left (d x + c\right )^{4} + {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{3} + A a^{2} + {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [B] time = 1.32, size = 7062, normalized size = 8.47 \[ \text {output too large to display} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sqrt {\cos \left (d x + c\right )}\,{d x} \]
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 \[ \int \sqrt {\cos \left (c+d\,x\right )}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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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