Optimal. Leaf size=169 \[ \frac {2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}+\frac {6 A b E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}+\frac {10 b^2 B \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b d}+\frac {10 b B \sin (c+d x) \sqrt {b \cos (c+d x)}}{21 d} \]
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Rubi [A] time = 0.14, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {16, 2748, 2635, 2640, 2639, 2642, 2641} \[ \frac {2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}+\frac {6 A b E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}+\frac {10 b^2 B \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b d}+\frac {10 b B \sin (c+d x) \sqrt {b \cos (c+d x)}}{21 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2635
Rule 2639
Rule 2640
Rule 2641
Rule 2642
Rule 2748
Rubi steps
\begin {align*} \int \cos (c+d x) (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx &=\frac {\int (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx}{b}\\ &=\frac {A \int (b \cos (c+d x))^{5/2} \, dx}{b}+\frac {B \int (b \cos (c+d x))^{7/2} \, dx}{b^2}\\ &=\frac {2 A (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac {1}{5} (3 A b) \int \sqrt {b \cos (c+d x)} \, dx+\frac {1}{7} (5 B) \int (b \cos (c+d x))^{3/2} \, dx\\ &=\frac {10 b B \sqrt {b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 A (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac {1}{21} \left (5 b^2 B\right ) \int \frac {1}{\sqrt {b \cos (c+d x)}} \, dx+\frac {\left (3 A b \sqrt {b \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)}}\\ &=\frac {6 A b \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}+\frac {10 b B \sqrt {b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 A (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac {\left (5 b^2 B \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 \sqrt {b \cos (c+d x)}}\\ &=\frac {6 A b \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}+\frac {10 b^2 B \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {10 b B \sqrt {b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 A (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 103, normalized size = 0.61 \[ \frac {(b \cos (c+d x))^{5/2} \left (2 \sin (c+d x) \sqrt {\cos (c+d x)} (42 A \cos (c+d x)+15 B \cos (2 (c+d x))+65 B)+252 A E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+100 B F\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{210 b d \cos ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B b \cos \left (d x + c\right )^{3} + A b \cos \left (d x + c\right )^{2}\right )} \sqrt {b \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {3}{2}} \cos \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.79, size = 301, normalized size = 1.78 \[ -\frac {2 \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, b^{2} \left (240 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-168 A -360 B \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (168 A +280 B \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-42 A -80 B \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-63 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+25 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{105 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d} \]
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 \[ \int {\left (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {3}{2}} \cos \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \cos \left (c+d\,x\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (A+B\,\cos \left (c+d\,x\right )\right ) \,d x \]
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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