Optimal. Leaf size=125 \[ \frac {30 b^3 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{77 d \sqrt {b \cos (c+d x)}}+\frac {2 \sin (c+d x) (b \cos (c+d x))^{9/2}}{11 b^2 d}+\frac {30 b^2 \sin (c+d x) \sqrt {b \cos (c+d x)}}{77 d}+\frac {18 \sin (c+d x) (b \cos (c+d x))^{5/2}}{77 d} \]
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Rubi [A] time = 0.08, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 2635, 2642, 2641} \[ \frac {2 \sin (c+d x) (b \cos (c+d x))^{9/2}}{11 b^2 d}+\frac {30 b^2 \sin (c+d x) \sqrt {b \cos (c+d x)}}{77 d}+\frac {30 b^3 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{77 d \sqrt {b \cos (c+d x)}}+\frac {18 \sin (c+d x) (b \cos (c+d x))^{5/2}}{77 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2635
Rule 2641
Rule 2642
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (b \cos (c+d x))^{5/2} \, dx &=\frac {\int (b \cos (c+d x))^{11/2} \, dx}{b^3}\\ &=\frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^2 d}+\frac {9 \int (b \cos (c+d x))^{7/2} \, dx}{11 b}\\ &=\frac {18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}+\frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^2 d}+\frac {1}{77} (45 b) \int (b \cos (c+d x))^{3/2} \, dx\\ &=\frac {30 b^2 \sqrt {b \cos (c+d x)} \sin (c+d x)}{77 d}+\frac {18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}+\frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^2 d}+\frac {1}{77} \left (15 b^3\right ) \int \frac {1}{\sqrt {b \cos (c+d x)}} \, dx\\ &=\frac {30 b^2 \sqrt {b \cos (c+d x)} \sin (c+d x)}{77 d}+\frac {18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}+\frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^2 d}+\frac {\left (15 b^3 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{77 \sqrt {b \cos (c+d x)}}\\ &=\frac {30 b^3 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{77 d \sqrt {b \cos (c+d x)}}+\frac {30 b^2 \sqrt {b \cos (c+d x)} \sin (c+d x)}{77 d}+\frac {18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}+\frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^2 d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 83, normalized size = 0.66 \[ \frac {(b \cos (c+d x))^{5/2} \left (240 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+(290 \sin (c+d x)+57 \sin (3 (c+d x))+7 \sin (5 (c+d x))) \sqrt {\cos (c+d x)}\right )}{616 d \cos ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \cos \left (d x + c\right )} b^{2} \cos \left (d x + c\right )^{5}, x\right ) \]
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 \[ \int \left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}} \cos \left (d x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 236, normalized size = 1.89 \[ -\frac {2 \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, b^{3} \left (448 \left (\cos ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1568 \left (\cos ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2384 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2040 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1084 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-370 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+62 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{77 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d} \]
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 \[ \int \left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}} \cos \left (d x + c\right )^{3}\,{d x} \]
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 \[ \int {\cos \left (c+d\,x\right )}^3\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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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