Optimal. Leaf size=63 \[ \frac {x \sqrt {\cos (c+d x)}}{2 \sqrt {b \cos (c+d x)}}+\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {b \cos (c+d x)}} \]
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Rubi [A] time = 0.01, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {17, 2635, 8} \[ \frac {x \sqrt {\cos (c+d x)}}{2 \sqrt {b \cos (c+d x)}}+\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 17
Rule 2635
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx &=\frac {\sqrt {\cos (c+d x)} \int \cos ^2(c+d x) \, dx}{\sqrt {b \cos (c+d x)}}\\ &=\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt {b \cos (c+d x)}}+\frac {\sqrt {\cos (c+d x)} \int 1 \, dx}{2 \sqrt {b \cos (c+d x)}}\\ &=\frac {x \sqrt {\cos (c+d x)}}{2 \sqrt {b \cos (c+d x)}}+\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt {b \cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 45, normalized size = 0.71 \[ \frac {(2 (c+d x)+\sin (2 (c+d x))) \sqrt {\cos (c+d x)}}{4 d \sqrt {b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 157, normalized size = 2.49 \[ \left [\frac {2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - \sqrt {-b} \log \left (2 \, b \cos \left (d x + c\right )^{2} + 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right )}{4 \, b d}, \frac {\sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + \sqrt {b} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right )}{2 \, b d}\right ] \]
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 \[ \int \frac {\cos \left (d x + c\right )^{\frac {5}{2}}}{\sqrt {b \cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 42, normalized size = 0.67 \[ \frac {\left (\cos \left (d x +c \right ) \sin \left (d x +c \right )+d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{2 d \sqrt {b \cos \left (d x +c \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 25, normalized size = 0.40 \[ \frac {2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )}{4 \, \sqrt {b} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 65, normalized size = 1.03 \[ \frac {\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (\sin \left (c+d\,x\right )+\sin \left (3\,c+3\,d\,x\right )+4\,d\,x\,\cos \left (c+d\,x\right )\right )}{4\,b\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
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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