Optimal. Leaf size=61 \[ \frac {1}{3} \text {Int}\left (\frac {x}{\sqrt {\cos (a+b x)}},x\right )+\frac {4 \cos ^{\frac {3}{2}}(a+b x)}{9 b^2}+\frac {2 x \sin (a+b x) \sqrt {\cos (a+b x)}}{3 b} \]
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Rubi [A] time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x \cos ^{\frac {3}{2}}(a+b x) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x \cos ^{\frac {3}{2}}(a+b x) \, dx &=\frac {4 \cos ^{\frac {3}{2}}(a+b x)}{9 b^2}+\frac {2 x \sqrt {\cos (a+b x)} \sin (a+b x)}{3 b}+\frac {1}{3} \int \frac {x}{\sqrt {\cos (a+b x)}} \, dx\\ \end {align*}
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Mathematica [A] time = 1.94, size = 0, normalized size = 0.00 \[ \int x \cos ^{\frac {3}{2}}(a+b x) \, dx \]
Verification is Not applicable to the result.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \cos \left (b x + a\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 0, normalized size = 0.00 \[ \int x \left (\cos ^{\frac {3}{2}}\left (b x +a \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \cos \left (b x + a\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\cos \left (a+b\,x\right )}^{3/2} \,d x \]
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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