Integrand size = 12, antiderivative size = 12 \[ \int x \cos ^{\frac {3}{2}}(a+b x) \, dx=\frac {4 \cos ^{\frac {3}{2}}(a+b x)}{9 b^2}+\frac {2 x \sqrt {\cos (a+b x)} \sin (a+b x)}{3 b}+\frac {1}{3} \text {Int}\left (\frac {x}{\sqrt {\cos (a+b x)}},x\right ) \]
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Not integrable
Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x \cos ^{\frac {3}{2}}(a+b x) \, dx=\int x \cos ^{\frac {3}{2}}(a+b x) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \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*}
Not integrable
Time = 2.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int x \cos ^{\frac {3}{2}}(a+b x) \, dx=\int x \cos ^{\frac {3}{2}}(a+b x) \, dx \]
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Not integrable
Time = 0.13 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83
\[\int x \left (\cos ^{\frac {3}{2}}\left (b x +a \right )\right )d x\]
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Exception generated. \[ \int x \cos ^{\frac {3}{2}}(a+b x) \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 87.66 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int x \cos ^{\frac {3}{2}}(a+b x) \, dx=\int x \cos ^{\frac {3}{2}}{\left (a + b x \right )}\, dx \]
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Not integrable
Time = 0.70 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int x \cos ^{\frac {3}{2}}(a+b x) \, dx=\int { x \cos \left (b x + a\right )^{\frac {3}{2}} \,d x } \]
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Not integrable
Time = 0.37 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int x \cos ^{\frac {3}{2}}(a+b x) \, dx=\int { x \cos \left (b x + a\right )^{\frac {3}{2}} \,d x } \]
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Not integrable
Time = 13.73 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int x \cos ^{\frac {3}{2}}(a+b x) \, dx=\int x\,{\cos \left (a+b\,x\right )}^{3/2} \,d x \]
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