Integrand size = 18, antiderivative size = 60 \[ \int x \cos ^{\frac {3}{2}}(a+b x) \sin (a+b x) \, dx=-\frac {2 x \cos ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {12 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{25 b^2}+\frac {4 \cos ^{\frac {3}{2}}(a+b x) \sin (a+b x)}{25 b^2} \]
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Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3525, 2715, 2719} \[ \int x \cos ^{\frac {3}{2}}(a+b x) \sin (a+b x) \, dx=\frac {12 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{25 b^2}+\frac {4 \sin (a+b x) \cos ^{\frac {3}{2}}(a+b x)}{25 b^2}-\frac {2 x \cos ^{\frac {5}{2}}(a+b x)}{5 b} \]
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Rule 2715
Rule 2719
Rule 3525
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x \cos ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {2 \int \cos ^{\frac {5}{2}}(a+b x) \, dx}{5 b} \\ & = -\frac {2 x \cos ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {4 \cos ^{\frac {3}{2}}(a+b x) \sin (a+b x)}{25 b^2}+\frac {6 \int \sqrt {\cos (a+b x)} \, dx}{25 b} \\ & = -\frac {2 x \cos ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {12 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{25 b^2}+\frac {4 \cos ^{\frac {3}{2}}(a+b x) \sin (a+b x)}{25 b^2} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.85 \[ \int x \cos ^{\frac {3}{2}}(a+b x) \sin (a+b x) \, dx=-\frac {2 \left (-6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+\cos ^{\frac {3}{2}}(a+b x) (5 b x \cos (a+b x)-2 \sin (a+b x))\right )}{25 b^2} \]
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\[\int x \cos \left (x b +a \right )^{\frac {3}{2}} \sin \left (x b +a \right )d x\]
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Exception generated. \[ \int x \cos ^{\frac {3}{2}}(a+b x) \sin (a+b x) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \cos ^{\frac {3}{2}}(a+b x) \sin (a+b x) \, dx=\text {Timed out} \]
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\[ \int x \cos ^{\frac {3}{2}}(a+b x) \sin (a+b x) \, dx=\int { x \cos \left (b x + a\right )^{\frac {3}{2}} \sin \left (b x + a\right ) \,d x } \]
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\[ \int x \cos ^{\frac {3}{2}}(a+b x) \sin (a+b x) \, dx=\int { x \cos \left (b x + a\right )^{\frac {3}{2}} \sin \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x \cos ^{\frac {3}{2}}(a+b x) \sin (a+b x) \, dx=\int x\,{\cos \left (a+b\,x\right )}^{3/2}\,\sin \left (a+b\,x\right ) \,d x \]
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