Integrand size = 18, antiderivative size = 88 \[ \int x \cos (a+b x) \sin ^{\frac {5}{2}}(a+b x) \, dx=-\frac {20 \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right )}{147 b^2}+\frac {20 \cos (a+b x) \sqrt {\sin (a+b x)}}{147 b^2}+\frac {4 \cos (a+b x) \sin ^{\frac {5}{2}}(a+b x)}{49 b^2}+\frac {2 x \sin ^{\frac {7}{2}}(a+b x)}{7 b} \]
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Time = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3524, 2715, 2720} \[ \int x \cos (a+b x) \sin ^{\frac {5}{2}}(a+b x) \, dx=-\frac {20 \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right )}{147 b^2}+\frac {4 \sin ^{\frac {5}{2}}(a+b x) \cos (a+b x)}{49 b^2}+\frac {20 \sqrt {\sin (a+b x)} \cos (a+b x)}{147 b^2}+\frac {2 x \sin ^{\frac {7}{2}}(a+b x)}{7 b} \]
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Rule 2715
Rule 2720
Rule 3524
Rubi steps \begin{align*} \text {integral}& = \frac {2 x \sin ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {2 \int \sin ^{\frac {7}{2}}(a+b x) \, dx}{7 b} \\ & = \frac {4 \cos (a+b x) \sin ^{\frac {5}{2}}(a+b x)}{49 b^2}+\frac {2 x \sin ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {10 \int \sin ^{\frac {3}{2}}(a+b x) \, dx}{49 b} \\ & = \frac {20 \cos (a+b x) \sqrt {\sin (a+b x)}}{147 b^2}+\frac {4 \cos (a+b x) \sin ^{\frac {5}{2}}(a+b x)}{49 b^2}+\frac {2 x \sin ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {10 \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx}{147 b} \\ & = -\frac {20 \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right )}{147 b^2}+\frac {20 \cos (a+b x) \sqrt {\sin (a+b x)}}{147 b^2}+\frac {4 \cos (a+b x) \sin ^{\frac {5}{2}}(a+b x)}{49 b^2}+\frac {2 x \sin ^{\frac {7}{2}}(a+b x)}{7 b} \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.76 \[ \int x \cos (a+b x) \sin ^{\frac {5}{2}}(a+b x) \, dx=\frac {40 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right )+\sqrt {\sin (a+b x)} \left (46 \cos (a+b x)-6 \cos (3 (a+b x))+84 b x \sin ^3(a+b x)\right )}{294 b^2} \]
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\[\int x \cos \left (x b +a \right ) \sin \left (x b +a \right )^{\frac {5}{2}}d x\]
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Exception generated. \[ \int x \cos (a+b x) \sin ^{\frac {5}{2}}(a+b x) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \cos (a+b x) \sin ^{\frac {5}{2}}(a+b x) \, dx=\text {Timed out} \]
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\[ \int x \cos (a+b x) \sin ^{\frac {5}{2}}(a+b x) \, dx=\int { x \cos \left (b x + a\right ) \sin \left (b x + a\right )^{\frac {5}{2}} \,d x } \]
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\[ \int x \cos (a+b x) \sin ^{\frac {5}{2}}(a+b x) \, dx=\int { x \cos \left (b x + a\right ) \sin \left (b x + a\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int x \cos (a+b x) \sin ^{\frac {5}{2}}(a+b x) \, dx=\int x\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^{5/2} \,d x \]
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