Optimal. Leaf size=88 \[ -\frac{20 \text{EllipticF}\left (\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right ),2\right )}{147 b^2}+\frac{4 \sin ^{\frac{5}{2}}(a+b x) \cos (a+b x)}{49 b^2}+\frac{20 \sqrt{\sin (a+b x)} \cos (a+b x)}{147 b^2}+\frac{2 x \sin ^{\frac{7}{2}}(a+b x)}{7 b} \]
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Rubi [A] time = 0.0442303, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3443, 2635, 2641} \[ -\frac{20 F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{147 b^2}+\frac{4 \sin ^{\frac{5}{2}}(a+b x) \cos (a+b x)}{49 b^2}+\frac{20 \sqrt{\sin (a+b x)} \cos (a+b x)}{147 b^2}+\frac{2 x \sin ^{\frac{7}{2}}(a+b x)}{7 b} \]
Antiderivative was successfully verified.
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Rule 3443
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int x \cos (a+b x) \sin ^{\frac{5}{2}}(a+b x) \, dx &=\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 F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\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*}
Mathematica [A] time = 0.54583, size = 67, normalized size = 0.76 \[ \frac{40 \text{EllipticF}\left (\frac{1}{4} (-2 a-2 b x+\pi ),2\right )+\sqrt{\sin (a+b x)} \left (84 b x \sin ^3(a+b x)+46 \cos (a+b x)-6 \cos (3 (a+b x))\right )}{294 b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.104, size = 0, normalized size = 0. \begin{align*} \int x\cos \left ( bx+a \right ) \left ( \sin \left ( bx+a \right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cos \left (b x + a\right ) \sin \left (b x + a\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cos \left (b x + a\right ) \sin \left (b x + a\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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