Optimal. Leaf size=65 \[ -\frac{4 \text{EllipticF}\left (\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right ),2\right )}{9 b^2}+\frac{4 \sqrt{\sin (a+b x)} \cos (a+b x)}{9 b^2}+\frac{2 x \sin ^{\frac{3}{2}}(a+b x)}{3 b} \]
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Rubi [A] time = 0.0315743, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3443, 2635, 2641} \[ -\frac{4 F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{9 b^2}+\frac{4 \sqrt{\sin (a+b x)} \cos (a+b x)}{9 b^2}+\frac{2 x \sin ^{\frac{3}{2}}(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 3443
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int x \cos (a+b x) \sqrt{\sin (a+b x)} \, dx &=\frac{2 x \sin ^{\frac{3}{2}}(a+b x)}{3 b}-\frac{2 \int \sin ^{\frac{3}{2}}(a+b x) \, dx}{3 b}\\ &=\frac{4 \cos (a+b x) \sqrt{\sin (a+b x)}}{9 b^2}+\frac{2 x \sin ^{\frac{3}{2}}(a+b x)}{3 b}-\frac{2 \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx}{9 b}\\ &=-\frac{4 F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right )}{9 b^2}+\frac{4 \cos (a+b x) \sqrt{\sin (a+b x)}}{9 b^2}+\frac{2 x \sin ^{\frac{3}{2}}(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.188767, size = 56, normalized size = 0.86 \[ \frac{4 \text{EllipticF}\left (\frac{1}{4} (-2 a-2 b x+\pi ),2\right )+2 \sqrt{\sin (a+b x)} (3 b x \sin (a+b x)+2 \cos (a+b x))}{9 b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.095, size = 0, normalized size = 0. \begin{align*} \int x\cos \left ( bx+a \right ) \sqrt{\sin \left ( bx+a \right ) }\, 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 ) \sqrt{\sin \left (b x + a\right )}\,{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{\sin{\left (a + b x \right )}} \cos{\left (a + b x \right )}\, dx \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 ) \sqrt{\sin \left (b x + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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