Optimal. Leaf size=47 \[ \frac{6 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{5 b}-\frac{2 \sin ^{\frac{3}{2}}(a+b x) \cos (a+b x)}{5 b} \]
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Rubi [A] time = 0.0169634, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2635, 2639} \[ \frac{6 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{5 b}-\frac{2 \sin ^{\frac{3}{2}}(a+b x) \cos (a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rubi steps
\begin{align*} \int \sin ^{\frac{5}{2}}(a+b x) \, dx &=-\frac{2 \cos (a+b x) \sin ^{\frac{3}{2}}(a+b x)}{5 b}+\frac{3}{5} \int \sqrt{\sin (a+b x)} \, dx\\ &=\frac{6 E\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right )}{5 b}-\frac{2 \cos (a+b x) \sin ^{\frac{3}{2}}(a+b x)}{5 b}\\ \end{align*}
Mathematica [A] time = 0.0867625, size = 44, normalized size = 0.94 \[ -\frac{\sqrt{\sin (a+b x)} \sin (2 (a+b x))+6 E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )}{5 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 142, normalized size = 3. \begin{align*}{\frac{1}{b\cos \left ( bx+a \right ) } \left ({\frac{2\, \left ( \sin \left ( bx+a \right ) \right ) ^{4}}{5}}-{\frac{2\, \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{5}}-{\frac{6}{5}\sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) }{\it EllipticE} \left ( \sqrt{\sin \left ( bx+a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) }+{\frac{3}{5}\sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( bx+a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) } \right ){\frac{1}{\sqrt{\sin \left ( bx+a \right ) }}}} \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 \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sqrt{\sin \left (b x + a\right )}, x\right ) \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 \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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