Optimal. Leaf size=69 \[ \frac{\sin ^{\frac{7}{2}}(2 a+2 b x)}{14 b}+\frac{3 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{10 b}-\frac{\sin ^{\frac{3}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{10 b} \]
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Rubi [A] time = 0.0453035, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4297, 2635, 2639} \[ \frac{\sin ^{\frac{7}{2}}(2 a+2 b x)}{14 b}+\frac{3 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{10 b}-\frac{\sin ^{\frac{3}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{10 b} \]
Antiderivative was successfully verified.
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Rule 4297
Rule 2635
Rule 2639
Rubi steps
\begin{align*} \int \cos ^2(a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x) \, dx &=\frac{\sin ^{\frac{7}{2}}(2 a+2 b x)}{14 b}+\frac{1}{2} \int \sin ^{\frac{5}{2}}(2 a+2 b x) \, dx\\ &=-\frac{\cos (2 a+2 b x) \sin ^{\frac{3}{2}}(2 a+2 b x)}{10 b}+\frac{\sin ^{\frac{7}{2}}(2 a+2 b x)}{14 b}+\frac{3}{10} \int \sqrt{\sin (2 a+2 b x)} \, dx\\ &=\frac{3 E\left (\left .a-\frac{\pi }{4}+b x\right |2\right )}{10 b}-\frac{\cos (2 a+2 b x) \sin ^{\frac{3}{2}}(2 a+2 b x)}{10 b}+\frac{\sin ^{\frac{7}{2}}(2 a+2 b x)}{14 b}\\ \end{align*}
Mathematica [A] time = 0.208149, size = 66, normalized size = 0.96 \[ \frac{\sqrt{\sin (2 (a+b x))} (15 \sin (2 (a+b x))-14 \sin (4 (a+b x))-5 \sin (6 (a+b x)))+84 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{280 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 110.597, size = 336654858, normalized size = 4879055.9 \begin{align*} \text{output too large to display} \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 \cos \left (b x + a\right )^{2} \sin \left (2 \, b x + 2 \, 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 (2 \, b x + 2 \, a\right )^{2} \cos \left (b x + a\right )^{2} - \cos \left (b x + a\right )^{2}\right )} \sqrt{\sin \left (2 \, b x + 2 \, 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(-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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