Optimal. Leaf size=69 \[ \frac{\text{EllipticF}\left (a+b x-\frac{\pi }{4},2\right )}{6 b}+\frac{\sin ^{\frac{5}{2}}(2 a+2 b x)}{10 b}-\frac{\sqrt{\sin (2 a+2 b x)} \cos (2 a+2 b x)}{6 b} \]
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Rubi [A] time = 0.045327, 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, 2641} \[ \frac{\sin ^{\frac{5}{2}}(2 a+2 b x)}{10 b}+\frac{F\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{6 b}-\frac{\sqrt{\sin (2 a+2 b x)} \cos (2 a+2 b x)}{6 b} \]
Antiderivative was successfully verified.
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Rule 4297
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \cos ^2(a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x) \, dx &=\frac{\sin ^{\frac{5}{2}}(2 a+2 b x)}{10 b}+\frac{1}{2} \int \sin ^{\frac{3}{2}}(2 a+2 b x) \, dx\\ &=-\frac{\cos (2 a+2 b x) \sqrt{\sin (2 a+2 b x)}}{6 b}+\frac{\sin ^{\frac{5}{2}}(2 a+2 b x)}{10 b}+\frac{1}{6} \int \frac{1}{\sqrt{\sin (2 a+2 b x)}} \, dx\\ &=\frac{F\left (\left .a-\frac{\pi }{4}+b x\right |2\right )}{6 b}-\frac{\cos (2 a+2 b x) \sqrt{\sin (2 a+2 b x)}}{6 b}+\frac{\sin ^{\frac{5}{2}}(2 a+2 b x)}{10 b}\\ \end{align*}
Mathematica [A] time = 0.347827, size = 76, normalized size = 1.1 \[ \frac{20 \sqrt{\sin (2 (a+b x))} \text{EllipticF}\left (a+b x-\frac{\pi }{4},2\right )+9 \sin (2 (a+b x))-10 \sin (4 (a+b x))-3 \sin (6 (a+b x))}{120 b \sqrt{\sin (2 (a+b x))}} \]
Antiderivative was successfully verified.
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Maple [B] time = 33.345, size = 174364218, normalized size = 2527017.7 \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{3}{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 (\cos \left (b x + a\right )^{2} \sin \left (2 \, b x + 2 \, a\right )^{\frac{3}{2}}, 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 \cos \left (b x + a\right )^{2} \sin \left (2 \, b x + 2 \, a\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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