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 = 0.05, antiderivative size = 69, normalized size of antiderivative = 1.00, 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 = {4298, 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 2635
Rule 2639
Rule 4298
Rubi steps
\begin {align*} \int \sin ^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*}
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Mathematica [A] time = 0.22, 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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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left ({\left (\cos \left (b x + a\right )^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right )^{2} - \cos \left (b x + a\right )^{2} + 1\right )} \sqrt {\sin \left (2 \, b x + 2 \, a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 167.06, size = 278615779, normalized size = 4037909.84 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin \left (2 \, b x + 2 \, a\right )^{\frac {5}{2}} \sin \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\sin \left (a+b\,x\right )}^2\,{\sin \left (2\,a+2\,b\,x\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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