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 = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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*}
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Mathematica [A] time = 0.08, 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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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sqrt {\sin \left (b x + a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin \left (b x + a\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 142, normalized size = 3.02 \[ \frac {\frac {2 \left (\sin ^{4}\left (b x +a \right )\right )}{5}-\frac {2 \left (\sin ^{2}\left (b x +a \right )\right )}{5}-\frac {6 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \EllipticE \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {3 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \EllipticF \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{5}}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b} \]
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 \[ \int \sin \left (b x + a\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 42, normalized size = 0.89 \[ -\frac {\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^{7/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {3}{2};\ {\cos \left (a+b\,x\right )}^2\right )}{b\,{\left ({\sin \left (a+b\,x\right )}^2\right )}^{7/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin ^{\frac {5}{2}}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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