Optimal. Leaf size=47 \[ \frac {2 F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{3 b}-\frac {2 \sqrt {\sin (a+b x)} \cos (a+b x)}{3 b} \]
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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, 2641} \[ \frac {2 F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{3 b}-\frac {2 \sqrt {\sin (a+b x)} \cos (a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rubi steps
\begin {align*} \int \sin ^{\frac {3}{2}}(a+b x) \, dx &=-\frac {2 \cos (a+b x) \sqrt {\sin (a+b x)}}{3 b}+\frac {1}{3} \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx\\ &=\frac {2 F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{3 b}-\frac {2 \cos (a+b x) \sqrt {\sin (a+b x)}}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 40, normalized size = 0.85 \[ -\frac {2 \left (F\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )+\sqrt {\sin (a+b x)} \cos (a+b x)\right )}{3 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sin \left (b x + a\right )^{\frac {3}{2}}, x\right ) \]
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 \[ \int \sin \left (b x + a\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 88, normalized size = 1.87 \[ \frac {\frac {\sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \EllipticF \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{3}-\frac {2 \left (\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3}}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b} \]
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 \[ \int \sin \left (b x + a\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.44, size = 42, normalized size = 0.89 \[ -\frac {\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{2};\ {\cos \left (a+b\,x\right )}^2\right )}{b\,{\left ({\sin \left (a+b\,x\right )}^2\right )}^{5/4}} \]
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 \[ \int \sin ^{\frac {3}{2}}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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