Optimal. Leaf size=49 \[ -\frac{3}{2} \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{x}\right )+x^{3/2} \sin (x)+\frac{3}{2} \sqrt{x} \cos (x) \]
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Rubi [A] time = 0.0572119, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3296, 3304, 3352} \[ -\frac{3}{2} \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{x}\right )+x^{3/2} \sin (x)+\frac{3}{2} \sqrt{x} \cos (x) \]
Antiderivative was successfully verified.
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Rule 3296
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int x^{3/2} \cos (x) \, dx &=x^{3/2} \sin (x)-\frac{3}{2} \int \sqrt{x} \sin (x) \, dx\\ &=\frac{3}{2} \sqrt{x} \cos (x)+x^{3/2} \sin (x)-\frac{3}{4} \int \frac{\cos (x)}{\sqrt{x}} \, dx\\ &=\frac{3}{2} \sqrt{x} \cos (x)+x^{3/2} \sin (x)-\frac{3}{2} \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{3}{2} \sqrt{x} \cos (x)-\frac{3}{2} \sqrt{\frac{\pi }{2}} C\left (\sqrt{\frac{2}{\pi }} \sqrt{x}\right )+x^{3/2} \sin (x)\\ \end{align*}
Mathematica [C] time = 0.0133136, size = 55, normalized size = 1.12 \[ \frac{\sqrt{x} \text{Gamma}\left (\frac{5}{2},-i x\right )}{2 \sqrt{-i x}}+\frac{\sqrt{x} \text{Gamma}\left (\frac{5}{2},i x\right )}{2 \sqrt{i x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 34, normalized size = 0.7 \begin{align*}{x}^{{\frac{3}{2}}}\sin \left ( x \right ) -{\frac{3\,\sqrt{2}\sqrt{\pi }}{4}{\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt{x}} \right ) }+{\frac{3\,\cos \left ( x \right ) }{2}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.78026, size = 100, normalized size = 2.04 \begin{align*} x^{\frac{3}{2}} \sin \left (x\right ) + \frac{1}{32} \, \sqrt{\pi }{\left (\left (3 i - 3\right ) \, \sqrt{2} \operatorname{erf}\left (\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x}\right ) + \left (3 i + 3\right ) \, \sqrt{2} \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x}\right ) - \left (3 i + 3\right ) \, \sqrt{2} \operatorname{erf}\left (\sqrt{-i} \sqrt{x}\right ) + \left (3 i - 3\right ) \, \sqrt{2} \operatorname{erf}\left (\left (-1\right )^{\frac{1}{4}} \sqrt{x}\right )\right )} + \frac{3}{2} \, \sqrt{x} \cos \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5896, size = 134, normalized size = 2.73 \begin{align*} -\frac{3}{4} \, \sqrt{2} \sqrt{\pi } \operatorname{C}\left (\frac{\sqrt{2} \sqrt{x}}{\sqrt{\pi }}\right ) + \frac{1}{2} \,{\left (2 \, x \sin \left (x\right ) + 3 \, \cos \left (x\right )\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 16.2414, size = 83, normalized size = 1.69 \begin{align*} \frac{5 x^{\frac{3}{2}} \sin{\left (x \right )} \Gamma \left (\frac{5}{4}\right )}{4 \Gamma \left (\frac{9}{4}\right )} + \frac{15 \sqrt{x} \cos{\left (x \right )} \Gamma \left (\frac{5}{4}\right )}{8 \Gamma \left (\frac{9}{4}\right )} - \frac{15 \sqrt{2} \sqrt{\pi } C\left (\frac{\sqrt{2} \sqrt{x}}{\sqrt{\pi }}\right ) \Gamma \left (\frac{5}{4}\right )}{16 \Gamma \left (\frac{9}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.14162, size = 93, normalized size = 1.9 \begin{align*} \left (\frac{3}{16} i + \frac{3}{16}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x}\right ) - \left (\frac{3}{16} i - \frac{3}{16}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x}\right ) - \frac{1}{4} \,{\left (2 i \, x^{\frac{3}{2}} - 3 \, \sqrt{x}\right )} e^{\left (i \, x\right )} - \frac{1}{4} \,{\left (-2 i \, x^{\frac{3}{2}} - 3 \, \sqrt{x}\right )} e^{\left (-i \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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