Optimal. Leaf size=31 \[ \frac{1}{5} \left (4-x^2\right )^{5/2}-\frac{4}{3} \left (4-x^2\right )^{3/2} \]
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Rubi [A] time = 0.0142777, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{1}{5} \left (4-x^2\right )^{5/2}-\frac{4}{3} \left (4-x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^3 \sqrt{4-x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sqrt{4-x} x \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (4 \sqrt{4-x}-(4-x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=-\frac{4}{3} \left (4-x^2\right )^{3/2}+\frac{1}{5} \left (4-x^2\right )^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0094979, size = 22, normalized size = 0.71 \[ -\frac{1}{15} \left (4-x^2\right )^{3/2} \left (3 x^2+8\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 25, normalized size = 0.8 \begin{align*}{\frac{ \left ( -2+x \right ) \left ( 2+x \right ) \left ( 3\,{x}^{2}+8 \right ) }{15}\sqrt{-{x}^{2}+4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40397, size = 35, normalized size = 1.13 \begin{align*} -\frac{1}{5} \,{\left (-x^{2} + 4\right )}^{\frac{3}{2}} x^{2} - \frac{8}{15} \,{\left (-x^{2} + 4\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89788, size = 57, normalized size = 1.84 \begin{align*} \frac{1}{15} \,{\left (3 \, x^{4} - 4 \, x^{2} - 32\right )} \sqrt{-x^{2} + 4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.568048, size = 39, normalized size = 1.26 \begin{align*} \frac{x^{4} \sqrt{4 - x^{2}}}{5} - \frac{4 x^{2} \sqrt{4 - x^{2}}}{15} - \frac{32 \sqrt{4 - x^{2}}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05564, size = 41, normalized size = 1.32 \begin{align*} \frac{1}{5} \,{\left (x^{2} - 4\right )}^{2} \sqrt{-x^{2} + 4} - \frac{4}{3} \,{\left (-x^{2} + 4\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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