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 = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 43
Rule 266
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.41, size = 23, normalized size = 0.74 \[ \frac {1}{15} \, {\left (3 \, x^{4} - 4 \, x^{2} - 32\right )} \sqrt {-x^{2} + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.88, size = 30, normalized size = 0.97 \[ \frac {1}{5} \, {\left (x^{2} - 4\right )}^{2} \sqrt {-x^{2} + 4} - \frac {4}{3} \, {\left (-x^{2} + 4\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.81 \[ \frac {\left (x -2\right ) \left (x +2\right ) \left (3 x^{2}+8\right ) \sqrt {-x^{2}+4}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 26, normalized size = 0.84 \[ -\frac {1}{5} \, {\left (-x^{2} + 4\right )}^{\frac {3}{2}} x^{2} - \frac {8}{15} \, {\left (-x^{2} + 4\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 23, normalized size = 0.74 \[ -\sqrt {4-x^2}\,\left (-\frac {x^4}{5}+\frac {4\,x^2}{15}+\frac {32}{15}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.64, size = 39, normalized size = 1.26 \[ \frac {x^{4} \sqrt {4 - x^{2}}}{5} - \frac {4 x^{2} \sqrt {4 - x^{2}}}{15} - \frac {32 \sqrt {4 - x^{2}}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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