Optimal. Leaf size=31 \[ \frac {1}{48} \left (9-4 x^2\right )^{3/2}-\frac {9}{16} \sqrt {9-4 x^2} \]
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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} \[ \frac {1}{48} \left (9-4 x^2\right )^{3/2}-\frac {9}{16} \sqrt {9-4 x^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {9-4 x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\sqrt {9-4 x}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {9}{4 \sqrt {9-4 x}}-\frac {1}{4} \sqrt {9-4 x}\right ) \, dx,x,x^2\right )\\ &=-\frac {9}{16} \sqrt {9-4 x^2}+\frac {1}{48} \left (9-4 x^2\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 0.71 \[ -\frac {1}{24} \sqrt {9-4 x^2} \left (2 x^2+9\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 18, normalized size = 0.58 \[ -\frac {1}{24} \, {\left (2 \, x^{2} + 9\right )} \sqrt {-4 \, x^{2} + 9} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 23, normalized size = 0.74 \[ \frac {1}{48} \, {\left (-4 \, x^{2} + 9\right )}^{\frac {3}{2}} - \frac {9}{16} \, \sqrt {-4 \, x^{2} + 9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 29, normalized size = 0.94 \[ \frac {\left (2 x -3\right ) \left (2 x +3\right ) \left (2 x^{2}+9\right )}{24 \sqrt {-4 x^{2}+9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.90, size = 26, normalized size = 0.84 \[ -\frac {1}{12} \, \sqrt {-4 \, x^{2} + 9} x^{2} - \frac {3}{8} \, \sqrt {-4 \, x^{2} + 9} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 18, normalized size = 0.58 \[ -\frac {\sqrt {\frac {9}{4}-x^2}\,\left (\frac {x^2}{3}+\frac {3}{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.38, size = 29, normalized size = 0.94 \[ - \frac {x^{2} \sqrt {9 - 4 x^{2}}}{12} - \frac {3 \sqrt {9 - 4 x^{2}}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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