Optimal. Leaf size=31 \[ -\frac {4}{243} \left (4-9 x^2\right )^{3/2}+\frac {1}{405} \left (4-9 x^2\right )^{5/2} \]
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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 = {272, 45}
\begin {gather*} \frac {1}{405} \left (4-9 x^2\right )^{5/2}-\frac {4}{243} \left (4-9 x^2\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^3 \sqrt {4-9 x^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \sqrt {4-9 x} x \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {4}{9} \sqrt {4-9 x}-\frac {1}{9} (4-9 x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=-\frac {4}{243} \left (4-9 x^2\right )^{3/2}+\frac {1}{405} \left (4-9 x^2\right )^{5/2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 22, normalized size = 0.71 \begin {gather*} \frac {\left (-8-27 x^2\right ) \left (4-9 x^2\right )^{3/2}}{1215} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.06, size = 23, normalized size = 0.74 \begin {gather*} \frac {\left (-32-36 x^2+243 x^4\right ) \sqrt {4-9 x^2}}{1215} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 27, normalized size = 0.87
method | result | size |
trager | \(\left (\frac {1}{5} x^{4}-\frac {4}{135} x^{2}-\frac {32}{1215}\right ) \sqrt {-9 x^{2}+4}\) | \(23\) |
default | \(-\frac {x^{2} \left (-9 x^{2}+4\right )^{\frac {3}{2}}}{45}-\frac {8 \left (-9 x^{2}+4\right )^{\frac {3}{2}}}{1215}\) | \(27\) |
gosper | \(\frac {\left (-2+3 x \right ) \left (2+3 x \right ) \left (27 x^{2}+8\right ) \sqrt {-9 x^{2}+4}}{1215}\) | \(29\) |
risch | \(-\frac {\left (243 x^{4}-36 x^{2}-32\right ) \left (9 x^{2}-4\right )}{1215 \sqrt {-9 x^{2}+4}}\) | \(31\) |
meijerg | \(-\frac {8 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-\frac {9 x^{2}}{4}\right )^{\frac {3}{2}} \left (\frac {27 x^{2}}{4}+2\right )}{15}\right )}{81 \sqrt {\pi }}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 26, normalized size = 0.84 \begin {gather*} -\frac {1}{45} \, {\left (-9 \, x^{2} + 4\right )}^{\frac {3}{2}} x^{2} - \frac {8}{1215} \, {\left (-9 \, x^{2} + 4\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 23, normalized size = 0.74 \begin {gather*} \frac {1}{1215} \, {\left (243 \, x^{4} - 36 \, x^{2} - 32\right )} \sqrt {-9 \, x^{2} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 44, normalized size = 1.42 \begin {gather*} \frac {x^{4} \sqrt {4 - 9 x^{2}}}{5} - \frac {4 x^{2} \sqrt {4 - 9 x^{2}}}{135} - \frac {32 \sqrt {4 - 9 x^{2}}}{1215} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 54, normalized size = 1.74 \begin {gather*} \frac {\frac {1}{5} \sqrt {-9 x^{2}+4} \left (-9 x^{2}+4\right )^{2}-\frac {4}{3} \sqrt {-9 x^{2}+4} \left (-9 x^{2}+4\right )}{81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 23, normalized size = 0.74 \begin {gather*} -\frac {\sqrt {\frac {4}{9}-x^2}\,\left (-\frac {9\,x^4}{5}+\frac {4\,x^2}{15}+\frac {32}{135}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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