Integrand size = 15, antiderivative size = 31 \[ \int x^3 \sqrt {4-9 x^2} \, dx=-\frac {4}{243} \left (4-9 x^2\right )^{3/2}+\frac {1}{405} \left (4-9 x^2\right )^{5/2} \]
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Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int x^3 \sqrt {4-9 x^2} \, dx=\frac {1}{405} \left (4-9 x^2\right )^{5/2}-\frac {4}{243} \left (4-9 x^2\right )^{3/2} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int x^3 \sqrt {4-9 x^2} \, dx=\frac {\left (-8-27 x^2\right ) \left (4-9 x^2\right )^{3/2}}{1215} \]
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Time = 0.15 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.61
method | result | size |
pseudoelliptic | \(-\frac {\left (27 x^{2}+8\right ) \left (-9 x^{2}+4\right )^{\frac {3}{2}}}{1215}\) | \(19\) |
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 | \(-\frac {8 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-\frac {9 x^{2}}{4}\right )^{\frac {3}{2}} \left (\frac {27 x^{2}}{4}+2\right )}{15}\right )}{81 \sqrt {\pi }}\) | \(33\) |
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Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int x^3 \sqrt {4-9 x^2} \, dx=\frac {1}{1215} \, {\left (243 \, x^{4} - 36 \, x^{2} - 32\right )} \sqrt {-9 \, x^{2} + 4} \]
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Time = 0.15 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int x^3 \sqrt {4-9 x^2} \, dx=\frac {x^{4} \sqrt {4 - 9 x^{2}}}{5} - \frac {4 x^{2} \sqrt {4 - 9 x^{2}}}{135} - \frac {32 \sqrt {4 - 9 x^{2}}}{1215} \]
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Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int x^3 \sqrt {4-9 x^2} \, dx=-\frac {1}{45} \, {\left (-9 \, x^{2} + 4\right )}^{\frac {3}{2}} x^{2} - \frac {8}{1215} \, {\left (-9 \, x^{2} + 4\right )}^{\frac {3}{2}} \]
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Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int x^3 \sqrt {4-9 x^2} \, dx=\frac {1}{405} \, {\left (9 \, x^{2} - 4\right )}^{2} \sqrt {-9 \, x^{2} + 4} - \frac {4}{243} \, {\left (-9 \, x^{2} + 4\right )}^{\frac {3}{2}} \]
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Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int x^3 \sqrt {4-9 x^2} \, dx=-\frac {\sqrt {\frac {4}{9}-x^2}\,\left (-\frac {9\,x^4}{5}+\frac {4\,x^2}{15}+\frac {32}{135}\right )}{3} \]
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