Integrand size = 15, antiderivative size = 31 \[ \int \frac {x^5}{\sqrt {1-x^3}} \, dx=-\frac {2}{3} \sqrt {1-x^3}+\frac {2}{9} \left (1-x^3\right )^{3/2} \]
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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} \[ \int \frac {x^5}{\sqrt {1-x^3}} \, dx=\frac {2}{9} \left (1-x^3\right )^{3/2}-\frac {2 \sqrt {1-x^3}}{3} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {x}{\sqrt {1-x}} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (\frac {1}{\sqrt {1-x}}-\sqrt {1-x}\right ) \, dx,x,x^3\right ) \\ & = -\frac {2}{3} \sqrt {1-x^3}+\frac {2}{9} \left (1-x^3\right )^{3/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {x^5}{\sqrt {1-x^3}} \, dx=-\frac {2}{9} \sqrt {1-x^3} \left (2+x^3\right ) \]
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Time = 3.82 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55
method | result | size |
pseudoelliptic | \(-\frac {2 \left (x^{3}+2\right ) \sqrt {-x^{3}+1}}{9}\) | \(17\) |
trager | \(\left (-\frac {2 x^{3}}{9}-\frac {4}{9}\right ) \sqrt {-x^{3}+1}\) | \(18\) |
risch | \(\frac {2 \left (x^{3}+2\right ) \left (x^{3}-1\right )}{9 \sqrt {-x^{3}+1}}\) | \(22\) |
gosper | \(\frac {2 \left (-1+x \right ) \left (x^{2}+x +1\right ) \left (x^{3}+2\right )}{9 \sqrt {-x^{3}+1}}\) | \(26\) |
default | \(-\frac {2 x^{3} \sqrt {-x^{3}+1}}{9}-\frac {4 \sqrt {-x^{3}+1}}{9}\) | \(27\) |
elliptic | \(-\frac {2 x^{3} \sqrt {-x^{3}+1}}{9}-\frac {4 \sqrt {-x^{3}+1}}{9}\) | \(27\) |
meijerg | \(\frac {\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 x^{3}+8\right ) \sqrt {-x^{3}+1}}{6}}{3 \sqrt {\pi }}\) | \(33\) |
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Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.52 \[ \int \frac {x^5}{\sqrt {1-x^3}} \, dx=-\frac {2}{9} \, {\left (x^{3} + 2\right )} \sqrt {-x^{3} + 1} \]
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Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {x^5}{\sqrt {1-x^3}} \, dx=- \frac {2 x^{3} \sqrt {1 - x^{3}}}{9} - \frac {4 \sqrt {1 - x^{3}}}{9} \]
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Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {x^5}{\sqrt {1-x^3}} \, dx=\frac {2}{9} \, {\left (-x^{3} + 1\right )}^{\frac {3}{2}} - \frac {2}{3} \, \sqrt {-x^{3} + 1} \]
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Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {x^5}{\sqrt {1-x^3}} \, dx=\frac {2}{9} \, {\left (-x^{3} + 1\right )}^{\frac {3}{2}} - \frac {2}{3} \, \sqrt {-x^{3} + 1} \]
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Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.52 \[ \int \frac {x^5}{\sqrt {1-x^3}} \, dx=-\frac {2\,\sqrt {1-x^3}\,\left (x^3+2\right )}{9} \]
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