Integrand size = 15, antiderivative size = 18 \[ \int \frac {\sqrt {-x+x^4}}{x^6} \, dx=\frac {2 \left (-x+x^4\right )^{3/2}}{9 x^6} \]
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Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2039} \[ \int \frac {\sqrt {-x+x^4}}{x^6} \, dx=\frac {2 \left (x^4-x\right )^{3/2}}{9 x^6} \]
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Rule 2039
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (-x+x^4\right )^{3/2}}{9 x^6} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-x+x^4}}{x^6} \, dx=\frac {2 \left (x \left (-1+x^3\right )\right )^{3/2}}{9 x^6} \]
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Time = 2.60 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11
method | result | size |
trager | \(\frac {2 \left (x^{3}-1\right ) \sqrt {x^{4}-x}}{9 x^{5}}\) | \(20\) |
pseudoelliptic | \(\frac {2 \left (x^{3}-1\right ) \sqrt {x^{4}-x}}{9 x^{5}}\) | \(20\) |
gosper | \(\frac {2 \left (x -1\right ) \left (x^{2}+x +1\right ) \sqrt {x^{4}-x}}{9 x^{5}}\) | \(24\) |
risch | \(\frac {\frac {2}{9} x^{6}-\frac {4}{9} x^{3}+\frac {2}{9}}{x^{4} \sqrt {x \left (x^{3}-1\right )}}\) | \(25\) |
default | \(-\frac {2 \sqrt {x^{4}-x}}{9 x^{5}}+\frac {2 \sqrt {x^{4}-x}}{9 x^{2}}\) | \(30\) |
elliptic | \(-\frac {2 \sqrt {x^{4}-x}}{9 x^{5}}+\frac {2 \sqrt {x^{4}-x}}{9 x^{2}}\) | \(30\) |
meijerg | \(-\frac {2 \sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, \left (-x^{3}+1\right )^{\frac {3}{2}}}{9 \sqrt {-\operatorname {signum}\left (x^{3}-1\right )}\, x^{\frac {9}{2}}}\) | \(33\) |
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none
Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {-x+x^4}}{x^6} \, dx=\frac {2 \, \sqrt {x^{4} - x} {\left (x^{3} - 1\right )}}{9 \, x^{5}} \]
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\[ \int \frac {\sqrt {-x+x^4}}{x^6} \, dx=\int \frac {\sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )}}{x^{6}}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt {-x+x^4}}{x^6} \, dx=\frac {2 \, {\left (x^{4} - x\right )} \sqrt {x^{2} + x + 1} \sqrt {x - 1}}{9 \, x^{\frac {11}{2}}} \]
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none
Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {-x+x^4}}{x^6} \, dx=\frac {2}{9} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {3}{2}} \]
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Time = 5.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {-x+x^4}}{x^6} \, dx=\frac {2\,\sqrt {x^4-x}\,\left (x^3-1\right )}{9\,x^5} \]
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