Integrand size = 15, antiderivative size = 25 \[ \int \frac {1}{x^5 \sqrt {-x+x^4}} \, dx=\frac {2 \left (1+2 x^3\right ) \sqrt {-x+x^4}}{9 x^5} \]
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Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2041, 2039} \[ \int \frac {1}{x^5 \sqrt {-x+x^4}} \, dx=\frac {2 \sqrt {x^4-x}}{9 x^5}+\frac {4 \sqrt {x^4-x}}{9 x^2} \]
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Rule 2039
Rule 2041
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {-x+x^4}}{9 x^5}+\frac {2}{3} \int \frac {1}{x^2 \sqrt {-x+x^4}} \, dx \\ & = \frac {2 \sqrt {-x+x^4}}{9 x^5}+\frac {4 \sqrt {-x+x^4}}{9 x^2} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^5 \sqrt {-x+x^4}} \, dx=\frac {2 \sqrt {x \left (-1+x^3\right )} \left (1+2 x^3\right )}{9 x^5} \]
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Time = 2.90 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
method | result | size |
trager | \(\frac {2 \left (2 x^{3}+1\right ) \sqrt {x^{4}-x}}{9 x^{5}}\) | \(22\) |
pseudoelliptic | \(\frac {2 \left (2 x^{3}+1\right ) \sqrt {x^{4}-x}}{9 x^{5}}\) | \(22\) |
risch | \(\frac {-\frac {2}{9} x^{3}-\frac {2}{9}+\frac {4}{9} x^{6}}{x^{4} \sqrt {x \left (x^{3}-1\right )}}\) | \(27\) |
default | \(\frac {2 \sqrt {x^{4}-x}}{9 x^{5}}+\frac {4 \sqrt {x^{4}-x}}{9 x^{2}}\) | \(30\) |
elliptic | \(\frac {2 \sqrt {x^{4}-x}}{9 x^{5}}+\frac {4 \sqrt {x^{4}-x}}{9 x^{2}}\) | \(30\) |
gosper | \(\frac {2 \left (x -1\right ) \left (x^{2}+x +1\right ) \left (2 x^{3}+1\right )}{9 x^{4} \sqrt {x^{4}-x}}\) | \(31\) |
meijerg | \(-\frac {2 \sqrt {-\operatorname {signum}\left (x^{3}-1\right )}\, \left (2 x^{3}+1\right ) \sqrt {-x^{3}+1}}{9 \sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, x^{\frac {9}{2}}}\) | \(40\) |
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none
Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^5 \sqrt {-x+x^4}} \, dx=\frac {2 \, \sqrt {x^{4} - x} {\left (2 \, x^{3} + 1\right )}}{9 \, x^{5}} \]
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\[ \int \frac {1}{x^5 \sqrt {-x+x^4}} \, dx=\int \frac {1}{x^{5} \sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )}}\, dx \]
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none
Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {1}{x^5 \sqrt {-x+x^4}} \, dx=\frac {2 \, {\left (2 \, x^{7} - x^{4} - x\right )}}{9 \, \sqrt {x^{2} + x + 1} \sqrt {x - 1} x^{\frac {11}{2}}} \]
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none
Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^5 \sqrt {-x+x^4}} \, dx=-\frac {2}{9} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {3}{2}} + \frac {2}{3} \, \sqrt {-\frac {1}{x^{3}} + 1} \]
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Time = 5.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^5 \sqrt {-x+x^4}} \, dx=\frac {2\,\sqrt {x^4-x}\,\left (2\,x^3+1\right )}{9\,x^5} \]
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