Integrand size = 15, antiderivative size = 18 \[ \int \frac {\sqrt [4]{-x+x^4}}{x^5} \, dx=\frac {4 \left (-x+x^4\right )^{5/4}}{15 x^5} \]
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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} \[ \int \frac {\sqrt [4]{-x+x^4}}{x^5} \, dx=\frac {4 \left (x^4-x\right )^{5/4}}{15 x^5} \]
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Rule 2039
Rubi steps \begin{align*} \text {integral}& = \frac {4 \left (-x+x^4\right )^{5/4}}{15 x^5} \\ \end{align*}
Time = 9.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{-x+x^4}}{x^5} \, dx=\frac {4 \left (x \left (-1+x^3\right )\right )^{5/4}}{15 x^5} \]
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Time = 0.80 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11
method | result | size |
trager | \(\frac {4 \left (x^{3}-1\right ) \left (x^{4}-x \right )^{\frac {1}{4}}}{15 x^{4}}\) | \(20\) |
pseudoelliptic | \(\frac {4 \left (x^{3}-1\right ) \left (x^{4}-x \right )^{\frac {1}{4}}}{15 x^{4}}\) | \(20\) |
gosper | \(\frac {4 \left (x -1\right ) \left (x^{2}+x +1\right ) \left (x^{4}-x \right )^{\frac {1}{4}}}{15 x^{4}}\) | \(24\) |
risch | \(\frac {4 {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (x^{6}-2 x^{3}+1\right )}{15 x^{4} \left (x^{3}-1\right )}\) | \(32\) |
meijerg | \(-\frac {4 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} \left (-x^{3}+1\right )^{\frac {5}{4}}}{15 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} x^{\frac {15}{4}}}\) | \(33\) |
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none
Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt [4]{-x+x^4}}{x^5} \, dx=\frac {4 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} {\left (x^{3} - 1\right )}}{15 \, x^{4}} \]
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\[ \int \frac {\sqrt [4]{-x+x^4}}{x^5} \, dx=\int \frac {\sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}}{x^{5}}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt [4]{-x+x^4}}{x^5} \, dx=\frac {4 \, {\left (x^{4} - x\right )} {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}}}{15 \, x^{\frac {19}{4}}} \]
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none
Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt [4]{-x+x^4}}{x^5} \, dx=\frac {4}{15} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {5}{4}} \]
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Time = 5.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {\sqrt [4]{-x+x^4}}{x^5} \, dx=-\frac {4\,{\left (x^4-x\right )}^{1/4}-4\,x^3\,{\left (x^4-x\right )}^{1/4}}{15\,x^4} \]
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