Integrand size = 15, antiderivative size = 18 \[ \int \frac {\sqrt [4]{x^2+x^4}}{x^4} \, dx=-\frac {2 \left (x^2+x^4\right )^{5/4}}{5 x^5} \]
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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} \[ \int \frac {\sqrt [4]{x^2+x^4}}{x^4} \, dx=-\frac {2 \left (x^4+x^2\right )^{5/4}}{5 x^5} \]
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Rule 2039
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (x^2+x^4\right )^{5/4}}{5 x^5} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt [4]{x^2+x^4}}{x^4} \, dx=-\frac {2 \left (1+x^2\right ) \sqrt [4]{x^2+x^4}}{5 x^3} \]
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Time = 0.79 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72
method | result | size |
meijerg | \(-\frac {2 \left (x^{2}+1\right )^{\frac {5}{4}}}{5 x^{\frac {5}{2}}}\) | \(13\) |
gosper | \(-\frac {2 \left (x^{2}+1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{4}}}{5 x^{3}}\) | \(20\) |
trager | \(-\frac {2 \left (x^{2}+1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{4}}}{5 x^{3}}\) | \(20\) |
pseudoelliptic | \(-\frac {2 \left (x^{2}+1\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{5 x^{3}}\) | \(22\) |
risch | \(-\frac {2 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \left (x^{4}+2 x^{2}+1\right )}{5 x^{3} \left (x^{2}+1\right )}\) | \(34\) |
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none
Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt [4]{x^2+x^4}}{x^4} \, dx=-\frac {2 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}}{5 \, x^{3}} \]
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\[ \int \frac {\sqrt [4]{x^2+x^4}}{x^4} \, dx=\int \frac {\sqrt [4]{x^{2} \left (x^{2} + 1\right )}}{x^{4}}\, dx \]
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none
Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt [4]{x^2+x^4}}{x^4} \, dx=-\frac {2 \, {\left (x^{3} + x\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}}{5 \, x^{\frac {7}{2}}} \]
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none
Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.50 \[ \int \frac {\sqrt [4]{x^2+x^4}}{x^4} \, dx=-\frac {2}{5} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {5}{4}} \]
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Time = 5.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.61 \[ \int \frac {\sqrt [4]{x^2+x^4}}{x^4} \, dx=-\frac {2\,{\left (x^4+x^2\right )}^{1/4}}{5\,x}-\frac {2\,{\left (x^4+x^2\right )}^{1/4}}{5\,x^3} \]
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