Integrand size = 13, antiderivative size = 16 \[ \int \frac {1}{x^2 \sqrt {x+x^4}} \, dx=-\frac {2 \sqrt {x+x^4}}{3 x^2} \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2039} \[ \int \frac {1}{x^2 \sqrt {x+x^4}} \, dx=-\frac {2 \sqrt {x^4+x}}{3 x^2} \]
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Rule 2039
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {x+x^4}}{3 x^2} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {x+x^4}} \, dx=-\frac {2 \sqrt {x+x^4}}{3 x^2} \]
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Time = 3.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
default | \(-\frac {2 \sqrt {x^{4}+x}}{3 x^{2}}\) | \(13\) |
trager | \(-\frac {2 \sqrt {x^{4}+x}}{3 x^{2}}\) | \(13\) |
meijerg | \(-\frac {2 \sqrt {x^{3}+1}}{3 x^{\frac {3}{2}}}\) | \(13\) |
elliptic | \(-\frac {2 \sqrt {x^{4}+x}}{3 x^{2}}\) | \(13\) |
pseudoelliptic | \(-\frac {2 \sqrt {x^{4}+x}}{3 x^{2}}\) | \(13\) |
risch | \(-\frac {2 \left (x^{3}+1\right )}{3 x \sqrt {x \left (x^{3}+1\right )}}\) | \(20\) |
gosper | \(-\frac {2 \left (1+x \right ) \left (x^{2}-x +1\right )}{3 x \sqrt {x^{4}+x}}\) | \(24\) |
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Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^2 \sqrt {x+x^4}} \, dx=-\frac {2 \, \sqrt {x^{4} + x}}{3 \, x^{2}} \]
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\[ \int \frac {1}{x^2 \sqrt {x+x^4}} \, dx=\int \frac {1}{x^{2} \sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (12) = 24\).
Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56 \[ \int \frac {1}{x^2 \sqrt {x+x^4}} \, dx=-\frac {2 \, {\left (x^{4} + x\right )}}{3 \, \sqrt {x^{2} - x + 1} \sqrt {x + 1} x^{\frac {5}{2}}} \]
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Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^2 \sqrt {x+x^4}} \, dx=-\frac {2}{3} \, \sqrt {\frac {1}{x^{3}} + 1} \]
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Time = 5.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^2 \sqrt {x+x^4}} \, dx=-\frac {2\,\sqrt {x^4+x}}{3\,x^2} \]
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