Integrand size = 11, antiderivative size = 39 \[ \int \frac {1}{x^6 \left (5+x^2\right )} \, dx=-\frac {1}{25 x^5}+\frac {1}{75 x^3}-\frac {1}{125 x}-\frac {\arctan \left (\frac {x}{\sqrt {5}}\right )}{125 \sqrt {5}} \]
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Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {331, 209} \[ \int \frac {1}{x^6 \left (5+x^2\right )} \, dx=-\frac {\arctan \left (\frac {x}{\sqrt {5}}\right )}{125 \sqrt {5}}-\frac {1}{25 x^5}+\frac {1}{75 x^3}-\frac {1}{125 x} \]
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Rule 209
Rule 331
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{25 x^5}-\frac {1}{5} \int \frac {1}{x^4 \left (5+x^2\right )} \, dx \\ & = -\frac {1}{25 x^5}+\frac {1}{75 x^3}+\frac {1}{25} \int \frac {1}{x^2 \left (5+x^2\right )} \, dx \\ & = -\frac {1}{25 x^5}+\frac {1}{75 x^3}-\frac {1}{125 x}-\frac {1}{125} \int \frac {1}{5+x^2} \, dx \\ & = -\frac {1}{25 x^5}+\frac {1}{75 x^3}-\frac {1}{125 x}-\frac {\arctan \left (\frac {x}{\sqrt {5}}\right )}{125 \sqrt {5}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^6 \left (5+x^2\right )} \, dx=-\frac {1}{25 x^5}+\frac {1}{75 x^3}-\frac {1}{125 x}-\frac {\arctan \left (\frac {x}{\sqrt {5}}\right )}{125 \sqrt {5}} \]
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Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.74
method | result | size |
default | \(-\frac {1}{25 x^{5}}+\frac {1}{75 x^{3}}-\frac {1}{125 x}-\frac {\arctan \left (\frac {x \sqrt {5}}{5}\right ) \sqrt {5}}{625}\) | \(29\) |
risch | \(\frac {-\frac {1}{125} x^{4}+\frac {1}{75} x^{2}-\frac {1}{25}}{x^{5}}-\frac {\arctan \left (\frac {x \sqrt {5}}{5}\right ) \sqrt {5}}{625}\) | \(30\) |
meijerg | \(\frac {\sqrt {5}\, \left (-\frac {2 \sqrt {5}}{x}+\frac {10 \sqrt {5}}{3 x^{3}}-\frac {10 \sqrt {5}}{x^{5}}-2 \arctan \left (\frac {x \sqrt {5}}{5}\right )\right )}{1250}\) | \(40\) |
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Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^6 \left (5+x^2\right )} \, dx=-\frac {3 \, \sqrt {5} x^{5} \arctan \left (\frac {1}{5} \, \sqrt {5} x\right ) + 15 \, x^{4} - 25 \, x^{2} + 75}{1875 \, x^{5}} \]
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Time = 0.05 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^6 \left (5+x^2\right )} \, dx=- \frac {\sqrt {5} \operatorname {atan}{\left (\frac {\sqrt {5} x}{5} \right )}}{625} + \frac {- 3 x^{4} + 5 x^{2} - 15}{375 x^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^6 \left (5+x^2\right )} \, dx=-\frac {1}{625} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} x\right ) - \frac {3 \, x^{4} - 5 \, x^{2} + 15}{375 \, x^{5}} \]
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Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^6 \left (5+x^2\right )} \, dx=-\frac {1}{625} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} x\right ) - \frac {3 \, x^{4} - 5 \, x^{2} + 15}{375 \, x^{5}} \]
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Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^6 \left (5+x^2\right )} \, dx=-\frac {\frac {x^4}{125}-\frac {x^2}{75}+\frac {1}{25}}{x^5}-\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {\sqrt {5}\,x}{5}\right )}{625} \]
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