Integrand size = 11, antiderivative size = 13 \[ \int x \sqrt [4]{1+x^2} \, dx=\frac {2}{5} \left (1+x^2\right )^{5/4} \]
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Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {267} \[ \int x \sqrt [4]{1+x^2} \, dx=\frac {2}{5} \left (x^2+1\right )^{5/4} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = \frac {2}{5} \left (1+x^2\right )^{5/4} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int x \sqrt [4]{1+x^2} \, dx=\frac {2}{5} \left (1+x^2\right )^{5/4} \]
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Time = 0.76 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77
method | result | size |
gosper | \(\frac {2 \left (x^{2}+1\right )^{\frac {5}{4}}}{5}\) | \(10\) |
derivativedivides | \(\frac {2 \left (x^{2}+1\right )^{\frac {5}{4}}}{5}\) | \(10\) |
default | \(\frac {2 \left (x^{2}+1\right )^{\frac {5}{4}}}{5}\) | \(10\) |
risch | \(\frac {2 \left (x^{2}+1\right )^{\frac {5}{4}}}{5}\) | \(10\) |
pseudoelliptic | \(\frac {2 \left (x^{2}+1\right )^{\frac {5}{4}}}{5}\) | \(10\) |
trager | \(\left (\frac {2}{5}+\frac {2 x^{2}}{5}\right ) \left (x^{2}+1\right )^{\frac {1}{4}}\) | \(16\) |
meijerg | \(\frac {x^{2} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, 1\right ], \left [2\right ], -x^{2}\right )}{2}\) | \(17\) |
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none
Time = 0.22 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x \sqrt [4]{1+x^2} \, dx=\frac {2}{5} \, {\left (x^{2} + 1\right )}^{\frac {5}{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (10) = 20\).
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 2.00 \[ \int x \sqrt [4]{1+x^2} \, dx=\frac {2 x^{2} \sqrt [4]{x^{2} + 1}}{5} + \frac {2 \sqrt [4]{x^{2} + 1}}{5} \]
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none
Time = 0.21 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x \sqrt [4]{1+x^2} \, dx=\frac {2}{5} \, {\left (x^{2} + 1\right )}^{\frac {5}{4}} \]
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none
Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x \sqrt [4]{1+x^2} \, dx=\frac {2}{5} \, {\left (x^{2} + 1\right )}^{\frac {5}{4}} \]
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Time = 5.03 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x \sqrt [4]{1+x^2} \, dx=\frac {2\,{\left (x^2+1\right )}^{5/4}}{5} \]
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