Integrand size = 13, antiderivative size = 13 \[ \int x^3 \sqrt [4]{1+x^4} \, dx=\frac {1}{5} \left (1+x^4\right )^{5/4} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \[ \int x^3 \sqrt [4]{1+x^4} \, dx=\frac {1}{5} \left (x^4+1\right )^{5/4} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \left (1+x^4\right )^{5/4} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int x^3 \sqrt [4]{1+x^4} \, dx=\frac {1}{5} \left (1+x^4\right )^{5/4} \]
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Time = 0.76 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77
method | result | size |
gosper | \(\frac {\left (x^{4}+1\right )^{\frac {5}{4}}}{5}\) | \(10\) |
derivativedivides | \(\frac {\left (x^{4}+1\right )^{\frac {5}{4}}}{5}\) | \(10\) |
default | \(\frac {\left (x^{4}+1\right )^{\frac {5}{4}}}{5}\) | \(10\) |
risch | \(\frac {\left (x^{4}+1\right )^{\frac {5}{4}}}{5}\) | \(10\) |
pseudoelliptic | \(\frac {\left (x^{4}+1\right )^{\frac {5}{4}}}{5}\) | \(10\) |
trager | \(\left (\frac {x^{4}}{5}+\frac {1}{5}\right ) \left (x^{4}+1\right )^{\frac {1}{4}}\) | \(16\) |
meijerg | \(\frac {x^{4} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, 1\right ], \left [2\right ], -x^{4}\right )}{4}\) | \(17\) |
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Time = 0.23 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt [4]{1+x^4} \, dx=\frac {1}{5} \, {\left (x^{4} + 1\right )}^{\frac {5}{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (8) = 16\).
Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.69 \[ \int x^3 \sqrt [4]{1+x^4} \, dx=\frac {x^{4} \sqrt [4]{x^{4} + 1}}{5} + \frac {\sqrt [4]{x^{4} + 1}}{5} \]
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Time = 0.17 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt [4]{1+x^4} \, dx=\frac {1}{5} \, {\left (x^{4} + 1\right )}^{\frac {5}{4}} \]
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Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt [4]{1+x^4} \, dx=\frac {1}{5} \, {\left (x^{4} + 1\right )}^{\frac {5}{4}} \]
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Time = 5.14 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt [4]{1+x^4} \, dx=\frac {{\left (x^4+1\right )}^{5/4}}{5} \]
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