Integrand size = 13, antiderivative size = 13 \[ \int x^3 \sqrt {5+x^4} \, dx=\frac {1}{6} \left (5+x^4\right )^{3/2} \]
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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} \[ \int x^3 \sqrt {5+x^4} \, dx=\frac {1}{6} \left (x^4+5\right )^{3/2} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \left (5+x^4\right )^{3/2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int x^3 \sqrt {5+x^4} \, dx=\frac {1}{6} \left (5+x^4\right )^{3/2} \]
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Time = 4.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77
method | result | size |
gosper | \(\frac {\left (x^{4}+5\right )^{\frac {3}{2}}}{6}\) | \(10\) |
derivativedivides | \(\frac {\left (x^{4}+5\right )^{\frac {3}{2}}}{6}\) | \(10\) |
default | \(\frac {\left (x^{4}+5\right )^{\frac {3}{2}}}{6}\) | \(10\) |
risch | \(\frac {\left (x^{4}+5\right )^{\frac {3}{2}}}{6}\) | \(10\) |
pseudoelliptic | \(\frac {\left (x^{4}+5\right )^{\frac {3}{2}}}{6}\) | \(10\) |
trager | \(\left (\frac {x^{4}}{6}+\frac {5}{6}\right ) \sqrt {x^{4}+5}\) | \(16\) |
meijerg | \(-\frac {5 \sqrt {5}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (2+\frac {2 x^{4}}{5}\right ) \sqrt {1+\frac {x^{4}}{5}}}{3}\right )}{8 \sqrt {\pi }}\) | \(36\) |
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none
Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt {5+x^4} \, dx=\frac {1}{6} \, {\left (x^{4} + 5\right )}^{\frac {3}{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (8) = 16\).
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.85 \[ \int x^3 \sqrt {5+x^4} \, dx=\frac {x^{4} \sqrt {x^{4} + 5}}{6} + \frac {5 \sqrt {x^{4} + 5}}{6} \]
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none
Time = 0.21 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt {5+x^4} \, dx=\frac {1}{6} \, {\left (x^{4} + 5\right )}^{\frac {3}{2}} \]
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Time = 0.30 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt {5+x^4} \, dx=\frac {1}{6} \, {\left (x^{4} + 5\right )}^{\frac {3}{2}} \]
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Time = 5.57 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt {5+x^4} \, dx=\frac {{\left (x^4+5\right )}^{3/2}}{6} \]
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