Integrand size = 11, antiderivative size = 13 \[ \int x \sqrt {1+x^2} \, dx=\frac {1}{3} \left (1+x^2\right )^{3/2} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {267} \[ \int x \sqrt {1+x^2} \, dx=\frac {1}{3} \left (x^2+1\right )^{3/2} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \left (1+x^2\right )^{3/2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int x \sqrt {1+x^2} \, dx=\frac {1}{3} \left (1+x^2\right )^{3/2} \]
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Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77
method | result | size |
gosper | \(\frac {\left (x^{2}+1\right )^{\frac {3}{2}}}{3}\) | \(10\) |
derivativedivides | \(\frac {\left (x^{2}+1\right )^{\frac {3}{2}}}{3}\) | \(10\) |
default | \(\frac {\left (x^{2}+1\right )^{\frac {3}{2}}}{3}\) | \(10\) |
risch | \(\frac {\left (x^{2}+1\right )^{\frac {3}{2}}}{3}\) | \(10\) |
pseudoelliptic | \(\frac {\left (x^{2}+1\right )^{\frac {3}{2}}}{3}\) | \(10\) |
trager | \(\left (\frac {x^{2}}{3}+\frac {1}{3}\right ) \sqrt {x^{2}+1}\) | \(16\) |
meijerg | \(-\frac {\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (2 x^{2}+2\right ) \sqrt {x^{2}+1}}{3}}{4 \sqrt {\pi }}\) | \(31\) |
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none
Time = 0.24 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x \sqrt {1+x^2} \, dx=\frac {1}{3} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (8) = 16\).
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.69 \[ \int x \sqrt {1+x^2} \, dx=\frac {x^{2} \sqrt {x^{2} + 1}}{3} + \frac {\sqrt {x^{2} + 1}}{3} \]
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none
Time = 0.19 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x \sqrt {1+x^2} \, dx=\frac {1}{3} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} \]
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none
Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x \sqrt {1+x^2} \, dx=\frac {1}{3} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} \]
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Time = 0.03 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x \sqrt {1+x^2} \, dx=\frac {{\left (x^2+1\right )}^{3/2}}{3} \]
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