Integrand size = 17, antiderivative size = 15 \[ \int \frac {\sqrt {1+\sqrt {x}}}{\sqrt {x}} \, dx=\frac {4}{3} \left (1+\sqrt {x}\right )^{3/2} \]
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Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {267} \[ \int \frac {\sqrt {1+\sqrt {x}}}{\sqrt {x}} \, dx=\frac {4}{3} \left (\sqrt {x}+1\right )^{3/2} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = \frac {4}{3} \left (1+\sqrt {x}\right )^{3/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+\sqrt {x}}}{\sqrt {x}} \, dx=\frac {4}{3} \left (1+\sqrt {x}\right )^{3/2} \]
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Time = 6.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67
method | result | size |
derivativedivides | \(\frac {4 \left (\sqrt {x}+1\right )^{\frac {3}{2}}}{3}\) | \(10\) |
default | \(\frac {4 \left (\sqrt {x}+1\right )^{\frac {3}{2}}}{3}\) | \(10\) |
meijerg | \(-\frac {\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (2 \sqrt {x}+2\right ) \sqrt {\sqrt {x}+1}}{3}}{\sqrt {\pi }}\) | \(31\) |
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none
Time = 0.26 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {1+\sqrt {x}}}{\sqrt {x}} \, dx=\frac {4}{3} \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).
Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.07 \[ \int \frac {\sqrt {1+\sqrt {x}}}{\sqrt {x}} \, dx=\frac {4 \sqrt {x} \sqrt {\sqrt {x} + 1}}{3} + \frac {4 \sqrt {\sqrt {x} + 1}}{3} \]
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none
Time = 0.19 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {1+\sqrt {x}}}{\sqrt {x}} \, dx=\frac {4}{3} \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} \]
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none
Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {1+\sqrt {x}}}{\sqrt {x}} \, dx=\frac {4}{3} \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} \]
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Time = 6.15 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {1+\sqrt {x}}}{\sqrt {x}} \, dx=\frac {4\,{\left (\sqrt {x}+1\right )}^{3/2}}{3} \]
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