Integrand size = 13, antiderivative size = 68 \[ \int \sqrt {\sqrt {-1+x}+x} \, dx=-\frac {1}{4} \left (1+2 \sqrt {-1+x}\right ) \sqrt {\sqrt {-1+x}+x}+\frac {2}{3} \left (\sqrt {-1+x}+x\right )^{3/2}-\frac {3}{8} \text {arcsinh}\left (\frac {1+2 \sqrt {-1+x}}{\sqrt {3}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {654, 626, 633, 221} \[ \int \sqrt {\sqrt {-1+x}+x} \, dx=-\frac {3}{8} \text {arcsinh}\left (\frac {2 \sqrt {x-1}+1}{\sqrt {3}}\right )+\frac {2}{3} \left (x+\sqrt {x-1}\right )^{3/2}-\frac {1}{4} \left (2 \sqrt {x-1}+1\right ) \sqrt {x+\sqrt {x-1}} \]
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Rule 221
Rule 626
Rule 633
Rule 654
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x \sqrt {1+x+x^2} \, dx,x,\sqrt {-1+x}\right ) \\ & = \frac {2}{3} \left (\sqrt {-1+x}+x\right )^{3/2}-\text {Subst}\left (\int \sqrt {1+x+x^2} \, dx,x,\sqrt {-1+x}\right ) \\ & = -\frac {1}{4} \left (1+2 \sqrt {-1+x}\right ) \sqrt {\sqrt {-1+x}+x}+\frac {2}{3} \left (\sqrt {-1+x}+x\right )^{3/2}-\frac {3}{8} \text {Subst}\left (\int \frac {1}{\sqrt {1+x+x^2}} \, dx,x,\sqrt {-1+x}\right ) \\ & = -\frac {1}{4} \left (1+2 \sqrt {-1+x}\right ) \sqrt {\sqrt {-1+x}+x}+\frac {2}{3} \left (\sqrt {-1+x}+x\right )^{3/2}-\frac {1}{8} \sqrt {3} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 \sqrt {-1+x}\right ) \\ & = -\frac {1}{4} \left (1+2 \sqrt {-1+x}\right ) \sqrt {\sqrt {-1+x}+x}+\frac {2}{3} \left (\sqrt {-1+x}+x\right )^{3/2}-\frac {3}{8} \sinh ^{-1}\left (\frac {1+2 \sqrt {-1+x}}{\sqrt {3}}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.96 \[ \int \sqrt {\sqrt {-1+x}+x} \, dx=\frac {1}{12} \left (5+2 \sqrt {-1+x}+8 (-1+x)\right ) \sqrt {\sqrt {-1+x}+x}+\frac {3}{8} \log \left (-1-2 \sqrt {-1+x}+2 \sqrt {\sqrt {-1+x}+x}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(\frac {2 \left (x +\sqrt {x -1}\right )^{\frac {3}{2}}}{3}-\frac {\left (1+2 \sqrt {x -1}\right ) \sqrt {x +\sqrt {x -1}}}{4}-\frac {3 \,\operatorname {arcsinh}\left (\frac {2 \sqrt {3}\, \left (\sqrt {x -1}+\frac {1}{2}\right )}{3}\right )}{8}\) | \(48\) |
default | \(\frac {2 \left (x +\sqrt {x -1}\right )^{\frac {3}{2}}}{3}-\frac {\left (1+2 \sqrt {x -1}\right ) \sqrt {x +\sqrt {x -1}}}{4}-\frac {3 \,\operatorname {arcsinh}\left (\frac {2 \sqrt {3}\, \left (\sqrt {x -1}+\frac {1}{2}\right )}{3}\right )}{8}\) | \(48\) |
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Time = 0.61 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.87 \[ \int \sqrt {\sqrt {-1+x}+x} \, dx=\frac {1}{12} \, {\left (8 \, x + 2 \, \sqrt {x - 1} - 3\right )} \sqrt {x + \sqrt {x - 1}} + \frac {3}{16} \, \log \left (-4 \, \sqrt {x + \sqrt {x - 1}} {\left (2 \, \sqrt {x - 1} + 1\right )} + 8 \, x + 8 \, \sqrt {x - 1} - 3\right ) \]
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Time = 0.33 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.75 \[ \int \sqrt {\sqrt {-1+x}+x} \, dx=2 \sqrt {x + \sqrt {x - 1}} \left (\frac {x}{3} + \frac {\sqrt {x - 1}}{12} - \frac {1}{8}\right ) - \frac {3 \operatorname {asinh}{\left (\frac {2 \sqrt {3} \left (\sqrt {x - 1} + \frac {1}{2}\right )}{3} \right )}}{8} \]
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\[ \int \sqrt {\sqrt {-1+x}+x} \, dx=\int { \sqrt {x + \sqrt {x - 1}} \,d x } \]
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Time = 0.36 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78 \[ \int \sqrt {\sqrt {-1+x}+x} \, dx=\frac {1}{12} \, {\left (2 \, \sqrt {x - 1} {\left (4 \, \sqrt {x - 1} + 1\right )} + 5\right )} \sqrt {x + \sqrt {x - 1}} + \frac {3}{8} \, \log \left (2 \, \sqrt {x + \sqrt {x - 1}} - 2 \, \sqrt {x - 1} - 1\right ) \]
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Timed out. \[ \int \sqrt {\sqrt {-1+x}+x} \, dx=\int \sqrt {x+\sqrt {x-1}} \,d x \]
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