Integrand size = 17, antiderivative size = 21 \[ \int \frac {1}{-\sqrt {-1+x}+\sqrt {x}} \, dx=\frac {2}{3} (-1+x)^{3/2}+\frac {2 x^{3/2}}{3} \]
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Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2131, 30, 32} \[ \int \frac {1}{-\sqrt {-1+x}+\sqrt {x}} \, dx=\frac {2 x^{3/2}}{3}+\frac {2}{3} (x-1)^{3/2} \]
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Rule 30
Rule 32
Rule 2131
Rubi steps \begin{align*} \text {integral}& = \int \sqrt {-1+x} \, dx+\int \sqrt {x} \, dx \\ & = \frac {2}{3} (-1+x)^{3/2}+\frac {2 x^{3/2}}{3} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {1}{-\sqrt {-1+x}+\sqrt {x}} \, dx=\frac {2}{3} \left ((-1+x)^{3/2}+x^{3/2}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67
method | result | size |
default | \(\frac {2 \left (-1+x \right )^{\frac {3}{2}}}{3}+\frac {2 x^{\frac {3}{2}}}{3}\) | \(14\) |
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none
Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \frac {1}{-\sqrt {-1+x}+\sqrt {x}} \, dx=\frac {2}{3} \, {\left (x - 1\right )}^{\frac {3}{2}} + \frac {2}{3} \, x^{\frac {3}{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (17) = 34\).
Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.00 \[ \int \frac {1}{-\sqrt {-1+x}+\sqrt {x}} \, dx=\frac {2 \sqrt {x} \sqrt {x - 1}}{- 3 \sqrt {x} + 3 \sqrt {x - 1}} - \frac {4 x}{- 3 \sqrt {x} + 3 \sqrt {x - 1}} + \frac {2}{- 3 \sqrt {x} + 3 \sqrt {x - 1}} \]
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\[ \int \frac {1}{-\sqrt {-1+x}+\sqrt {x}} \, dx=\int { -\frac {1}{\sqrt {x - 1} - \sqrt {x}} \,d x } \]
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none
Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \frac {1}{-\sqrt {-1+x}+\sqrt {x}} \, dx=\frac {2}{3} \, {\left (x - 1\right )}^{\frac {3}{2}} + \frac {2}{3} \, x^{\frac {3}{2}} \]
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Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {1}{-\sqrt {-1+x}+\sqrt {x}} \, dx=\frac {2\,x\,\sqrt {x-1}}{3}-\frac {2\,\sqrt {x-1}}{3}+\frac {2\,x^{3/2}}{3} \]
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