Integrand size = 15, antiderivative size = 27 \[ \int \frac {1}{-\sqrt [4]{x}+\sqrt {x}} \, dx=4 \sqrt [4]{x}+2 \sqrt {x}+4 \log \left (1-\sqrt [4]{x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1607, 272, 45} \[ \int \frac {1}{-\sqrt [4]{x}+\sqrt {x}} \, dx=2 \sqrt {x}+4 \sqrt [4]{x}+4 \log \left (1-\sqrt [4]{x}\right ) \]
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Rule 45
Rule 272
Rule 1607
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (-1+\sqrt [4]{x}\right ) \sqrt [4]{x}} \, dx \\ & = 4 \text {Subst}\left (\int \frac {x^2}{-1+x} \, dx,x,\sqrt [4]{x}\right ) \\ & = 4 \text {Subst}\left (\int \left (1+\frac {1}{-1+x}+x\right ) \, dx,x,\sqrt [4]{x}\right ) \\ & = 4 \sqrt [4]{x}+2 \sqrt {x}+4 \log \left (1-\sqrt [4]{x}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {1}{-\sqrt [4]{x}+\sqrt {x}} \, dx=2 \left (2+\sqrt [4]{x}\right ) \sqrt [4]{x}+4 \log \left (-1+\sqrt [4]{x}\right ) \]
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Time = 0.96 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(2 \sqrt {x}+4 x^{\frac {1}{4}}+4 \ln \left (x^{\frac {1}{4}}-1\right )\) | \(20\) |
default | \(2 \sqrt {x}+4 x^{\frac {1}{4}}+4 \ln \left (x^{\frac {1}{4}}-1\right )\) | \(20\) |
meijerg | \(\frac {2 x^{\frac {1}{4}} \left (6+3 x^{\frac {1}{4}}\right )}{3}+4 \ln \left (1-x^{\frac {1}{4}}\right )\) | \(24\) |
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Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {1}{-\sqrt [4]{x}+\sqrt {x}} \, dx=2 \, \sqrt {x} + 4 \, x^{\frac {1}{4}} + 4 \, \log \left (x^{\frac {1}{4}} - 1\right ) \]
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Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {1}{-\sqrt [4]{x}+\sqrt {x}} \, dx=4 \sqrt [4]{x} + 2 \sqrt {x} + 4 \log {\left (\sqrt [4]{x} - 1 \right )} \]
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Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {1}{-\sqrt [4]{x}+\sqrt {x}} \, dx=2 \, \sqrt {x} + 4 \, x^{\frac {1}{4}} + 4 \, \log \left (x^{\frac {1}{4}} - 1\right ) \]
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Time = 0.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {1}{-\sqrt [4]{x}+\sqrt {x}} \, dx=2 \, \sqrt {x} + 4 \, x^{\frac {1}{4}} + 4 \, \log \left ({\left | x^{\frac {1}{4}} - 1 \right |}\right ) \]
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Time = 16.55 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {1}{-\sqrt [4]{x}+\sqrt {x}} \, dx=4\,\ln \left (x^{1/4}-1\right )+2\,\sqrt {x}+4\,x^{1/4} \]
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