Optimal. Leaf size=62 \[ 2 \sqrt {x}+\frac {4 \tan ^{-1}\left (\frac {1-2 \sqrt [4]{x}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {4}{3} \log \left (1+\sqrt [4]{x}\right )-\frac {2}{3} \log \left (1-\sqrt [4]{x}+\sqrt {x}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {1607, 348, 327,
298, 31, 648, 632, 210, 642} \begin {gather*} \frac {4 \text {ArcTan}\left (\frac {1-2 \sqrt [4]{x}}{\sqrt {3}}\right )}{\sqrt {3}}+2 \sqrt {x}+\frac {4}{3} \log \left (\sqrt [4]{x}+1\right )-\frac {2}{3} \log \left (\sqrt {x}-\sqrt [4]{x}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 298
Rule 327
Rule 348
Rule 632
Rule 642
Rule 648
Rule 1607
Rubi steps
\begin {align*} \int \frac {1}{\frac {1}{\sqrt [4]{x}}+\sqrt {x}} \, dx &=\int \frac {\sqrt [4]{x}}{1+x^{3/4}} \, dx\\ &=4 \text {Subst}\left (\int \frac {x^4}{1+x^3} \, dx,x,\sqrt [4]{x}\right )\\ &=2 \sqrt {x}-4 \text {Subst}\left (\int \frac {x}{1+x^3} \, dx,x,\sqrt [4]{x}\right )\\ &=2 \sqrt {x}+\frac {4}{3} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [4]{x}\right )-\frac {4}{3} \text {Subst}\left (\int \frac {1+x}{1-x+x^2} \, dx,x,\sqrt [4]{x}\right )\\ &=2 \sqrt {x}+\frac {4}{3} \log \left (1+\sqrt [4]{x}\right )-\frac {2}{3} \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\sqrt [4]{x}\right )-2 \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [4]{x}\right )\\ &=2 \sqrt {x}+\frac {4}{3} \log \left (1+\sqrt [4]{x}\right )-\frac {2}{3} \log \left (1-\sqrt [4]{x}+\sqrt {x}\right )+4 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [4]{x}\right )\\ &=2 \sqrt {x}+\frac {4 \tan ^{-1}\left (\frac {1-2 \sqrt [4]{x}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {4}{3} \log \left (1+\sqrt [4]{x}\right )-\frac {2}{3} \log \left (1-\sqrt [4]{x}+\sqrt {x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 62, normalized size = 1.00 \begin {gather*} \frac {2}{3} \left (3 \sqrt {x}+2 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [4]{x}}{\sqrt {3}}\right )+2 \log \left (1+\sqrt [4]{x}\right )-\log \left (1-\sqrt [4]{x}+\sqrt {x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 46, normalized size = 0.74
method | result | size |
derivativedivides | \(2 \sqrt {x}+\frac {4 \ln \left (1+x^{\frac {1}{4}}\right )}{3}-\frac {2 \ln \left (1-x^{\frac {1}{4}}+\sqrt {x}\right )}{3}-\frac {4 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{4}}-1\right ) \sqrt {3}}{3}\right )}{3}\) | \(46\) |
default | \(2 \sqrt {x}+\frac {4 \ln \left (1+x^{\frac {1}{4}}\right )}{3}-\frac {2 \ln \left (1-x^{\frac {1}{4}}+\sqrt {x}\right )}{3}-\frac {4 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{4}}-1\right ) \sqrt {3}}{3}\right )}{3}\) | \(46\) |
meijerg | \(2 \sqrt {x}-\frac {4 \sqrt {x}\, \left (-\frac {\ln \left (1+x^{\frac {1}{4}}\right )}{\sqrt {x}}+\frac {\ln \left (1-x^{\frac {1}{4}}+\sqrt {x}\right )}{2 \sqrt {x}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x^{\frac {1}{4}}}{2-x^{\frac {1}{4}}}\right )}{\sqrt {x}}\right )}{3}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 45, normalized size = 0.73 \begin {gather*} -\frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {1}{4}} - 1\right )}\right ) + 2 \, \sqrt {x} - \frac {2}{3} \, \log \left (\sqrt {x} - x^{\frac {1}{4}} + 1\right ) + \frac {4}{3} \, \log \left (x^{\frac {1}{4}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 47, normalized size = 0.76 \begin {gather*} -\frac {4}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} x^{\frac {1}{4}} - \frac {1}{3} \, \sqrt {3}\right ) + 2 \, \sqrt {x} - \frac {2}{3} \, \log \left (\sqrt {x} - x^{\frac {1}{4}} + 1\right ) + \frac {4}{3} \, \log \left (x^{\frac {1}{4}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 68, normalized size = 1.10 \begin {gather*} 2 \sqrt {x} + \frac {4 \log {\left (\sqrt [4]{x} + 1 \right )}}{3} - \frac {2 \log {\left (- 4 \sqrt [4]{x} + 4 \sqrt {x} + 4 \right )}}{3} - \frac {4 \sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [4]{x}}{3} - \frac {\sqrt {3}}{3} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.07, size = 45, normalized size = 0.73 \begin {gather*} -\frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {1}{4}} - 1\right )}\right ) + 2 \, \sqrt {x} - \frac {2}{3} \, \log \left (\sqrt {x} - x^{\frac {1}{4}} + 1\right ) + \frac {4}{3} \, \log \left (x^{\frac {1}{4}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.08, size = 73, normalized size = 1.18 \begin {gather*} \frac {4\,\ln \left (16\,x^{1/4}+16\right )}{3}+\ln \left (9\,{\left (-\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )}^2+16\,x^{1/4}\right )\,\left (-\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )-\ln \left (9\,{\left (\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )}^2+16\,x^{1/4}\right )\,\left (\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )+2\,\sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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