Optimal. Leaf size=62 \[ 2 \sqrt {x}+\frac {4}{3} \log \left (\sqrt [4]{x}+1\right )-\frac {2}{3} \log \left (\sqrt {x}-\sqrt [4]{x}+1\right )+\frac {4 \tan ^{-1}\left (\frac {1-2 \sqrt [4]{x}}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.04, 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 = {1593, 341, 321, 292, 31, 634, 618, 204, 628} \[ 2 \sqrt {x}+\frac {4}{3} \log \left (\sqrt [4]{x}+1\right )-\frac {2}{3} \log \left (\sqrt {x}-\sqrt [4]{x}+1\right )+\frac {4 \tan ^{-1}\left (\frac {1-2 \sqrt [4]{x}}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 321
Rule 341
Rule 618
Rule 628
Rule 634
Rule 1593
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 \operatorname {Subst}\left (\int \frac {x^4}{1+x^3} \, dx,x,\sqrt [4]{x}\right )\\ &=2 \sqrt {x}-4 \operatorname {Subst}\left (\int \frac {x}{1+x^3} \, dx,x,\sqrt [4]{x}\right )\\ &=2 \sqrt {x}+\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [4]{x}\right )-\frac {4}{3} \operatorname {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} \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\sqrt [4]{x}\right )-2 \operatorname {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 \operatorname {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 [C] time = 0.01, size = 24, normalized size = 0.39 \[ -2 \sqrt {x} \left (\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-x^{3/4}\right )-1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 47, normalized size = 0.76 \[ -\frac {4}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} x^{\frac {1}{4}} - \frac {1}{3} \, \sqrt {3}\right ) + 2 \, \sqrt {x} - \frac {2}{3} \, \log \left (\sqrt {x} - x^{\frac {1}{4}} + 1\right ) + \frac {4}{3} \, \log \left (x^{\frac {1}{4}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 45, normalized size = 0.73 \[ -\frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {1}{4}} - 1\right )}\right ) + 2 \, \sqrt {x} - \frac {2}{3} \, \log \left (\sqrt {x} - x^{\frac {1}{4}} + 1\right ) + \frac {4}{3} \, \log \left (x^{\frac {1}{4}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 46, normalized size = 0.74 \[ -\frac {4 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{4}}-1\right ) \sqrt {3}}{3}\right )}{3}+\frac {4 \ln \left (x^{\frac {1}{4}}+1\right )}{3}-\frac {2 \ln \left (\sqrt {x}-x^{\frac {1}{4}}+1\right )}{3}+2 \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 45, normalized size = 0.73 \[ -\frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {1}{4}} - 1\right )}\right ) + 2 \, \sqrt {x} - \frac {2}{3} \, \log \left (\sqrt {x} - x^{\frac {1}{4}} + 1\right ) + \frac {4}{3} \, \log \left (x^{\frac {1}{4}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 73, normalized size = 1.18 \[ \frac {4\,\ln \left (16\,x^{1/4}+16\right )}{3}+\ln \left (9\,{\left (-\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )}^2+16\,x^{1/4}\right )\,\left (-\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )-\ln \left (9\,{\left (\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )}^2+16\,x^{1/4}\right )\,\left (\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )+2\,\sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.65, size = 68, normalized size = 1.10 \[ 2 \sqrt {x} + \frac {4 \log {\left (\sqrt [4]{x} + 1 \right )}}{3} - \frac {2 \log {\left (- 4 \sqrt [4]{x} + 4 \sqrt {x} + 4 \right )}}{3} - \frac {4 \sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [4]{x}}{3} - \frac {\sqrt {3}}{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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