Optimal. Leaf size=108 \[ 2 \sqrt {x}-\frac {3 \log \left (\sqrt [3]{x}-\sqrt {2} \sqrt [6]{x}+1\right )}{2 \sqrt {2}}+\frac {3 \log \left (\sqrt [3]{x}+\sqrt {2} \sqrt [6]{x}+1\right )}{2 \sqrt {2}}+\frac {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}-\frac {3 \tan ^{-1}\left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.08, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {1584, 341, 321, 329, 297, 1162, 617, 204, 1165, 628} \[ 2 \sqrt {x}-\frac {3 \log \left (\sqrt [3]{x}-\sqrt {2} \sqrt [6]{x}+1\right )}{2 \sqrt {2}}+\frac {3 \log \left (\sqrt [3]{x}+\sqrt {2} \sqrt [6]{x}+1\right )}{2 \sqrt {2}}+\frac {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}-\frac {3 \tan ^{-1}\left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 321
Rule 329
Rule 341
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1584
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{\sqrt [3]{x}+x} \, dx &=\int \frac {\sqrt [6]{x}}{1+x^{2/3}} \, dx\\ &=3 \operatorname {Subst}\left (\int \frac {x^{5/2}}{1+x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=2 \sqrt {x}-3 \operatorname {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=2 \sqrt {x}-6 \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt [6]{x}\right )\\ &=2 \sqrt {x}+3 \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt [6]{x}\right )-3 \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt [6]{x}\right )\\ &=2 \sqrt {x}-\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt [6]{x}\right )}{2 \sqrt {2}}-\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt [6]{x}\right )}{2 \sqrt {2}}\\ &=2 \sqrt {x}-\frac {3 \log \left (1-\sqrt {2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt {2}}+\frac {3 \log \left (1+\sqrt {2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt {2}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}\\ &=2 \sqrt {x}+\frac {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}-\frac {3 \tan ^{-1}\left (1+\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}-\frac {3 \log \left (1-\sqrt {2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt {2}}+\frac {3 \log \left (1+\sqrt {2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 24, normalized size = 0.22 \[ -2 \sqrt {x} \left (\, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-x^{2/3}\right )-1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 120, normalized size = 1.11 \[ 3 \, \sqrt {2} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1} - \sqrt {2} x^{\frac {1}{6}} - 1\right ) + 3 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} x^{\frac {1}{6}} + 4 \, x^{\frac {1}{3}} + 4} - \sqrt {2} x^{\frac {1}{6}} + 1\right ) + \frac {3}{4} \, \sqrt {2} \log \left (4 \, \sqrt {2} x^{\frac {1}{6}} + 4 \, x^{\frac {1}{3}} + 4\right ) - \frac {3}{4} \, \sqrt {2} \log \left (-4 \, \sqrt {2} x^{\frac {1}{6}} + 4 \, x^{\frac {1}{3}} + 4\right ) + 2 \, \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 83, normalized size = 0.77 \[ -\frac {3}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, x^{\frac {1}{6}}\right )}\right ) - \frac {3}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, x^{\frac {1}{6}}\right )}\right ) + \frac {3}{4} \, \sqrt {2} \log \left (\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) - \frac {3}{4} \, \sqrt {2} \log \left (-\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) + 2 \, \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 71, normalized size = 0.66 \[ -\frac {3 \sqrt {2}\, \arctan \left (\sqrt {2}\, x^{\frac {1}{6}}-1\right )}{2}-\frac {3 \sqrt {2}\, \arctan \left (\sqrt {2}\, x^{\frac {1}{6}}+1\right )}{2}-\frac {3 \sqrt {2}\, \ln \left (\frac {x^{\frac {1}{3}}-\sqrt {2}\, x^{\frac {1}{6}}+1}{x^{\frac {1}{3}}+\sqrt {2}\, x^{\frac {1}{6}}+1}\right )}{4}+2 \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.96, size = 83, normalized size = 0.77 \[ -\frac {3}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, x^{\frac {1}{6}}\right )}\right ) - \frac {3}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, x^{\frac {1}{6}}\right )}\right ) + \frac {3}{4} \, \sqrt {2} \log \left (\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) - \frac {3}{4} \, \sqrt {2} \log \left (-\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) + 2 \, \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 42, normalized size = 0.39 \[ 2\,\sqrt {x}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x^{1/6}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {3}{2}+\frac {3}{2}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x^{1/6}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {3}{2}-\frac {3}{2}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.18, size = 110, normalized size = 1.02 \[ 2 \sqrt {x} - \frac {3 \sqrt {2} \log {\left (- 4 \sqrt {2} \sqrt [6]{x} + 4 \sqrt [3]{x} + 4 \right )}}{4} + \frac {3 \sqrt {2} \log {\left (4 \sqrt {2} \sqrt [6]{x} + 4 \sqrt [3]{x} + 4 \right )}}{4} - \frac {3 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt [6]{x} - 1 \right )}}{2} - \frac {3 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt [6]{x} + 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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