Optimal. Leaf size=73 \[ \frac {3 x^{2/3}}{2}-\frac {12 x^{7/12}}{7}-\frac {12 x^{5/12}}{5}+2 \sqrt {x}+3 \sqrt [3]{x}-4 \sqrt [4]{x}+6 \sqrt [6]{x}-12 \sqrt [12]{x}+12 \log \left (\sqrt [12]{x}+1\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1593, 266, 43} \[ \frac {3 x^{2/3}}{2}-\frac {12 x^{7/12}}{7}-\frac {12 x^{5/12}}{5}+2 \sqrt {x}+3 \sqrt [3]{x}-4 \sqrt [4]{x}+6 \sqrt [6]{x}-12 \sqrt [12]{x}+12 \log \left (\sqrt [12]{x}+1\right ) \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1593
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{x}+\sqrt [3]{x}} \, dx &=\int \frac {1}{\left (1+\sqrt [12]{x}\right ) \sqrt [4]{x}} \, dx\\ &=12 \operatorname {Subst}\left (\int \frac {x^8}{1+x} \, dx,x,\sqrt [12]{x}\right )\\ &=12 \operatorname {Subst}\left (\int \left (-1+x-x^2+x^3-x^4+x^5-x^6+x^7+\frac {1}{1+x}\right ) \, dx,x,\sqrt [12]{x}\right )\\ &=-12 \sqrt [12]{x}+6 \sqrt [6]{x}-4 \sqrt [4]{x}+3 \sqrt [3]{x}-\frac {12 x^{5/12}}{5}+2 \sqrt {x}-\frac {12 x^{7/12}}{7}+\frac {3 x^{2/3}}{2}+12 \log \left (1+\sqrt [12]{x}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 73, normalized size = 1.00 \[ \frac {3 x^{2/3}}{2}-\frac {12 x^{7/12}}{7}-\frac {12 x^{5/12}}{5}+2 \sqrt {x}+3 \sqrt [3]{x}-4 \sqrt [4]{x}+6 \sqrt [6]{x}-12 \sqrt [12]{x}+12 \log \left (\sqrt [12]{x}+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 49, normalized size = 0.67 \[ \frac {3}{2} \, x^{\frac {2}{3}} - \frac {12}{7} \, x^{\frac {7}{12}} + 2 \, \sqrt {x} - \frac {12}{5} \, x^{\frac {5}{12}} + 3 \, x^{\frac {1}{3}} - 4 \, x^{\frac {1}{4}} + 6 \, x^{\frac {1}{6}} - 12 \, x^{\frac {1}{12}} + 12 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 49, normalized size = 0.67 \[ \frac {3}{2} \, x^{\frac {2}{3}} - \frac {12}{7} \, x^{\frac {7}{12}} + 2 \, \sqrt {x} - \frac {12}{5} \, x^{\frac {5}{12}} + 3 \, x^{\frac {1}{3}} - 4 \, x^{\frac {1}{4}} + 6 \, x^{\frac {1}{6}} - 12 \, x^{\frac {1}{12}} + 12 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 173, normalized size = 2.37 \[ \ln \left (x -1\right )+\ln \left (\sqrt {x}-1\right )-\ln \left (\sqrt {x}+1\right )+2 \ln \left (x^{\frac {1}{4}}+1\right )-2 \ln \left (x^{\frac {1}{6}}+1\right )+4 \ln \left (x^{\frac {1}{12}}+1\right )+2 \ln \left (x^{\frac {1}{3}}-1\right )-2 \ln \left (x^{\frac {1}{4}}-1\right )+2 \ln \left (x^{\frac {1}{6}}-1\right )-4 \ln \left (x^{\frac {1}{12}}-1\right )+\ln \left (x^{\frac {1}{3}}-x^{\frac {1}{6}}+1\right )-2 \ln \left (x^{\frac {1}{6}}-x^{\frac {1}{12}}+1\right )-\ln \left (x^{\frac {1}{3}}+x^{\frac {1}{6}}+1\right )+2 \ln \left (x^{\frac {1}{6}}+x^{\frac {1}{12}}+1\right )-\ln \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}+1\right )+\frac {3 x^{\frac {2}{3}}}{2}-\frac {12 x^{\frac {7}{12}}}{7}+2 \sqrt {x}-\frac {12 x^{\frac {5}{12}}}{5}+3 x^{\frac {1}{3}}-4 x^{\frac {1}{4}}+6 x^{\frac {1}{6}}-12 x^{\frac {1}{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 49, normalized size = 0.67 \[ \frac {3}{2} \, x^{\frac {2}{3}} - \frac {12}{7} \, x^{\frac {7}{12}} + 2 \, \sqrt {x} - \frac {12}{5} \, x^{\frac {5}{12}} + 3 \, x^{\frac {1}{3}} - 4 \, x^{\frac {1}{4}} + 6 \, x^{\frac {1}{6}} - 12 \, x^{\frac {1}{12}} + 12 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 49, normalized size = 0.67 \[ 12\,\ln \left (x^{1/12}+1\right )+2\,\sqrt {x}+3\,x^{1/3}-4\,x^{1/4}+\frac {3\,x^{2/3}}{2}+6\,x^{1/6}-12\,x^{1/12}-\frac {12\,x^{5/12}}{5}-\frac {12\,x^{7/12}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt [4]{x} + \sqrt [3]{x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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