Optimal. Leaf size=76 \[ \frac {6 x^{5/6}}{5}-\frac {12 x^{7/12}}{7}+3 \sqrt [3]{x}-12 \sqrt [12]{x}+6 \log \left (\sqrt [12]{x}+1\right )-2 \log \left (\sqrt [4]{x}+1\right )-4 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [12]{x}}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {1584, 341, 50, 58, 618, 204, 31} \[ \frac {6 x^{5/6}}{5}-\frac {12 x^{7/12}}{7}+3 \sqrt [3]{x}-12 \sqrt [12]{x}+6 \log \left (\sqrt [12]{x}+1\right )-2 \log \left (\sqrt [4]{x}+1\right )-4 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [12]{x}}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 58
Rule 204
Rule 341
Rule 618
Rule 1584
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{x}}{\sqrt [4]{x}+\sqrt {x}} \, dx &=\int \frac {\sqrt [12]{x}}{1+\sqrt [4]{x}} \, dx\\ &=4 \operatorname {Subst}\left (\int \frac {x^{10/3}}{1+x} \, dx,x,\sqrt [4]{x}\right )\\ &=\frac {6 x^{5/6}}{5}-4 \operatorname {Subst}\left (\int \frac {x^{7/3}}{1+x} \, dx,x,\sqrt [4]{x}\right )\\ &=-\frac {12 x^{7/12}}{7}+\frac {6 x^{5/6}}{5}+4 \operatorname {Subst}\left (\int \frac {x^{4/3}}{1+x} \, dx,x,\sqrt [4]{x}\right )\\ &=3 \sqrt [3]{x}-\frac {12 x^{7/12}}{7}+\frac {6 x^{5/6}}{5}-4 \operatorname {Subst}\left (\int \frac {\sqrt [3]{x}}{1+x} \, dx,x,\sqrt [4]{x}\right )\\ &=-12 \sqrt [12]{x}+3 \sqrt [3]{x}-\frac {12 x^{7/12}}{7}+\frac {6 x^{5/6}}{5}+4 \operatorname {Subst}\left (\int \frac {1}{x^{2/3} (1+x)} \, dx,x,\sqrt [4]{x}\right )\\ &=-12 \sqrt [12]{x}+3 \sqrt [3]{x}-\frac {12 x^{7/12}}{7}+\frac {6 x^{5/6}}{5}-2 \log \left (1+\sqrt [4]{x}\right )+6 \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [12]{x}\right )+6 \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [12]{x}\right )\\ &=-12 \sqrt [12]{x}+3 \sqrt [3]{x}-\frac {12 x^{7/12}}{7}+\frac {6 x^{5/6}}{5}+6 \log \left (1+\sqrt [12]{x}\right )-2 \log \left (1+\sqrt [4]{x}\right )-12 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [12]{x}\right )\\ &=-12 \sqrt [12]{x}+3 \sqrt [3]{x}-\frac {12 x^{7/12}}{7}+\frac {6 x^{5/6}}{5}-4 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [12]{x}}{\sqrt {3}}\right )+6 \log \left (1+\sqrt [12]{x}\right )-2 \log \left (1+\sqrt [4]{x}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 83, normalized size = 1.09 \[ \frac {6 x^{5/6}}{5}-\frac {12 x^{7/12}}{7}+3 \sqrt [3]{x}-12 \sqrt [12]{x}+4 \log \left (\sqrt [12]{x}+1\right )-2 \log \left (\sqrt [6]{x}-\sqrt [12]{x}+1\right )+4 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [12]{x}-1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 62, normalized size = 0.82 \[ 4 \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} x^{\frac {1}{12}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {6}{5} \, x^{\frac {5}{6}} - \frac {12}{7} \, x^{\frac {7}{12}} + 3 \, x^{\frac {1}{3}} - 12 \, x^{\frac {1}{12}} - 2 \, \log \left (x^{\frac {1}{6}} - x^{\frac {1}{12}} + 1\right ) + 4 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 60, normalized size = 0.79 \[ 4 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {1}{12}} - 1\right )}\right ) + \frac {6}{5} \, x^{\frac {5}{6}} - \frac {12}{7} \, x^{\frac {7}{12}} + 3 \, x^{\frac {1}{3}} - 12 \, x^{\frac {1}{12}} - 2 \, \log \left (x^{\frac {1}{6}} - x^{\frac {1}{12}} + 1\right ) + 4 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 61, normalized size = 0.80 \[ 4 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{12}}-1\right ) \sqrt {3}}{3}\right )+4 \ln \left (x^{\frac {1}{12}}+1\right )-2 \ln \left (x^{\frac {1}{6}}-x^{\frac {1}{12}}+1\right )+\frac {6 x^{\frac {5}{6}}}{5}-\frac {12 x^{\frac {7}{12}}}{7}+3 x^{\frac {1}{3}}-12 x^{\frac {1}{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 60, normalized size = 0.79 \[ 4 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {1}{12}} - 1\right )}\right ) + \frac {6}{5} \, x^{\frac {5}{6}} - \frac {12}{7} \, x^{\frac {7}{12}} + 3 \, x^{\frac {1}{3}} - 12 \, x^{\frac {1}{12}} - 2 \, \log \left (x^{\frac {1}{6}} - x^{\frac {1}{12}} + 1\right ) + 4 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.08, size = 78, normalized size = 1.03 \[ 4\,\ln \left (144\,x^{1/12}+144\right )-\ln \left (18-36\,x^{1/12}+\sqrt {3}\,18{}\mathrm {i}\right )\,\left (2+\sqrt {3}\,2{}\mathrm {i}\right )+\ln \left (36\,x^{1/12}-18+\sqrt {3}\,18{}\mathrm {i}\right )\,\left (-2+\sqrt {3}\,2{}\mathrm {i}\right )+3\,x^{1/3}+\frac {6\,x^{5/6}}{5}-12\,x^{1/12}-\frac {12\,x^{7/12}}{7} \]
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 \[ \int \frac {\sqrt [3]{x}}{\sqrt [4]{x} + \sqrt {x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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