Optimal. Leaf size=42 \[ \frac {3 x^{2/3}}{2}-3 \sqrt [3]{x}+6 \sqrt [6]{x}+3 \log \left (\sqrt [3]{x}+1\right )-6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]
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Rubi [A] time = 0.15, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6688, 1593, 1802, 635, 203, 260} \[ \frac {3 x^{2/3}}{2}-3 \sqrt [3]{x}+6 \sqrt [6]{x}+3 \log \left (\sqrt [3]{x}+1\right )-6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 1593
Rule 1802
Rule 6688
Rubi steps
\begin {align*} \int \frac {1+\sqrt {x}}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx &=\int \frac {1+\frac {1}{\sqrt {x}}}{1+\sqrt [3]{x}} \, dx\\ &=6 \operatorname {Subst}\left (\int \frac {x^2+x^5}{1+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \operatorname {Subst}\left (\int \frac {x^2 \left (1+x^3\right )}{1+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \operatorname {Subst}\left (\int \left (1-x+x^3-\frac {1-x}{1+x^2}\right ) \, dx,x,\sqrt [6]{x}\right )\\ &=6 \sqrt [6]{x}-3 \sqrt [3]{x}+\frac {3 x^{2/3}}{2}-6 \operatorname {Subst}\left (\int \frac {1-x}{1+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \sqrt [6]{x}-3 \sqrt [3]{x}+\frac {3 x^{2/3}}{2}-6 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [6]{x}\right )+6 \operatorname {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \sqrt [6]{x}-3 \sqrt [3]{x}+\frac {3 x^{2/3}}{2}-6 \tan ^{-1}\left (\sqrt [6]{x}\right )+3 \log \left (1+\sqrt [3]{x}\right )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 54, normalized size = 1.29 \[ \frac {3 x^{2/3}}{2}-3 \sqrt [3]{x}+6 \sqrt [6]{x}+(3+3 i) \log \left (-\sqrt [6]{x}+i\right )+(3-3 i) \log \left (\sqrt [6]{x}+i\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 30, normalized size = 0.71 \[ \frac {3}{2} \, x^{\frac {2}{3}} - 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} - 6 \, \arctan \left (x^{\frac {1}{6}}\right ) + 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 30, normalized size = 0.71 \[ \frac {3}{2} \, x^{\frac {2}{3}} - 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} - 6 \, \arctan \left (x^{\frac {1}{6}}\right ) + 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 48, normalized size = 1.14 \[ -6 \arctan \left (x^{\frac {1}{6}}\right )+\ln \left (x +1\right )+2 \ln \left (x^{\frac {1}{3}}+1\right )-\ln \left (x^{\frac {2}{3}}-x^{\frac {1}{3}}+1\right )+\frac {3 x^{\frac {2}{3}}}{2}-3 x^{\frac {1}{3}}+6 x^{\frac {1}{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.39, size = 30, normalized size = 0.71 \[ \frac {3}{2} \, x^{\frac {2}{3}} - 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} - 6 \, \arctan \left (x^{\frac {1}{6}}\right ) + 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 42, normalized size = 1.00 \[ \frac {3\,x^{2/3}}{2}+3\,\ln \left (\left (-6+x^{1/6}\,6{}\mathrm {i}\right )\,\left (6+x^{1/6}\,6{}\mathrm {i}\right )\right )-3\,x^{1/3}-6\,\mathrm {atan}\left (x^{1/6}\right )+6\,x^{1/6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 23.13, size = 39, normalized size = 0.93 \[ 6 \sqrt [6]{x} + \frac {3 x^{\frac {2}{3}}}{2} - 3 \sqrt [3]{x} + 3 \log {\left (\sqrt [3]{x} + 1 \right )} - 6 \operatorname {atan}{\left (\sqrt [6]{x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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