Optimal. Leaf size=26 \[ 3 \sqrt [3]{x}-3 \log \left (\sqrt [3]{x}+1\right )+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1593, 1819, 1810, 635, 203, 260} \[ 3 \sqrt [3]{x}-3 \log \left (\sqrt [3]{x}+1\right )+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 1593
Rule 1810
Rule 1819
Rubi steps
\begin {align*} \int \frac {1+\sqrt {x}}{x^{5/6}+x^{7/6}} \, dx &=\int \frac {1+\sqrt {x}}{\left (1+\sqrt [3]{x}\right ) x^{5/6}} \, dx\\ &=6 \operatorname {Subst}\left (\int \frac {1+x^3}{1+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \operatorname {Subst}\left (\int \left (x+\frac {1-x}{1+x^2}\right ) \, dx,x,\sqrt [6]{x}\right )\\ &=3 \sqrt [3]{x}+6 \operatorname {Subst}\left (\int \frac {1-x}{1+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=3 \sqrt [3]{x}+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 )\\ &=3 \sqrt [3]{x}+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 = 38, normalized size = 1.46 \[ 3 \sqrt [3]{x}+(-3-3 i) \log \left (-\sqrt [6]{x}+i\right )-(3-3 i) \log \left (\sqrt [6]{x}+i\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 20, normalized size = 0.77 \[ 3 \, x^{\frac {1}{3}} + 6 \, \arctan \left (x^{\frac {1}{6}}\right ) - 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 20, normalized size = 0.77 \[ 3 \, x^{\frac {1}{3}} + 6 \, \arctan \left (x^{\frac {1}{6}}\right ) - 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 21, normalized size = 0.81 \[ 6 \arctan \left (x^{\frac {1}{6}}\right )-3 \ln \left (x^{\frac {1}{3}}+1\right )+3 x^{\frac {1}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.29, size = 20, normalized size = 0.77 \[ 3 \, x^{\frac {1}{3}} + 6 \, \arctan \left (x^{\frac {1}{6}}\right ) - 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.36, size = 22, normalized size = 0.85 \[ 6\,\mathrm {atan}\left (x^{1/6}\right )-3\,\ln \left (36\,x^{1/3}+36\right )+3\,x^{1/3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.66, size = 24, normalized size = 0.92 \[ 3 \sqrt [3]{x} - 3 \log {\left (\sqrt [3]{x} + 1 \right )} + 6 \operatorname {atan}{\left (\sqrt [6]{x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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