Optimal. Leaf size=26 \[ 3 \sqrt [3]{x}+6 \tan ^{-1}\left (\sqrt [6]{x}\right )-3 \log \left (1+\sqrt [3]{x}\right ) \]
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Rubi [A]
time = 0.03, 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 = {1607, 1833,
1824, 649, 209, 266} \begin {gather*} 6 \text {ArcTan}\left (\sqrt [6]{x}\right )+3 \sqrt [3]{x}-3 \log \left (\sqrt [3]{x}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 266
Rule 649
Rule 1607
Rule 1824
Rule 1833
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 \text {Subst}\left (\int \frac {1+x^3}{1+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \text {Subst}\left (\int \left (x+\frac {1-x}{1+x^2}\right ) \, dx,x,\sqrt [6]{x}\right )\\ &=3 \sqrt [3]{x}+6 \text {Subst}\left (\int \frac {1-x}{1+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=3 \sqrt [3]{x}+6 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [6]{x}\right )-6 \text {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 [A]
time = 0.03, size = 26, normalized size = 1.00 \begin {gather*} 3 \sqrt [3]{x}+6 \tan ^{-1}\left (\sqrt [6]{x}\right )-3 \log \left (1+\sqrt [3]{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 21, normalized size = 0.81
method | result | size |
derivativedivides | \(3 x^{\frac {1}{3}}+6 \arctan \left (x^{\frac {1}{6}}\right )-3 \ln \left (1+x^{\frac {1}{3}}\right )\) | \(21\) |
default | \(3 x^{\frac {1}{3}}+6 \arctan \left (x^{\frac {1}{6}}\right )-3 \ln \left (1+x^{\frac {1}{3}}\right )\) | \(21\) |
meijerg | \(3 x^{\frac {1}{3}}+6 \arctan \left (x^{\frac {1}{6}}\right )-3 \ln \left (1+x^{\frac {1}{3}}\right )\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 20, normalized size = 0.77 \begin {gather*} 3 \, x^{\frac {1}{3}} + 6 \, \arctan \left (x^{\frac {1}{6}}\right ) - 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 20, normalized size = 0.77 \begin {gather*} 3 \, x^{\frac {1}{3}} + 6 \, \arctan \left (x^{\frac {1}{6}}\right ) - 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.11, size = 24, normalized size = 0.92 \begin {gather*} 3 \sqrt [3]{x} - 3 \log {\left (\sqrt [3]{x} + 1 \right )} + 6 \operatorname {atan}{\left (\sqrt [6]{x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.39, size = 20, normalized size = 0.77 \begin {gather*} 3 \, x^{\frac {1}{3}} + 6 \, \arctan \left (x^{\frac {1}{6}}\right ) - 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \end {gather*}
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 \begin {gather*} 6\,\mathrm {atan}\left (x^{1/6}\right )-3\,\ln \left (36\,x^{1/3}+36\right )+3\,x^{1/3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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