Optimal. Leaf size=33 \[ 6 \sqrt [6]{1+x}+3 \sqrt [3]{1+x}+6 \log \left (1-\sqrt [6]{1+x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2037, 1607,
272, 45} \begin {gather*} 3 \sqrt [3]{x+1}+6 \sqrt [6]{x+1}+6 \log \left (1-\sqrt [6]{x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 1607
Rule 2037
Rubi steps
\begin {align*} \int \frac {1}{-\sqrt {1+x}+(1+x)^{2/3}} \, dx &=\text {Subst}\left (\int \frac {1}{-\sqrt {x}+x^{2/3}} \, dx,x,1+x\right )\\ &=\text {Subst}\left (\int \frac {1}{\left (-1+\sqrt [6]{x}\right ) \sqrt {x}} \, dx,x,1+x\right )\\ &=6 \text {Subst}\left (\int \frac {x^2}{-1+x} \, dx,x,\sqrt [6]{1+x}\right )\\ &=6 \text {Subst}\left (\int \left (1+\frac {1}{-1+x}+x\right ) \, dx,x,\sqrt [6]{1+x}\right )\\ &=6 \sqrt [6]{1+x}+3 \sqrt [3]{1+x}+6 \log \left (1-\sqrt [6]{1+x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 31, normalized size = 0.94 \begin {gather*} 3 \left (2 \sqrt [6]{1+x}+\sqrt [3]{1+x}+2 \log \left (-1+\sqrt [6]{1+x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(110\) vs.
\(2(27)=54\).
time = 0.03, size = 111, normalized size = 3.36
method | result | size |
derivativedivides | \(3 \left (1+x \right )^{\frac {1}{3}}+6 \left (1+x \right )^{\frac {1}{6}}+6 \ln \left (-1+\left (1+x \right )^{\frac {1}{6}}\right )\) | \(26\) |
default | \(6 \left (1+x \right )^{\frac {1}{6}}+3 \left (1+x \right )^{\frac {1}{3}}+\ln \left (x \right )+2 \ln \left (-1+\left (1+x \right )^{\frac {1}{6}}\right )-\ln \left (\left (1+x \right )^{\frac {1}{3}}+\left (1+x \right )^{\frac {1}{6}}+1\right )-2 \ln \left (\left (1+x \right )^{\frac {1}{6}}+1\right )+\ln \left (\left (1+x \right )^{\frac {1}{3}}-\left (1+x \right )^{\frac {1}{6}}+1\right )-\ln \left (1+\sqrt {1+x}\right )+\ln \left (-1+\sqrt {1+x}\right )+2 \ln \left (-1+\left (1+x \right )^{\frac {1}{3}}\right )-\ln \left (\left (1+x \right )^{\frac {2}{3}}+\left (1+x \right )^{\frac {1}{3}}+1\right )\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.67, size = 25, normalized size = 0.76 \begin {gather*} 3 \, {\left (x + 1\right )}^{\frac {1}{3}} + 6 \, {\left (x + 1\right )}^{\frac {1}{6}} + 6 \, \log \left ({\left (x + 1\right )}^{\frac {1}{6}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 25, normalized size = 0.76 \begin {gather*} 3 \, {\left (x + 1\right )}^{\frac {1}{3}} + 6 \, {\left (x + 1\right )}^{\frac {1}{6}} + 6 \, \log \left ({\left (x + 1\right )}^{\frac {1}{6}} - 1\right ) \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (x + 1\right )^{\frac {2}{3}} - \sqrt {x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.63, size = 26, normalized size = 0.79 \begin {gather*} 3 \, {\left (x + 1\right )}^{\frac {1}{3}} + 6 \, {\left (x + 1\right )}^{\frac {1}{6}} + 6 \, \log \left ({\left | {\left (x + 1\right )}^{\frac {1}{6}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 25, normalized size = 0.76 \begin {gather*} 6\,\ln \left ({\left (x+1\right )}^{1/6}-1\right )+3\,{\left (x+1\right )}^{1/3}+6\,{\left (x+1\right )}^{1/6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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