Optimal. Leaf size=33 \[ 3 \sqrt [3]{x+1}+6 \sqrt [6]{x+1}+6 \log \left (1-\sqrt [6]{x+1}\right ) \]
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Rubi [A] time = 0.02, 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 = {2012, 1593, 266, 43} \[ 3 \sqrt [3]{x+1}+6 \sqrt [6]{x+1}+6 \log \left (1-\sqrt [6]{x+1}\right ) \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1593
Rule 2012
Rubi steps
\begin {align*} \int \frac {1}{-\sqrt {1+x}+(1+x)^{2/3}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{-\sqrt {x}+x^{2/3}} \, dx,x,1+x\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{\left (-1+\sqrt [6]{x}\right ) \sqrt {x}} \, dx,x,1+x\right )\\ &=6 \operatorname {Subst}\left (\int \frac {x^2}{-1+x} \, dx,x,\sqrt [6]{1+x}\right )\\ &=6 \operatorname {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 = 0.02, size = 33, normalized size = 1.00 \[ 3 \left (\sqrt [3]{x+1}+2 \sqrt [6]{x+1}+2 \log \left (1-\sqrt [6]{x+1}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.01, size = 31, normalized size = 0.94 \[ 3 \sqrt [3]{x+1}+6 \sqrt [6]{x+1}+6 \log \left (\sqrt [6]{x+1}-1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 25, normalized size = 0.76 \[ 3 \, {\left (x + 1\right )}^{\frac {1}{3}} + 6 \, {\left (x + 1\right )}^{\frac {1}{6}} + 6 \, \log \left ({\left (x + 1\right )}^{\frac {1}{6}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.18, size = 26, normalized size = 0.79 \[ 3 \, {\left (x + 1\right )}^{\frac {1}{3}} + 6 \, {\left (x + 1\right )}^{\frac {1}{6}} + 6 \, \log \left ({\left | {\left (x + 1\right )}^{\frac {1}{6}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 26, normalized size = 0.79
method | result | size |
derivativedivides | \(3 \left (1+x \right )^{\frac {1}{3}}+6 \left (1+x \right )^{\frac {1}{6}}+6 \ln \left (\left (1+x \right )^{\frac {1}{6}}-1\right )\) | \(26\) |
default | \(6 \left (1+x \right )^{\frac {1}{6}}+3 \left (1+x \right )^{\frac {1}{3}}+\ln \relax (x )+2 \ln \left (\left (1+x \right )^{\frac {1}{6}}-1\right )-\ln \left (\left (1+x \right )^{\frac {1}{3}}+\left (1+x \right )^{\frac {1}{6}}+1\right )-2 \ln \left (1+\left (1+x \right )^{\frac {1}{6}}\right )+\ln \left (\left (1+x \right )^{\frac {1}{3}}-\left (1+x \right )^{\frac {1}{6}}+1\right )-\ln \left (1+\sqrt {1+x}\right )+\ln \left (-1+\sqrt {1+x}\right )+2 \ln \left (\left (1+x \right )^{\frac {1}{3}}-1\right )-\ln \left (\left (1+x \right )^{\frac {2}{3}}+\left (1+x \right )^{\frac {1}{3}}+1\right )\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 25, normalized size = 0.76 \[ 3 \, {\left (x + 1\right )}^{\frac {1}{3}} + 6 \, {\left (x + 1\right )}^{\frac {1}{6}} + 6 \, \log \left ({\left (x + 1\right )}^{\frac {1}{6}} - 1\right ) \]
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 \[ 6\,\ln \left ({\left (x+1\right )}^{1/6}-1\right )+3\,{\left (x+1\right )}^{1/3}+6\,{\left (x+1\right )}^{1/6} \]
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 \[ \int \frac {1}{\left (x + 1\right )^{\frac {2}{3}} - \sqrt {x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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