Optimal. Leaf size=33 \[ -x+\frac {4}{3} \sqrt {2+3 x}-\frac {4}{3} \log \left (1+\sqrt {2+3 x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {442, 383, 78}
\begin {gather*} -x+\frac {4}{3} \sqrt {3 x+2}-\frac {4}{3} \log \left (\sqrt {3 x+2}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 383
Rule 442
Rubi steps
\begin {align*} \int \frac {1-\sqrt {2+3 x}}{1+\sqrt {2+3 x}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1-\sqrt {x}}{1+\sqrt {x}} \, dx,x,2+3 x\right )\\ &=\frac {2}{3} \text {Subst}\left (\int \frac {(1-x) x}{1+x} \, dx,x,\sqrt {2+3 x}\right )\\ &=\frac {2}{3} \text {Subst}\left (\int \left (2-x-\frac {2}{1+x}\right ) \, dx,x,\sqrt {2+3 x}\right )\\ &=-x+\frac {4}{3} \sqrt {2+3 x}-\frac {4}{3} \log \left (1+\sqrt {2+3 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 34, normalized size = 1.03 \begin {gather*} \frac {1}{3} \left (-2-3 x+4 \sqrt {2+3 x}-4 \log \left (1+\sqrt {2+3 x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 27, normalized size = 0.82
method | result | size |
derivativedivides | \(-\frac {2}{3}-x +\frac {4 \sqrt {2+3 x}}{3}-\frac {4 \ln \left (1+\sqrt {2+3 x}\right )}{3}\) | \(27\) |
default | \(-\frac {2}{3}-x +\frac {4 \sqrt {2+3 x}}{3}-\frac {4 \ln \left (1+\sqrt {2+3 x}\right )}{3}\) | \(27\) |
trager | \(-x +\frac {4 \sqrt {2+3 x}}{3}-\frac {2 \ln \left (2 \sqrt {2+3 x}+3+3 x \right )}{3}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 26, normalized size = 0.79 \begin {gather*} -x + \frac {4}{3} \, \sqrt {3 \, x + 2} - \frac {4}{3} \, \log \left (\sqrt {3 \, x + 2} + 1\right ) - \frac {2}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 25, normalized size = 0.76 \begin {gather*} -x + \frac {4}{3} \, \sqrt {3 \, x + 2} - \frac {4}{3} \, \log \left (\sqrt {3 \, x + 2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 27, normalized size = 0.82 \begin {gather*} - x + \frac {4 \sqrt {3 x + 2}}{3} - \frac {4 \log {\left (\sqrt {3 x + 2} + 1 \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.11, size = 26, normalized size = 0.79 \begin {gather*} -x + \frac {4}{3} \, \sqrt {3 \, x + 2} - \frac {4}{3} \, \log \left (\sqrt {3 \, x + 2} + 1\right ) - \frac {2}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.10, size = 25, normalized size = 0.76 \begin {gather*} \frac {4\,\sqrt {3\,x+2}}{3}-\frac {4\,\ln \left (\sqrt {3\,x+2}+1\right )}{3}-x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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