Optimal. Leaf size=30 \[ \frac {x}{8}-\frac {x^2}{8}+\frac {x^3}{6}-\frac {1}{16} \log (1+2 x) \]
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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45}
\begin {gather*} \frac {x^3}{6}-\frac {x^2}{8}+\frac {x}{8}-\frac {1}{16} \log (2 x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {x^3}{1+2 x} \, dx &=\int \left (\frac {1}{8}-\frac {x}{4}+\frac {x^2}{2}-\frac {1}{8 (1+2 x)}\right ) \, dx\\ &=\frac {x}{8}-\frac {x^2}{8}+\frac {x^3}{6}-\frac {1}{16} \log (1+2 x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 27, normalized size = 0.90 \begin {gather*} \frac {1}{96} \left (11+12 x-12 x^2+16 x^3-6 \log (1+2 x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.87, size = 22, normalized size = 0.73 \begin {gather*} \frac {x}{8}-\frac {x^2}{8}+\frac {x^3}{6}-\frac {\text {Log}\left [1+2 x\right ]}{16} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 23, normalized size = 0.77
method | result | size |
default | \(\frac {x}{8}-\frac {x^{2}}{8}+\frac {x^{3}}{6}-\frac {\ln \left (1+2 x \right )}{16}\) | \(23\) |
norman | \(\frac {x}{8}-\frac {x^{2}}{8}+\frac {x^{3}}{6}-\frac {\ln \left (1+2 x \right )}{16}\) | \(23\) |
meijerg | \(\frac {x \left (16 x^{2}-12 x +12\right )}{96}-\frac {\ln \left (1+2 x \right )}{16}\) | \(23\) |
risch | \(\frac {x}{8}-\frac {x^{2}}{8}+\frac {x^{3}}{6}-\frac {\ln \left (1+2 x \right )}{16}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 22, normalized size = 0.73 \begin {gather*} \frac {1}{6} \, x^{3} - \frac {1}{8} \, x^{2} + \frac {1}{8} \, x - \frac {1}{16} \, \log \left (2 \, x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 22, normalized size = 0.73 \begin {gather*} \frac {1}{6} \, x^{3} - \frac {1}{8} \, x^{2} + \frac {1}{8} \, x - \frac {1}{16} \, \log \left (2 \, x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 20, normalized size = 0.67 \begin {gather*} \frac {x^{3}}{6} - \frac {x^{2}}{8} + \frac {x}{8} - \frac {\log {\left (2 x + 1 \right )}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 28, normalized size = 0.93 \begin {gather*} \frac {\frac {4}{3} x^{3}-x^{2}+x}{8}-\frac {\ln \left |2 x+1\right |}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 20, normalized size = 0.67 \begin {gather*} \frac {x}{8}-\frac {\ln \left (x+\frac {1}{2}\right )}{16}-\frac {x^2}{8}+\frac {x^3}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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