Optimal. Leaf size=14 \[ \frac {1}{x}+2 \log (x)+3 \log (2+x) \]
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Rubi [A]
time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1607, 907}
\begin {gather*} \frac {1}{x}+2 \log (x)+3 \log (x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 907
Rule 1607
Rubi steps
\begin {align*} \int \frac {-2+3 x+5 x^2}{2 x^2+x^3} \, dx &=\int \frac {-2+3 x+5 x^2}{x^2 (2+x)} \, dx\\ &=\int \left (-\frac {1}{x^2}+\frac {2}{x}+\frac {3}{2+x}\right ) \, dx\\ &=\frac {1}{x}+2 \log (x)+3 \log (2+x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {1}{x}+2 \log (x)+3 \log (2+x) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.69, size = 14, normalized size = 1.00 \begin {gather*} \frac {1}{x}+2 \text {Log}\left [x\right ]+3 \text {Log}\left [2+x\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 15, normalized size = 1.07
method | result | size |
default | \(\frac {1}{x}+2 \ln \left (x \right )+3 \ln \left (2+x \right )\) | \(15\) |
norman | \(\frac {1}{x}+2 \ln \left (x \right )+3 \ln \left (2+x \right )\) | \(15\) |
risch | \(\frac {1}{x}+2 \ln \left (x \right )+3 \ln \left (2+x \right )\) | \(15\) |
meijerg | \(\frac {1}{x}+2 \ln \left (x \right )-2 \ln \left (2\right )+3 \ln \left (1+\frac {x}{2}\right )\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 14, normalized size = 1.00 \begin {gather*} \frac {1}{x} + 3 \, \log \left (x + 2\right ) + 2 \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 18, normalized size = 1.29 \begin {gather*} \frac {3 \, x \log \left (x + 2\right ) + 2 \, x \log \left (x\right ) + 1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 14, normalized size = 1.00 \begin {gather*} 2 \log {\left (x \right )} + 3 \log {\left (x + 2 \right )} + \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 15, normalized size = 1.07 \begin {gather*} 3 \ln \left |x+2\right |+2 \ln \left |x\right |+\frac 1{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 14, normalized size = 1.00 \begin {gather*} 3\,\ln \left (x+2\right )+2\,\ln \left (x\right )+\frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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