Optimal. Leaf size=17 \[ \frac{1}{2} \log (x+2)+\frac{5}{2} \log (x+4) \]
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Rubi [A] time = 0.004653, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {632, 31} \[ \frac{1}{2} \log (x+2)+\frac{5}{2} \log (x+4) \]
Antiderivative was successfully verified.
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Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{7+3 x}{8+6 x+x^2} \, dx &=\frac{1}{2} \int \frac{1}{2+x} \, dx+\frac{5}{2} \int \frac{1}{4+x} \, dx\\ &=\frac{1}{2} \log (2+x)+\frac{5}{2} \log (4+x)\\ \end{align*}
Mathematica [A] time = 0.0046759, size = 17, normalized size = 1. \[ \frac{1}{2} \log (x+2)+\frac{5}{2} \log (x+4) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 14, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 2+x \right ) }{2}}+{\frac{5\,\ln \left ( 4+x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06368, size = 18, normalized size = 1.06 \begin{align*} \frac{5}{2} \, \log \left (x + 4\right ) + \frac{1}{2} \, \log \left (x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54145, size = 45, normalized size = 2.65 \begin{align*} \frac{5}{2} \, \log \left (x + 4\right ) + \frac{1}{2} \, \log \left (x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.236576, size = 14, normalized size = 0.82 \begin{align*} \frac{\log{\left (x + 2 \right )}}{2} + \frac{5 \log{\left (x + 4 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28152, size = 20, normalized size = 1.18 \begin{align*} \frac{5}{2} \, \log \left ({\left | x + 4 \right |}\right ) + \frac{1}{2} \, \log \left ({\left | x + 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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