Optimal. Leaf size=26 \[ \frac{x}{15}-\frac{4}{63} \log (3 x+2)+\frac{1}{175} \log (5 x+1) \]
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Rubi [A] time = 0.0147503, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {1386, 703, 632, 31} \[ \frac{x}{15}-\frac{4}{63} \log (3 x+2)+\frac{1}{175} \log (5 x+1) \]
Antiderivative was successfully verified.
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Rule 1386
Rule 703
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{x}{13+\frac{2}{x}+15 x} \, dx &=\int \frac{x^2}{2+13 x+15 x^2} \, dx\\ &=\frac{x}{15}+\frac{1}{15} \int \frac{-2-13 x}{2+13 x+15 x^2} \, dx\\ &=\frac{x}{15}+\frac{3}{35} \int \frac{1}{3+15 x} \, dx-\frac{20}{21} \int \frac{1}{10+15 x} \, dx\\ &=\frac{x}{15}-\frac{4}{63} \log (2+3 x)+\frac{1}{175} \log (1+5 x)\\ \end{align*}
Mathematica [A] time = 0.0044611, size = 26, normalized size = 1. \[ \frac{x}{15}-\frac{4}{63} \log (3 x+2)+\frac{1}{175} \log (5 x+1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 21, normalized size = 0.8 \begin{align*}{\frac{x}{15}}-{\frac{4\,\ln \left ( 2+3\,x \right ) }{63}}+{\frac{\ln \left ( 1+5\,x \right ) }{175}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04901, size = 27, normalized size = 1.04 \begin{align*} \frac{1}{15} \, x + \frac{1}{175} \, \log \left (5 \, x + 1\right ) - \frac{4}{63} \, \log \left (3 \, x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.14777, size = 66, normalized size = 2.54 \begin{align*} \frac{1}{15} \, x + \frac{1}{175} \, \log \left (5 \, x + 1\right ) - \frac{4}{63} \, \log \left (3 \, x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.105634, size = 20, normalized size = 0.77 \begin{align*} \frac{x}{15} + \frac{\log{\left (x + \frac{1}{5} \right )}}{175} - \frac{4 \log{\left (x + \frac{2}{3} \right )}}{63} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28539, size = 30, normalized size = 1.15 \begin{align*} \frac{1}{15} \, x + \frac{1}{175} \, \log \left ({\left | 5 \, x + 1 \right |}\right ) - \frac{4}{63} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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