Optimal. Leaf size=34 \[ -\frac{1}{2 x}-\frac{13 \log (x)}{4}-\frac{9}{28} \log (3 x+2)+\frac{25}{7} \log (5 x+1) \]
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Rubi [A] time = 0.0297196, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1386, 709, 800} \[ -\frac{1}{2 x}-\frac{13 \log (x)}{4}-\frac{9}{28} \log (3 x+2)+\frac{25}{7} \log (5 x+1) \]
Antiderivative was successfully verified.
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Rule 1386
Rule 709
Rule 800
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (13+\frac{2}{x}+15 x\right )} \, dx &=\int \frac{1}{x^2 \left (2+13 x+15 x^2\right )} \, dx\\ &=-\frac{1}{2 x}+\frac{1}{2} \int \frac{-13-15 x}{x \left (2+13 x+15 x^2\right )} \, dx\\ &=-\frac{1}{2 x}+\frac{1}{2} \int \left (-\frac{13}{2 x}-\frac{27}{14 (2+3 x)}+\frac{250}{7 (1+5 x)}\right ) \, dx\\ &=-\frac{1}{2 x}-\frac{13 \log (x)}{4}-\frac{9}{28} \log (2+3 x)+\frac{25}{7} \log (1+5 x)\\ \end{align*}
Mathematica [A] time = 0.0040524, size = 34, normalized size = 1. \[ -\frac{1}{2 x}-\frac{13 \log (x)}{4}-\frac{9}{28} \log (3 x+2)+\frac{25}{7} \log (5 x+1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 27, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,x}}-{\frac{13\,\ln \left ( x \right ) }{4}}-{\frac{9\,\ln \left ( 2+3\,x \right ) }{28}}+{\frac{25\,\ln \left ( 1+5\,x \right ) }{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.99963, size = 35, normalized size = 1.03 \begin{align*} -\frac{1}{2 \, x} + \frac{25}{7} \, \log \left (5 \, x + 1\right ) - \frac{9}{28} \, \log \left (3 \, x + 2\right ) - \frac{13}{4} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.2629, size = 90, normalized size = 2.65 \begin{align*} \frac{100 \, x \log \left (5 \, x + 1\right ) - 9 \, x \log \left (3 \, x + 2\right ) - 91 \, x \log \left (x\right ) - 14}{28 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.14262, size = 31, normalized size = 0.91 \begin{align*} - \frac{13 \log{\left (x \right )}}{4} + \frac{25 \log{\left (x + \frac{1}{5} \right )}}{7} - \frac{9 \log{\left (x + \frac{2}{3} \right )}}{28} - \frac{1}{2 x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12182, size = 39, normalized size = 1.15 \begin{align*} -\frac{1}{2 \, x} + \frac{25}{7} \, \log \left ({\left | 5 \, x + 1 \right |}\right ) - \frac{9}{28} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{13}{4} \, \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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