Optimal. Leaf size=41 \[ -\frac{1}{4 x^2}+\frac{13}{4 x}+\frac{139 \log (x)}{8}+\frac{27}{56} \log (3 x+2)-\frac{125}{7} \log (5 x+1) \]
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Rubi [A] time = 0.0338796, antiderivative size = 41, 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} \[ -\frac{1}{4 x^2}+\frac{13}{4 x}+\frac{139 \log (x)}{8}+\frac{27}{56} \log (3 x+2)-\frac{125}{7} \log (5 x+1) \]
Antiderivative was successfully verified.
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Rule 1386
Rule 709
Rule 800
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (13+\frac{2}{x}+15 x\right )} \, dx &=\int \frac{1}{x^3 \left (2+13 x+15 x^2\right )} \, dx\\ &=-\frac{1}{4 x^2}+\frac{1}{2} \int \frac{-13-15 x}{x^2 \left (2+13 x+15 x^2\right )} \, dx\\ &=-\frac{1}{4 x^2}+\frac{1}{2} \int \left (-\frac{13}{2 x^2}+\frac{139}{4 x}+\frac{81}{28 (2+3 x)}-\frac{1250}{7 (1+5 x)}\right ) \, dx\\ &=-\frac{1}{4 x^2}+\frac{13}{4 x}+\frac{139 \log (x)}{8}+\frac{27}{56} \log (2+3 x)-\frac{125}{7} \log (1+5 x)\\ \end{align*}
Mathematica [A] time = 0.0045968, size = 41, normalized size = 1. \[ -\frac{1}{4 x^2}+\frac{13}{4 x}+\frac{139 \log (x)}{8}+\frac{27}{56} \log (3 x+2)-\frac{125}{7} \log (5 x+1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 32, normalized size = 0.8 \begin{align*} -{\frac{1}{4\,{x}^{2}}}+{\frac{13}{4\,x}}+{\frac{139\,\ln \left ( x \right ) }{8}}+{\frac{27\,\ln \left ( 2+3\,x \right ) }{56}}-{\frac{125\,\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 = 0.9811, size = 42, normalized size = 1.02 \begin{align*} \frac{13 \, x - 1}{4 \, x^{2}} - \frac{125}{7} \, \log \left (5 \, x + 1\right ) + \frac{27}{56} \, \log \left (3 \, x + 2\right ) + \frac{139}{8} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25822, size = 117, normalized size = 2.85 \begin{align*} -\frac{1000 \, x^{2} \log \left (5 \, x + 1\right ) - 27 \, x^{2} \log \left (3 \, x + 2\right ) - 973 \, x^{2} \log \left (x\right ) - 182 \, x + 14}{56 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.153319, size = 36, normalized size = 0.88 \begin{align*} \frac{139 \log{\left (x \right )}}{8} - \frac{125 \log{\left (x + \frac{1}{5} \right )}}{7} + \frac{27 \log{\left (x + \frac{2}{3} \right )}}{56} + \frac{13 x - 1}{4 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14518, size = 46, normalized size = 1.12 \begin{align*} \frac{13 \, x - 1}{4 \, x^{2}} - \frac{125}{7} \, \log \left ({\left | 5 \, x + 1 \right |}\right ) + \frac{27}{56} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac{139}{8} \, \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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