Optimal. Leaf size=27 \[ \frac {\log (x)}{2}+\frac {3}{14} \log (2+3 x)-\frac {5}{7} \log (1+5 x) \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1368, 719, 29,
646, 31} \begin {gather*} \frac {\log (x)}{2}+\frac {3}{14} \log (3 x+2)-\frac {5}{7} \log (5 x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 646
Rule 719
Rule 1368
Rubi steps
\begin {align*} \int \frac {1}{\left (15+\frac {2}{x^2}+\frac {13}{x}\right ) x^3} \, dx &=\int \frac {1}{x \left (2+13 x+15 x^2\right )} \, dx\\ &=\frac {1}{2} \int \frac {1}{x} \, dx+\frac {1}{2} \int \frac {-13-15 x}{2+13 x+15 x^2} \, dx\\ &=\frac {\log (x)}{2}+\frac {45}{14} \int \frac {1}{10+15 x} \, dx-\frac {75}{7} \int \frac {1}{3+15 x} \, dx\\ &=\frac {\log (x)}{2}+\frac {3}{14} \log (2+3 x)-\frac {5}{7} \log (1+5 x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} \frac {\log (x)}{2}+\frac {3}{14} \log (2+3 x)-\frac {5}{7} \log (1+5 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 22, normalized size = 0.81
method | result | size |
default | \(\frac {\ln \left (x \right )}{2}+\frac {3 \ln \left (2+3 x \right )}{14}-\frac {5 \ln \left (1+5 x \right )}{7}\) | \(22\) |
norman | \(\frac {\ln \left (x \right )}{2}+\frac {3 \ln \left (2+3 x \right )}{14}-\frac {5 \ln \left (1+5 x \right )}{7}\) | \(22\) |
risch | \(\frac {\ln \left (x \right )}{2}+\frac {3 \ln \left (2+3 x \right )}{14}-\frac {5 \ln \left (1+5 x \right )}{7}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 21, normalized size = 0.78 \begin {gather*} -\frac {5}{7} \, \log \left (5 \, x + 1\right ) + \frac {3}{14} \, \log \left (3 \, x + 2\right ) + \frac {1}{2} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 21, normalized size = 0.78 \begin {gather*} -\frac {5}{7} \, \log \left (5 \, x + 1\right ) + \frac {3}{14} \, \log \left (3 \, x + 2\right ) + \frac {1}{2} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 24, normalized size = 0.89 \begin {gather*} \frac {\log {\left (x \right )}}{2} - \frac {5 \log {\left (x + \frac {1}{5} \right )}}{7} + \frac {3 \log {\left (x + \frac {2}{3} \right )}}{14} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.83, size = 24, normalized size = 0.89 \begin {gather*} -\frac {5}{7} \, \log \left ({\left | 5 \, x + 1 \right |}\right ) + \frac {3}{14} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.39, size = 17, normalized size = 0.63 \begin {gather*} \frac {3\,\ln \left (x+\frac {2}{3}\right )}{14}-\frac {5\,\ln \left (x+\frac {1}{5}\right )}{7}+\frac {\ln \left (x\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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