Optimal. Leaf size=32 \[ -\frac {1}{55 (3+5 x)}-\frac {7}{121} \log (1-2 x)+\frac {7}{121} \log (3+5 x) \]
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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78}
\begin {gather*} -\frac {1}{55 (5 x+3)}-\frac {7}{121} \log (1-2 x)+\frac {7}{121} \log (5 x+3) \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int \frac {2+3 x}{(1-2 x) (3+5 x)^2} \, dx &=\int \left (-\frac {14}{121 (-1+2 x)}+\frac {1}{11 (3+5 x)^2}+\frac {35}{121 (3+5 x)}\right ) \, dx\\ &=-\frac {1}{55 (3+5 x)}-\frac {7}{121} \log (1-2 x)+\frac {7}{121} \log (3+5 x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 30, normalized size = 0.94 \begin {gather*} \frac {1}{605} \left (-\frac {11}{3+5 x}-35 \log (5-10 x)+35 \log (3+5 x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 27, normalized size = 0.84
method | result | size |
risch | \(-\frac {1}{275 \left (x +\frac {3}{5}\right )}-\frac {7 \ln \left (-1+2 x \right )}{121}+\frac {7 \ln \left (3+5 x \right )}{121}\) | \(25\) |
default | \(-\frac {7 \ln \left (-1+2 x \right )}{121}-\frac {1}{55 \left (3+5 x \right )}+\frac {7 \ln \left (3+5 x \right )}{121}\) | \(27\) |
norman | \(\frac {x}{99+165 x}-\frac {7 \ln \left (-1+2 x \right )}{121}+\frac {7 \ln \left (3+5 x \right )}{121}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 26, normalized size = 0.81 \begin {gather*} -\frac {1}{55 \, {\left (5 \, x + 3\right )}} + \frac {7}{121} \, \log \left (5 \, x + 3\right ) - \frac {7}{121} \, \log \left (2 \, x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.74, size = 37, normalized size = 1.16 \begin {gather*} \frac {35 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 35 \, {\left (5 \, x + 3\right )} \log \left (2 \, x - 1\right ) - 11}{605 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 26, normalized size = 0.81 \begin {gather*} - \frac {7 \log {\left (x - \frac {1}{2} \right )}}{121} + \frac {7 \log {\left (x + \frac {3}{5} \right )}}{121} - \frac {1}{275 x + 165} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.46, size = 25, normalized size = 0.78 \begin {gather*} -\frac {1}{55 \, {\left (5 \, x + 3\right )}} - \frac {7}{121} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 18, normalized size = 0.56 \begin {gather*} \frac {14\,\mathrm {atanh}\left (\frac {20\,x}{11}+\frac {1}{11}\right )}{121}-\frac {1}{275\,\left (x+\frac {3}{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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