Optimal. Leaf size=43 \[ -\frac {1}{22 (3+5 x)^2}-\frac {2}{121 (3+5 x)}-\frac {4 \log (1-2 x)}{1331}+\frac {4 \log (3+5 x)}{1331} \]
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Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {46}
\begin {gather*} -\frac {2}{121 (5 x+3)}-\frac {1}{22 (5 x+3)^2}-\frac {4 \log (1-2 x)}{1331}+\frac {4 \log (5 x+3)}{1331} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x) (3+5 x)^3} \, dx &=\int \left (-\frac {8}{1331 (-1+2 x)}+\frac {5}{11 (3+5 x)^3}+\frac {10}{121 (3+5 x)^2}+\frac {20}{1331 (3+5 x)}\right ) \, dx\\ &=-\frac {1}{22 (3+5 x)^2}-\frac {2}{121 (3+5 x)}-\frac {4 \log (1-2 x)}{1331}+\frac {4 \log (3+5 x)}{1331}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 35, normalized size = 0.81 \begin {gather*} \frac {-\frac {11 (23+20 x)}{(3+5 x)^2}-8 \log (5-10 x)+8 \log (3+5 x)}{2662} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 36, normalized size = 0.84
method | result | size |
risch | \(\frac {-\frac {10 x}{121}-\frac {23}{242}}{\left (3+5 x \right )^{2}}-\frac {4 \ln \left (-1+2 x \right )}{1331}+\frac {4 \ln \left (3+5 x \right )}{1331}\) | \(32\) |
norman | \(\frac {\frac {85}{363} x +\frac {575}{2178} x^{2}}{\left (3+5 x \right )^{2}}-\frac {4 \ln \left (-1+2 x \right )}{1331}+\frac {4 \ln \left (3+5 x \right )}{1331}\) | \(35\) |
default | \(-\frac {4 \ln \left (-1+2 x \right )}{1331}-\frac {1}{22 \left (3+5 x \right )^{2}}-\frac {2}{121 \left (3+5 x \right )}+\frac {4 \ln \left (3+5 x \right )}{1331}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 36, normalized size = 0.84 \begin {gather*} -\frac {20 \, x + 23}{242 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {4}{1331} \, \log \left (5 \, x + 3\right ) - \frac {4}{1331} \, \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 = 1.28, size = 55, normalized size = 1.28 \begin {gather*} \frac {8 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 8 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (2 \, x - 1\right ) - 220 \, x - 253}{2662 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 34, normalized size = 0.79 \begin {gather*} - \frac {20 x + 23}{6050 x^{2} + 7260 x + 2178} - \frac {4 \log {\left (x - \frac {1}{2} \right )}}{1331} + \frac {4 \log {\left (x + \frac {3}{5} \right )}}{1331} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.26, size = 33, normalized size = 0.77 \begin {gather*} -\frac {20 \, x + 23}{242 \, {\left (5 \, x + 3\right )}^{2}} + \frac {4}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {4}{1331} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 26, normalized size = 0.60 \begin {gather*} \frac {8\,\mathrm {atanh}\left (\frac {20\,x}{11}+\frac {1}{11}\right )}{1331}-\frac {\frac {2\,x}{605}+\frac {23}{6050}}{x^2+\frac {6\,x}{5}+\frac {9}{25}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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