Optimal. Leaf size=46 \[ -\frac {1}{64 x}-\frac {3}{64 (2+3 x)^2}-\frac {3}{32 (2+3 x)}-\frac {9 \log (x)}{128}+\frac {9}{128} \log (2+3 x) \]
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Rubi [A]
time = 0.01, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {46}
\begin {gather*} -\frac {1}{64 x}-\frac {3}{32 (3 x+2)}-\frac {3}{64 (3 x+2)^2}-\frac {9 \log (x)}{128}+\frac {9}{128} \log (3 x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rubi steps
\begin {align*} \int \frac {1}{x^2 (4+6 x)^3} \, dx &=\int \left (\frac {1}{64 x^2}-\frac {9}{128 x}+\frac {9}{32 (2+3 x)^3}+\frac {9}{32 (2+3 x)^2}+\frac {27}{128 (2+3 x)}\right ) \, dx\\ &=-\frac {1}{64 x}-\frac {3}{64 (2+3 x)^2}-\frac {3}{32 (2+3 x)}-\frac {9 \log (x)}{128}+\frac {9}{128} \log (2+3 x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 39, normalized size = 0.85 \begin {gather*} \frac {1}{128} \left (-\frac {2 \left (4+27 x+27 x^2\right )}{x (2+3 x)^2}-9 \log (x)+9 \log (2+3 x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.86, size = 49, normalized size = 1.07 \begin {gather*} \frac {-8-54 x+9 x \left (4+12 x+9 x^2\right ) \left (\text {Log}\left [\frac {2}{3}+x\right ]-\text {Log}\left [x\right ]\right )-54 x^2}{128 x \left (4+12 x+9 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 37, normalized size = 0.80
method | result | size |
risch | \(\frac {-\frac {27}{64} x^{2}-\frac {27}{64} x -\frac {1}{16}}{x \left (2+3 x \right )^{2}}-\frac {9 \ln \left (x \right )}{128}+\frac {9 \ln \left (2+3 x \right )}{128}\) | \(36\) |
default | \(-\frac {1}{64 x}-\frac {3}{64 \left (2+3 x \right )^{2}}-\frac {3}{32 \left (2+3 x \right )}-\frac {9 \ln \left (x \right )}{128}+\frac {9 \ln \left (2+3 x \right )}{128}\) | \(37\) |
norman | \(\frac {-\frac {1}{16}+\frac {27}{32} x^{2}+\frac {243}{256} x^{3}}{x \left (2+3 x \right )^{2}}-\frac {9 \ln \left (x \right )}{128}+\frac {9 \ln \left (2+3 x \right )}{128}\) | \(37\) |
meijerg | \(-\frac {1}{64 x}-\frac {15}{256}-\frac {9 \ln \left (x \right )}{128}+\frac {9 \ln \left (2\right )}{128}-\frac {9 \ln \left (3\right )}{128}+\frac {9 x \left (\frac {15 x}{2}+6\right )}{512 \left (1+\frac {3 x}{2}\right )^{2}}+\frac {9 \ln \left (1+\frac {3 x}{2}\right )}{128}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 41, normalized size = 0.89 \begin {gather*} -\frac {27 \, x^{2} + 27 \, x + 4}{64 \, {\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )}} + \frac {9}{128} \, \log \left (3 \, x + 2\right ) - \frac {9}{128} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 68, normalized size = 1.48 \begin {gather*} -\frac {54 \, x^{2} - 9 \, {\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )} \log \left (3 \, x + 2\right ) + 9 \, {\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )} \log \left (x\right ) + 54 \, x + 8}{128 \, {\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 41, normalized size = 0.89 \begin {gather*} \frac {- 27 x^{2} - 27 x - 4}{576 x^{3} + 768 x^{2} + 256 x} - \frac {9 \log {\left (x \right )}}{128} + \frac {9 \log {\left (x + \frac {2}{3} \right )}}{128} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 45, normalized size = 0.98 \begin {gather*} -\frac {9}{128} \ln \left |x\right |+\frac {9}{128} \ln \left |3 x+2\right |+\frac {\frac {1}{256} \left (-108 x^{2}-108 x-16\right )}{x \left (3 x+2\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 35, normalized size = 0.76 \begin {gather*} \frac {9\,\mathrm {atanh}\left (3\,x+1\right )}{64}-\frac {\frac {3\,x^2}{64}+\frac {3\,x}{64}+\frac {1}{144}}{x^3+\frac {4\,x^2}{3}+\frac {4\,x}{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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