Optimal. Leaf size=67 \[ -\frac {1}{256 x^4}+\frac {3}{128 x^3}-\frac {27}{256 x^2}+\frac {135}{256 x}+\frac {81}{512 (2+3 x)^2}+\frac {405}{512 (2+3 x)}+\frac {1215 \log (x)}{1024}-\frac {1215 \log (2+3 x)}{1024} \]
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Rubi [A]
time = 0.02, antiderivative size = 67, 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}{256 x^4}+\frac {3}{128 x^3}-\frac {27}{256 x^2}+\frac {135}{256 x}+\frac {405}{512 (3 x+2)}+\frac {81}{512 (3 x+2)^2}+\frac {1215 \log (x)}{1024}-\frac {1215 \log (3 x+2)}{1024} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rubi steps
\begin {align*} \int \frac {1}{x^5 (4+6 x)^3} \, dx &=\int \left (\frac {1}{64 x^5}-\frac {9}{128 x^4}+\frac {27}{128 x^3}-\frac {135}{256 x^2}+\frac {1215}{1024 x}-\frac {243}{256 (2+3 x)^3}-\frac {1215}{512 (2+3 x)^2}-\frac {3645}{1024 (2+3 x)}\right ) \, dx\\ &=-\frac {1}{256 x^4}+\frac {3}{128 x^3}-\frac {27}{256 x^2}+\frac {135}{256 x}+\frac {81}{512 (2+3 x)^2}+\frac {405}{512 (2+3 x)}+\frac {1215 \log (x)}{1024}-\frac {1215 \log (2+3 x)}{1024}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 54, normalized size = 0.81 \begin {gather*} \frac {\frac {2 \left (-8+24 x-90 x^2+540 x^3+3645 x^4+3645 x^5\right )}{x^4 (2+3 x)^2}+1215 \log (x)-1215 \log (2+3 x)}{1024} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 52, normalized size = 0.78
method | result | size |
norman | \(\frac {-\frac {1}{64}-\frac {3645}{256} x^{5}-\frac {32805}{2048} x^{6}+\frac {3}{64} x -\frac {45}{256} x^{2}+\frac {135}{128} x^{3}}{x^{4} \left (2+3 x \right )^{2}}+\frac {1215 \ln \left (x \right )}{1024}-\frac {1215 \ln \left (2+3 x \right )}{1024}\) | \(50\) |
risch | \(\frac {\frac {3645}{512} x^{5}+\frac {3645}{512} x^{4}+\frac {135}{128} x^{3}-\frac {45}{256} x^{2}+\frac {3}{64} x -\frac {1}{64}}{x^{4} \left (2+3 x \right )^{2}}+\frac {1215 \ln \left (x \right )}{1024}-\frac {1215 \ln \left (2+3 x \right )}{1024}\) | \(51\) |
default | \(-\frac {1}{256 x^{4}}+\frac {3}{128 x^{3}}-\frac {27}{256 x^{2}}+\frac {135}{256 x}+\frac {81}{512 \left (2+3 x \right )^{2}}+\frac {405}{512 \left (2+3 x \right )}+\frac {1215 \ln \left (x \right )}{1024}-\frac {1215 \ln \left (2+3 x \right )}{1024}\) | \(52\) |
meijerg | \(-\frac {1}{256 x^{4}}+\frac {3}{128 x^{3}}-\frac {27}{256 x^{2}}+\frac {135}{256 x}+\frac {891}{2048}+\frac {1215 \ln \left (x \right )}{1024}-\frac {1215 \ln \left (2\right )}{1024}+\frac {1215 \ln \left (3\right )}{1024}-\frac {243 x \left (\frac {33 x}{2}+12\right )}{4096 \left (1+\frac {3 x}{2}\right )^{2}}-\frac {1215 \ln \left (1+\frac {3 x}{2}\right )}{1024}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 58, normalized size = 0.87 \begin {gather*} \frac {3645 \, x^{5} + 3645 \, x^{4} + 540 \, x^{3} - 90 \, x^{2} + 24 \, x - 8}{512 \, {\left (9 \, x^{6} + 12 \, x^{5} + 4 \, x^{4}\right )}} - \frac {1215}{1024} \, \log \left (3 \, x + 2\right ) + \frac {1215}{1024} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 89, normalized size = 1.33 \begin {gather*} \frac {7290 \, x^{5} + 7290 \, x^{4} + 1080 \, x^{3} - 180 \, x^{2} - 1215 \, {\left (9 \, x^{6} + 12 \, x^{5} + 4 \, x^{4}\right )} \log \left (3 \, x + 2\right ) + 1215 \, {\left (9 \, x^{6} + 12 \, x^{5} + 4 \, x^{4}\right )} \log \left (x\right ) + 48 \, x - 16}{1024 \, {\left (9 \, x^{6} + 12 \, x^{5} + 4 \, x^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 56, normalized size = 0.84 \begin {gather*} \frac {1215 \log {\left (x \right )}}{1024} - \frac {1215 \log {\left (x + \frac {2}{3} \right )}}{1024} + \frac {3645 x^{5} + 3645 x^{4} + 540 x^{3} - 90 x^{2} + 24 x - 8}{4608 x^{6} + 6144 x^{5} + 2048 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.65, size = 52, normalized size = 0.78 \begin {gather*} \frac {3645 \, x^{5} + 3645 \, x^{4} + 540 \, x^{3} - 90 \, x^{2} + 24 \, x - 8}{512 \, {\left (3 \, x + 2\right )}^{2} x^{4}} - \frac {1215}{1024} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {1215}{1024} \, \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 = 0.05, size = 51, normalized size = 0.76 \begin {gather*} \frac {\frac {405\,x^5}{512}+\frac {405\,x^4}{512}+\frac {15\,x^3}{128}-\frac {5\,x^2}{256}+\frac {x}{192}-\frac {1}{576}}{x^6+\frac {4\,x^5}{3}+\frac {4\,x^4}{9}}-\frac {1215\,\mathrm {atanh}\left (3\,x+1\right )}{512} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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