Optimal. Leaf size=44 \[ \frac {49}{243 (2+3 x)^3}-\frac {91}{54 (2+3 x)^2}+\frac {16}{9 (2+3 x)}+\frac {20}{81} \log (2+3 x) \]
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Rubi [A]
time = 0.01, antiderivative size = 44, 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 {16}{9 (3 x+2)}-\frac {91}{54 (3 x+2)^2}+\frac {49}{243 (3 x+2)^3}+\frac {20}{81} \log (3 x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int \frac {(1-2 x)^2 (3+5 x)}{(2+3 x)^4} \, dx &=\int \left (-\frac {49}{27 (2+3 x)^4}+\frac {91}{9 (2+3 x)^3}-\frac {16}{3 (2+3 x)^2}+\frac {20}{27 (2+3 x)}\right ) \, dx\\ &=\frac {49}{243 (2+3 x)^3}-\frac {91}{54 (2+3 x)^2}+\frac {16}{9 (2+3 x)}+\frac {20}{81} \log (2+3 x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 36, normalized size = 0.82 \begin {gather*} \frac {1916+7911 x+7776 x^2+120 (2+3 x)^3 \log (2+3 x)}{486 (2+3 x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 37, normalized size = 0.84
method | result | size |
risch | \(\frac {16 x^{2}+\frac {293}{18} x +\frac {958}{243}}{\left (2+3 x \right )^{3}}+\frac {20 \ln \left (2+3 x \right )}{81}\) | \(29\) |
norman | \(\frac {-\frac {191}{18} x^{2}-\frac {79}{54} x -\frac {479}{36} x^{3}}{\left (2+3 x \right )^{3}}+\frac {20 \ln \left (2+3 x \right )}{81}\) | \(32\) |
default | \(\frac {49}{243 \left (2+3 x \right )^{3}}-\frac {91}{54 \left (2+3 x \right )^{2}}+\frac {16}{9 \left (2+3 x \right )}+\frac {20 \ln \left (2+3 x \right )}{81}\) | \(37\) |
meijerg | \(\frac {x \left (\frac {9}{4} x^{2}+\frac {9}{2} x +3\right )}{16 \left (1+\frac {3 x}{2}\right )^{3}}-\frac {7 x^{2} \left (3+\frac {3 x}{2}\right )}{96 \left (1+\frac {3 x}{2}\right )^{3}}-\frac {x^{3}}{6 \left (1+\frac {3 x}{2}\right )^{3}}-\frac {5 x \left (\frac {99}{2} x^{2}+45 x +12\right )}{162 \left (1+\frac {3 x}{2}\right )^{3}}+\frac {20 \ln \left (1+\frac {3 x}{2}\right )}{81}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 38, normalized size = 0.86 \begin {gather*} \frac {7776 \, x^{2} + 7911 \, x + 1916}{486 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {20}{81} \, \log \left (3 \, x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.50, size = 52, normalized size = 1.18 \begin {gather*} \frac {7776 \, x^{2} + 120 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 7911 \, x + 1916}{486 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 34, normalized size = 0.77 \begin {gather*} \frac {7776 x^{2} + 7911 x + 1916}{13122 x^{3} + 26244 x^{2} + 17496 x + 3888} + \frac {20 \log {\left (3 x + 2 \right )}}{81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.05, size = 29, normalized size = 0.66 \begin {gather*} \frac {7776 \, x^{2} + 7911 \, x + 1916}{486 \, {\left (3 \, x + 2\right )}^{3}} + \frac {20}{81} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 33, normalized size = 0.75 \begin {gather*} \frac {20\,\ln \left (x+\frac {2}{3}\right )}{81}+\frac {\frac {16\,x^2}{27}+\frac {293\,x}{486}+\frac {958}{6561}}{x^3+2\,x^2+\frac {4\,x}{3}+\frac {8}{27}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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