Optimal. Leaf size=34 \[ -\frac {5+6 x}{2+5 x+3 x^2}+6 \log (1+x)-6 \log (2+3 x) \]
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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {628, 630, 31}
\begin {gather*} -\frac {6 x+5}{3 x^2+5 x+2}+6 \log (x+1)-6 \log (3 x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 628
Rule 630
Rubi steps
\begin {align*} \int \frac {1}{\left (2+5 x+3 x^2\right )^2} \, dx &=-\frac {5+6 x}{2+5 x+3 x^2}-6 \int \frac {1}{2+5 x+3 x^2} \, dx\\ &=-\frac {5+6 x}{2+5 x+3 x^2}-18 \int \frac {1}{2+3 x} \, dx+18 \int \frac {1}{3+3 x} \, dx\\ &=-\frac {5+6 x}{2+5 x+3 x^2}+6 \log (1+x)-6 \log (2+3 x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 33, normalized size = 0.97 \begin {gather*} \frac {-5-6 x}{2+5 x+3 x^2}+6 \log (1+x)-6 \log (2+3 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.61, size = 32, normalized size = 0.94
method | result | size |
default | \(-\frac {1}{x +1}+6 \ln \left (x +1\right )-\frac {3}{2+3 x}-6 \ln \left (2+3 x \right )\) | \(32\) |
risch | \(\frac {-2 x -\frac {5}{3}}{x^{2}+\frac {5}{3} x +\frac {2}{3}}+6 \ln \left (x +1\right )-6 \ln \left (2+3 x \right )\) | \(32\) |
norman | \(\frac {\frac {15}{2} x^{2}+\frac {13}{2} x}{3 x^{2}+5 x +2}+6 \ln \left (x +1\right )-6 \ln \left (2+3 x \right )\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 34, normalized size = 1.00 \begin {gather*} -\frac {6 \, x + 5}{3 \, x^{2} + 5 \, x + 2} - 6 \, \log \left (3 \, x + 2\right ) + 6 \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.35, size = 53, normalized size = 1.56 \begin {gather*} -\frac {6 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 6 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (x + 1\right ) + 6 \, x + 5}{3 \, x^{2} + 5 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 31, normalized size = 0.91 \begin {gather*} \frac {- 6 x - 5}{3 x^{2} + 5 x + 2} - 6 \log {\left (x + \frac {2}{3} \right )} + 6 \log {\left (x + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.70, size = 36, normalized size = 1.06 \begin {gather*} -\frac {6 \, x + 5}{3 \, x^{2} + 5 \, x + 2} - 6 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + 6 \, \log \left ({\left | x + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 34, normalized size = 1.00 \begin {gather*} -6\,\ln \left (\frac {3\,x+2}{x+1}\right )-\frac {2\,\left (3\,x+\frac {5}{2}\right )}{3\,x^2+5\,x+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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