Optimal. Leaf size=38 \[ -\frac {587 x+533}{9 \left (3 x^2+5 x+2\right )}+59 \log (x+1)-\frac {535}{9} \log (3 x+2) \]
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Rubi [A] time = 0.04, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {816, 1660, 632, 31} \begin {gather*} -\frac {587 x+533}{9 \left (3 x^2+5 x+2\right )}+59 \log (x+1)-\frac {535}{9} \log (3 x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 632
Rule 816
Rule 1660
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^2} \, dx &=\int \frac {45+51 x+8 x^2-4 x^3}{\left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac {533+587 x}{9 \left (2+5 x+3 x^2\right )}-\int \frac {\frac {181}{3}+\frac {4 x}{3}}{2+5 x+3 x^2} \, dx\\ &=-\frac {533+587 x}{9 \left (2+5 x+3 x^2\right )}+177 \int \frac {1}{3+3 x} \, dx-\frac {535}{3} \int \frac {1}{2+3 x} \, dx\\ &=-\frac {533+587 x}{9 \left (2+5 x+3 x^2\right )}+59 \log (1+x)-\frac {535}{9} \log (2+3 x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 38, normalized size = 1.00 \begin {gather*} -\frac {587 x+533}{27 x^2+45 x+18}-\frac {535}{9} \log (-6 x-4)+59 \log (-2 (x+1)) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 53, normalized size = 1.39 \begin {gather*} -\frac {535 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 531 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (x + 1\right ) + 587 \, x + 533}{9 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 36, normalized size = 0.95 \begin {gather*} -\frac {587 \, x + 533}{9 \, {\left (3 \, x + 2\right )} {\left (x + 1\right )}} - \frac {535}{9} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + 59 \, \log \left ({\left | x + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 32, normalized size = 0.84 \begin {gather*} -\frac {535 \ln \left (3 x +2\right )}{9}+59 \ln \left (x +1\right )-\frac {425}{9 \left (3 x +2\right )}-\frac {6}{x +1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 34, normalized size = 0.89 \begin {gather*} -\frac {587 \, x + 533}{9 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} - \frac {535}{9} \, \log \left (3 \, x + 2\right ) + 59 \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 30, normalized size = 0.79 \begin {gather*} 59\,\ln \left (x+1\right )-\frac {535\,\ln \left (x+\frac {2}{3}\right )}{9}-\frac {\frac {587\,x}{27}+\frac {533}{27}}{x^2+\frac {5\,x}{3}+\frac {2}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 31, normalized size = 0.82 \begin {gather*} - \frac {587 x + 533}{27 x^{2} + 45 x + 18} - \frac {535 \log {\left (x + \frac {2}{3} \right )}}{9} + 59 \log {\left (x + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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