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 = 0.0355339, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {816, 1660, 632, 31} \[ -\frac{587 x+533}{9 \left (3 x^2+5 x+2\right )}+59 \log (x+1)-\frac{535}{9} \log (3 x+2) \]
Antiderivative was successfully verified.
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Rule 816
Rule 1660
Rule 632
Rule 31
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*}
Mathematica [A] time = 0.0282554, size = 38, normalized size = 1. \[ -\frac{587 x+533}{27 x^2+45 x+18}-\frac{535}{9} \log (-6 x-4)+59 \log (-2 (x+1)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 32, normalized size = 0.8 \begin{align*} -6\, \left ( 1+x \right ) ^{-1}+59\,\ln \left ( 1+x \right ) -{\frac{425}{18+27\,x}}-{\frac{535\,\ln \left ( 2+3\,x \right ) }{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.3688, size = 46, normalized size = 1.21 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27635, size = 149, normalized size = 3.92 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.167587, size = 31, normalized size = 0.82 \begin{align*} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14735, size = 49, normalized size = 1.29 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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