Optimal. Leaf size=22 \[ -\frac{2 x}{3}-6 \log (x+1)+\frac{85}{9} \log (3 x+2) \]
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Rubi [A] time = 0.0145041, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {773, 632, 31} \[ -\frac{2 x}{3}-6 \log (x+1)+\frac{85}{9} \log (3 x+2) \]
Antiderivative was successfully verified.
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Rule 773
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{(5-x) (3+2 x)}{2+5 x+3 x^2} \, dx &=-\frac{2 x}{3}+\frac{1}{3} \int \frac{49+31 x}{2+5 x+3 x^2} \, dx\\ &=-\frac{2 x}{3}-18 \int \frac{1}{3+3 x} \, dx+\frac{85}{3} \int \frac{1}{2+3 x} \, dx\\ &=-\frac{2 x}{3}-6 \log (1+x)+\frac{85}{9} \log (2+3 x)\\ \end{align*}
Mathematica [A] time = 0.0071633, size = 22, normalized size = 1. \[ -\frac{2 x}{3}-6 \log (x+1)+\frac{85}{9} \log (3 x+2) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 19, normalized size = 0.9 \begin{align*} -{\frac{2\,x}{3}}-6\,\ln \left ( 1+x \right ) +{\frac{85\,\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 = 1.28362, size = 24, normalized size = 1.09 \begin{align*} -\frac{2}{3} \, x + \frac{85}{9} \, \log \left (3 \, x + 2\right ) - 6 \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36702, size = 58, normalized size = 2.64 \begin{align*} -\frac{2}{3} \, x + \frac{85}{9} \, \log \left (3 \, x + 2\right ) - 6 \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.119572, size = 20, normalized size = 0.91 \begin{align*} - \frac{2 x}{3} + \frac{85 \log{\left (x + \frac{2}{3} \right )}}{9} - 6 \log{\left (x + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15013, size = 27, normalized size = 1.23 \begin{align*} -\frac{2}{3} \, x + \frac{85}{9} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - 6 \, \log \left ({\left | x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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