Optimal. Leaf size=36 \[ -\frac{139 x+121}{3 \left (3 x^2+5 x+2\right )}+47 \log (x+1)-47 \log (3 x+2) \]
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Rubi [A] time = 0.0145328, antiderivative size = 36, 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 = {777, 616, 31} \[ -\frac{139 x+121}{3 \left (3 x^2+5 x+2\right )}+47 \log (x+1)-47 \log (3 x+2) \]
Antiderivative was successfully verified.
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Rule 777
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{(5-x) (3+2 x)}{\left (2+5 x+3 x^2\right )^2} \, dx &=-\frac{121+139 x}{3 \left (2+5 x+3 x^2\right )}-47 \int \frac{1}{2+5 x+3 x^2} \, dx\\ &=-\frac{121+139 x}{3 \left (2+5 x+3 x^2\right )}-141 \int \frac{1}{2+3 x} \, dx+141 \int \frac{1}{3+3 x} \, dx\\ &=-\frac{121+139 x}{3 \left (2+5 x+3 x^2\right )}+47 \log (1+x)-47 \log (2+3 x)\\ \end{align*}
Mathematica [A] time = 0.0143276, size = 34, normalized size = 0.94 \[ -\frac{139 x+121}{9 x^2+15 x+6}+47 \log (x+1)-47 \log (3 x+2) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 32, normalized size = 0.9 \begin{align*} -6\, \left ( 1+x \right ) ^{-1}+47\,\ln \left ( 1+x \right ) -{\frac{85}{6+9\,x}}-47\,\ln \left ( 2+3\,x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21612, size = 46, normalized size = 1.28 \begin{align*} -\frac{139 \, x + 121}{3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} - 47 \, \log \left (3 \, x + 2\right ) + 47 \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.21872, size = 149, normalized size = 4.14 \begin{align*} -\frac{141 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 141 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (x + 1\right ) + 139 \, x + 121}{3 \,{\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 = 0.142315, size = 29, normalized size = 0.81 \begin{align*} - \frac{139 x + 121}{9 x^{2} + 15 x + 6} - 47 \log{\left (x + \frac{2}{3} \right )} + 47 \log{\left (x + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13793, size = 49, normalized size = 1.36 \begin{align*} -\frac{139 \, x + 121}{3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} - 47 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + 47 \, \log \left ({\left | x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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