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 = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {777, 616, 31} \begin {gather*} -\frac {139 x+121}{3 \left (3 x^2+5 x+2\right )}+47 \log (x+1)-47 \log (3 x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 616
Rule 777
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*}
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Mathematica [A] time = 0.02, size = 34, normalized size = 0.94 \begin {gather*} -\frac {139 x+121}{9 x^2+15 x+6}+47 \log (x+1)-47 \log (3 x+2) \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)}{\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 = 0.38, size = 53, normalized size = 1.47 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 36, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 32, normalized size = 0.89 \begin {gather*} -47 \ln \left (3 x +2\right )+47 \ln \left (x +1\right )-\frac {85}{3 \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 = 0.61, size = 34, normalized size = 0.94 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.27, size = 26, normalized size = 0.72 \begin {gather*} 94\,\mathrm {atanh}\left (6\,x+5\right )-\frac {\frac {139\,x}{9}+\frac {121}{9}}{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 = 0.13, size = 29, normalized size = 0.81 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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