Optimal. Leaf size=43 \[ -\frac {2611 x+2449}{27 \left (3 x^2+5 x+2\right )}-\frac {8 x}{9}+71 \log (x+1)-\frac {1825}{27} \log (3 x+2) \]
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Rubi [A] time = 0.05, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {816, 1660, 1657, 632, 31} \begin {gather*} -\frac {2611 x+2449}{27 \left (3 x^2+5 x+2\right )}-\frac {8 x}{9}+71 \log (x+1)-\frac {1825}{27} \log (3 x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 632
Rule 816
Rule 1657
Rule 1660
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^2} \, dx &=\int \frac {\frac {13}{2} (3+2 x)^3-\frac {1}{2} (3+2 x)^4}{\left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac {2449+2611 x}{27 \left (2+5 x+3 x^2\right )}-\int \frac {\frac {563}{9}-\frac {52 x}{9}+\frac {8 x^2}{3}}{2+5 x+3 x^2} \, dx\\ &=-\frac {2449+2611 x}{27 \left (2+5 x+3 x^2\right )}-\int \left (\frac {8}{9}+\frac {547-92 x}{9 \left (2+5 x+3 x^2\right )}\right ) \, dx\\ &=-\frac {8 x}{9}-\frac {2449+2611 x}{27 \left (2+5 x+3 x^2\right )}-\frac {1}{9} \int \frac {547-92 x}{2+5 x+3 x^2} \, dx\\ &=-\frac {8 x}{9}-\frac {2449+2611 x}{27 \left (2+5 x+3 x^2\right )}-\frac {1825}{9} \int \frac {1}{2+3 x} \, dx+213 \int \frac {1}{3+3 x} \, dx\\ &=-\frac {8 x}{9}-\frac {2449+2611 x}{27 \left (2+5 x+3 x^2\right )}+71 \log (1+x)-\frac {1825}{27} \log (2+3 x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 47, normalized size = 1.09 \begin {gather*} -\frac {2611 x+2449}{81 x^2+135 x+54}-\frac {4}{9} (2 x+3)-\frac {1825}{27} \log (-6 x-4)+71 \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)^3}{\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 = 63, normalized size = 1.47 \begin {gather*} -\frac {72 \, x^{3} + 120 \, x^{2} + 1825 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 1917 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (x + 1\right ) + 2659 \, x + 2449}{27 \, {\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 = 39, normalized size = 0.91 \begin {gather*} -\frac {8}{9} \, x - \frac {2611 \, x + 2449}{27 \, {\left (3 \, x + 2\right )} {\left (x + 1\right )}} - \frac {1825}{27} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + 71 \, \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 = 35, normalized size = 0.81 \begin {gather*} -\frac {8 x}{9}-\frac {1825 \ln \left (3 x +2\right )}{27}+71 \ln \left (x +1\right )-\frac {2125}{27 \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 = 0.56, size = 37, normalized size = 0.86 \begin {gather*} -\frac {8}{9} \, x - \frac {2611 \, x + 2449}{27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} - \frac {1825}{27} \, \log \left (3 \, x + 2\right ) + 71 \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.26, size = 33, normalized size = 0.77 \begin {gather*} 71\,\ln \left (x+1\right )-\frac {8\,x}{9}-\frac {1825\,\ln \left (x+\frac {2}{3}\right )}{27}-\frac {\frac {2611\,x}{81}+\frac {2449}{81}}{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 = 0.16, size = 36, normalized size = 0.84 \begin {gather*} - \frac {8 x}{9} - \frac {2611 x + 2449}{81 x^{2} + 135 x + 54} - \frac {1825 \log {\left (x + \frac {2}{3} \right )}}{27} + 71 \log {\left (x + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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