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 = 0.01, antiderivative size = 22, 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 = {773, 632, 31} \begin {gather*} -\frac {2 x}{3}-6 \log (x+1)+\frac {85}{9} \log (3 x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 632
Rule 773
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*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} -\frac {2 x}{3}-6 \log (x+1)+\frac {85}{9} \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)}{2+5 x+3 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.38, size = 18, normalized size = 0.82 \begin {gather*} -\frac {2}{3} \, x + \frac {85}{9} \, \log \left (3 \, x + 2\right ) - 6 \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 20, normalized size = 0.91 \begin {gather*} -\frac {2}{3} \, x + \frac {85}{9} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - 6 \, \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 = 19, normalized size = 0.86 \begin {gather*} -\frac {2 x}{3}+\frac {85 \ln \left (3 x +2\right )}{9}-6 \ln \left (x +1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 18, normalized size = 0.82 \begin {gather*} -\frac {2}{3} \, x + \frac {85}{9} \, \log \left (3 \, x + 2\right ) - 6 \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 16, normalized size = 0.73 \begin {gather*} \frac {85\,\ln \left (x+\frac {2}{3}\right )}{9}-6\,\ln \left (x+1\right )-\frac {2\,x}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 20, normalized size = 0.91 \begin {gather*} - \frac {2 x}{3} + \frac {85 \log {\left (x + \frac {2}{3} \right )}}{9} - 6 \log {\left (x + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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