Optimal. Leaf size=34 \[ -\frac {35 x+29}{3 x^2+5 x+2}+35 \log (x+1)-35 \log (3 x+2) \]
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Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {638, 616, 31} \begin {gather*} -\frac {35 x+29}{3 x^2+5 x+2}+35 \log (x+1)-35 \log (3 x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 616
Rule 638
Rubi steps
\begin {align*} \int \frac {5-x}{\left (2+5 x+3 x^2\right )^2} \, dx &=-\frac {29+35 x}{2+5 x+3 x^2}-35 \int \frac {1}{2+5 x+3 x^2} \, dx\\ &=-\frac {29+35 x}{2+5 x+3 x^2}-105 \int \frac {1}{2+3 x} \, dx+105 \int \frac {1}{3+3 x} \, dx\\ &=-\frac {29+35 x}{2+5 x+3 x^2}+35 \log (1+x)-35 \log (2+3 x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.97 \begin {gather*} \frac {-35 x-29}{3 x^2+5 x+2}+35 \log (x+1)-35 \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}{\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 = 1.56 \begin {gather*} -\frac {35 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 35 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (x + 1\right ) + 35 \, x + 29}{3 \, x^{2} + 5 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 36, normalized size = 1.06 \begin {gather*} -\frac {35 \, x + 29}{3 \, x^{2} + 5 \, x + 2} - 35 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + 35 \, \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 = 0.94 \begin {gather*} -35 \ln \left (3 x +2\right )+35 \ln \left (x +1\right )-\frac {17}{3 x +2}-\frac {6}{x +1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 34, normalized size = 1.00 \begin {gather*} -\frac {35 \, x + 29}{3 \, x^{2} + 5 \, x + 2} - 35 \, \log \left (3 \, x + 2\right ) + 35 \, \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 = 26, normalized size = 0.76 \begin {gather*} 70\,\mathrm {atanh}\left (6\,x+5\right )-\frac {\frac {35\,x}{3}+\frac {29}{3}}{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 = 0.85 \begin {gather*} - \frac {35 x + 29}{3 x^{2} + 5 x + 2} - 35 \log {\left (x + \frac {2}{3} \right )} + 35 \log {\left (x + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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