Optimal. Leaf size=57 \[ \frac {105 (6 x+5)}{2 \left (3 x^2+5 x+2\right )}-\frac {35 x+29}{2 \left (3 x^2+5 x+2\right )^2}-315 \log (x+1)+315 \log (3 x+2) \]
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Rubi [A] time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {638, 614, 616, 31} \[ \frac {105 (6 x+5)}{2 \left (3 x^2+5 x+2\right )}-\frac {35 x+29}{2 \left (3 x^2+5 x+2\right )^2}-315 \log (x+1)+315 \log (3 x+2) \]
Antiderivative was successfully verified.
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Rule 31
Rule 614
Rule 616
Rule 638
Rubi steps
\begin {align*} \int \frac {5-x}{\left (2+5 x+3 x^2\right )^3} \, dx &=-\frac {29+35 x}{2 \left (2+5 x+3 x^2\right )^2}-\frac {105}{2} \int \frac {1}{\left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac {29+35 x}{2 \left (2+5 x+3 x^2\right )^2}+\frac {105 (5+6 x)}{2 \left (2+5 x+3 x^2\right )}+315 \int \frac {1}{2+5 x+3 x^2} \, dx\\ &=-\frac {29+35 x}{2 \left (2+5 x+3 x^2\right )^2}+\frac {105 (5+6 x)}{2 \left (2+5 x+3 x^2\right )}+945 \int \frac {1}{2+3 x} \, dx-945 \int \frac {1}{3+3 x} \, dx\\ &=-\frac {29+35 x}{2 \left (2+5 x+3 x^2\right )^2}+\frac {105 (5+6 x)}{2 \left (2+5 x+3 x^2\right )}-315 \log (1+x)+315 \log (2+3 x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 57, normalized size = 1.00 \[ \frac {-35 x-29}{2 \left (3 x^2+5 x+2\right )^2}+\frac {105 (6 x+5)}{2 \left (3 x^2+5 x+2\right )}-315 \log (x+1)+315 \log (3 x+2) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 93, normalized size = 1.63 \[ \frac {1890 \, x^{3} + 4725 \, x^{2} + 630 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 630 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (x + 1\right ) + 3850 \, x + 1021}{2 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 46, normalized size = 0.81 \[ \frac {1890 \, x^{3} + 4725 \, x^{2} + 3850 \, x + 1021}{2 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{2}} + 315 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - 315 \, \log \left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 48, normalized size = 0.84 \[ 315 \ln \left (3 x +2\right )-315 \ln \left (x +1\right )-\frac {51}{2 \left (3 x +2\right )^{2}}+\frac {156}{3 x +2}+\frac {3}{\left (x +1\right )^{2}}+\frac {53}{x +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 54, normalized size = 0.95 \[ \frac {1890 \, x^{3} + 4725 \, x^{2} + 3850 \, x + 1021}{2 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} + 315 \, \log \left (3 \, x + 2\right ) - 315 \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 45, normalized size = 0.79 \[ \frac {105\,x^3+\frac {525\,x^2}{2}+\frac {1925\,x}{9}+\frac {1021}{18}}{x^4+\frac {10\,x^3}{3}+\frac {37\,x^2}{9}+\frac {20\,x}{9}+\frac {4}{9}}-630\,\mathrm {atanh}\left (6\,x+5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 51, normalized size = 0.89 \[ - \frac {- 1890 x^{3} - 4725 x^{2} - 3850 x - 1021}{18 x^{4} + 60 x^{3} + 74 x^{2} + 40 x + 8} + 315 \log {\left (x + \frac {2}{3} \right )} - 315 \log {\left (x + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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