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 = 0.015495, antiderivative size = 57, normalized size of antiderivative = 1., 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 638
Rule 614
Rule 616
Rule 31
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*}
Mathematica [A] time = 0.0145249, size = 57, normalized size = 1. \[ \frac{-35 x-29}{2 \left (3 x^2+5 x+2\right )^2}+\frac{105 (6 x+5)}{2 \left (3 x^2+5 x+2\right )}-315 \log (x+1)+315 \log (3 x+2) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 48, normalized size = 0.8 \begin{align*} 3\, \left ( 1+x \right ) ^{-2}+53\, \left ( 1+x \right ) ^{-1}-315\,\ln \left ( 1+x \right ) -{\frac{51}{2\, \left ( 2+3\,x \right ) ^{2}}}+156\, \left ( 2+3\,x \right ) ^{-1}+315\,\ln \left ( 2+3\,x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00078, size = 73, normalized size = 1.28 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25189, size = 257, normalized size = 4.51 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.1704, size = 49, normalized size = 0.86 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18099, size = 62, normalized size = 1.09 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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