Optimal. Leaf size=69 \[ -\frac {(35 x+29) (2 x+3)^3}{2 \left (3 x^2+5 x+2\right )^2}+\frac {141 (8 x+7) (2 x+3)}{2 \left (3 x^2+5 x+2\right )}-705 \log (x+1)+705 \log (3 x+2) \]
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Rubi [A] time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {804, 722, 616, 31} \begin {gather*} -\frac {(35 x+29) (2 x+3)^3}{2 \left (3 x^2+5 x+2\right )^2}+\frac {141 (8 x+7) (2 x+3)}{2 \left (3 x^2+5 x+2\right )}-705 \log (x+1)+705 \log (3 x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 616
Rule 722
Rule 804
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^3} \, dx &=-\frac {(3+2 x)^3 (29+35 x)}{2 \left (2+5 x+3 x^2\right )^2}-\frac {141}{2} \int \frac {(3+2 x)^2}{\left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac {(3+2 x)^3 (29+35 x)}{2 \left (2+5 x+3 x^2\right )^2}+\frac {141 (3+2 x) (7+8 x)}{2 \left (2+5 x+3 x^2\right )}+705 \int \frac {1}{2+5 x+3 x^2} \, dx\\ &=-\frac {(3+2 x)^3 (29+35 x)}{2 \left (2+5 x+3 x^2\right )^2}+\frac {141 (3+2 x) (7+8 x)}{2 \left (2+5 x+3 x^2\right )}+2115 \int \frac {1}{2+3 x} \, dx-2115 \int \frac {1}{3+3 x} \, dx\\ &=-\frac {(3+2 x)^3 (29+35 x)}{2 \left (2+5 x+3 x^2\right )^2}+\frac {141 (3+2 x) (7+8 x)}{2 \left (2+5 x+3 x^2\right )}-705 \log (1+x)+705 \log (2+3 x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 59, normalized size = 0.86 \begin {gather*} -\frac {2611 x+2449}{54 \left (3 x^2+5 x+2\right )^2}+\frac {38118 x+31673}{54 \left (3 x^2+5 x+2\right )}+705 \log (-6 x-4)-705 \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 )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.38, size = 93, normalized size = 1.35 \begin {gather*} \frac {38118 \, x^{3} + 95203 \, x^{2} + 12690 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 12690 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (x + 1\right ) + 77330 \, x + 20299}{18 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 46, normalized size = 0.67 \begin {gather*} \frac {38118 \, x^{3} + 95203 \, x^{2} + 77330 \, x + 20299}{18 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{2}} + 705 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - 705 \, \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 = 48, normalized size = 0.70 \begin {gather*} 705 \ln \left (3 x +2\right )-705 \ln \left (x +1\right )-\frac {2125}{18 \left (3 x +2\right )^{2}}+\frac {3950}{9 \left (3 x +2\right )}+\frac {3}{\left (x +1\right )^{2}}+\frac {89}{x +1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 54, normalized size = 0.78 \begin {gather*} \frac {38118 \, x^{3} + 95203 \, x^{2} + 77330 \, x + 20299}{18 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} + 705 \, \log \left (3 \, x + 2\right ) - 705 \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 45, normalized size = 0.65 \begin {gather*} \frac {\frac {6353\,x^3}{27}+\frac {95203\,x^2}{162}+\frac {38665\,x}{81}+\frac {20299}{162}}{x^4+\frac {10\,x^3}{3}+\frac {37\,x^2}{9}+\frac {20\,x}{9}+\frac {4}{9}}-1410\,\mathrm {atanh}\left (6\,x+5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 51, normalized size = 0.74 \begin {gather*} - \frac {- 38118 x^{3} - 95203 x^{2} - 77330 x - 20299}{162 x^{4} + 540 x^{3} + 666 x^{2} + 360 x + 72} + 705 \log {\left (x + \frac {2}{3} \right )} - 705 \log {\left (x + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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