Optimal. Leaf size=64 \[ \frac {502254 x+398585}{486 \left (3 x^2+5 x+2\right )}-\frac {57499 x+56041}{486 \left (3 x^2+5 x+2\right )^2}-\frac {32 x}{27}-1085 \log (x+1)+\frac {29375}{27} \log (3 x+2) \]
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Rubi [A] time = 0.08, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {816, 1660, 1657, 632, 31} \begin {gather*} \frac {502254 x+398585}{486 \left (3 x^2+5 x+2\right )}-\frac {57499 x+56041}{486 \left (3 x^2+5 x+2\right )^2}-\frac {32 x}{27}-1085 \log (x+1)+\frac {29375}{27} \log (3 x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 632
Rule 816
Rule 1657
Rule 1660
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^5}{\left (2+5 x+3 x^2\right )^3} \, dx &=\int \frac {\frac {13}{2} (3+2 x)^5-\frac {1}{2} (3+2 x)^6}{\left (2+5 x+3 x^2\right )^3} \, dx\\ &=-\frac {56041+57499 x}{486 \left (2+5 x+3 x^2\right )^2}-\frac {1}{2} \int \frac {-\frac {72539}{243}-\frac {87920 x}{81}-\frac {9824 x^2}{27}+\frac {160 x^3}{9}+\frac {64 x^4}{3}}{\left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac {56041+57499 x}{486 \left (2+5 x+3 x^2\right )^2}+\frac {398585+502254 x}{486 \left (2+5 x+3 x^2\right )}+\frac {1}{2} \int \frac {\frac {58942}{27}+\frac {160 x}{27}-\frac {64 x^2}{9}}{2+5 x+3 x^2} \, dx\\ &=-\frac {56041+57499 x}{486 \left (2+5 x+3 x^2\right )^2}+\frac {398585+502254 x}{486 \left (2+5 x+3 x^2\right )}+\frac {1}{2} \int \left (-\frac {64}{27}+\frac {10 (1969+16 x)}{9 \left (2+5 x+3 x^2\right )}\right ) \, dx\\ &=-\frac {32 x}{27}-\frac {56041+57499 x}{486 \left (2+5 x+3 x^2\right )^2}+\frac {398585+502254 x}{486 \left (2+5 x+3 x^2\right )}+\frac {5}{9} \int \frac {1969+16 x}{2+5 x+3 x^2} \, dx\\ &=-\frac {32 x}{27}-\frac {56041+57499 x}{486 \left (2+5 x+3 x^2\right )^2}+\frac {398585+502254 x}{486 \left (2+5 x+3 x^2\right )}-3255 \int \frac {1}{3+3 x} \, dx+\frac {29375}{9} \int \frac {1}{2+3 x} \, dx\\ &=-\frac {32 x}{27}-\frac {56041+57499 x}{486 \left (2+5 x+3 x^2\right )^2}+\frac {398585+502254 x}{486 \left (2+5 x+3 x^2\right )}-1085 \log (1+x)+\frac {29375}{27} \log (2+3 x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 81, normalized size = 1.27 \begin {gather*} \frac {-1728 x^5-8352 x^4+486510 x^3+1221179 x^2+176250 \left (3 x^2+5 x+2\right )^2 \log (-6 x-4)-175770 \left (3 x^2+5 x+2\right )^2 \log (-2 (x+1))+973450 x+245891}{162 \left (3 x^2+5 x+2\right )^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)^5}{\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 = 0.39, size = 103, normalized size = 1.61 \begin {gather*} -\frac {1728 \, x^{5} + 5760 \, x^{4} - 495150 \, x^{3} - 1231835 \, x^{2} - 176250 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (3 \, x + 2\right ) + 175770 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (x + 1\right ) - 979210 \, x - 247043}{162 \, {\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 = 0.15, size = 49, normalized size = 0.77 \begin {gather*} -\frac {32}{27} \, x + \frac {502254 \, x^{3} + 1235675 \, x^{2} + 979978 \, x + 247043}{162 \, {\left (3 \, x + 2\right )}^{2} {\left (x + 1\right )}^{2}} + \frac {29375}{27} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - 1085 \, \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 = 0.06, size = 51, normalized size = 0.80 \begin {gather*} -\frac {32 x}{27}+\frac {29375 \ln \left (3 x +2\right )}{27}-1085 \ln \left (x +1\right )-\frac {53125}{162 \left (3 x +2\right )^{2}}+\frac {6250}{9 \left (3 x +2\right )}+\frac {3}{\left (x +1\right )^{2}}+\frac {113}{x +1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 57, normalized size = 0.89 \begin {gather*} -\frac {32}{27} \, x + \frac {502254 \, x^{3} + 1235675 \, x^{2} + 979978 \, x + 247043}{162 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} + \frac {29375}{27} \, \log \left (3 \, x + 2\right ) - 1085 \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.44, size = 52, normalized size = 0.81 \begin {gather*} \frac {29375\,\ln \left (x+\frac {2}{3}\right )}{27}-1085\,\ln \left (x+1\right )-\frac {32\,x}{27}+\frac {\frac {9301\,x^3}{27}+\frac {1235675\,x^2}{1458}+\frac {489989\,x}{729}+\frac {247043}{1458}}{x^4+\frac {10\,x^3}{3}+\frac {37\,x^2}{9}+\frac {20\,x}{9}+\frac {4}{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 58, normalized size = 0.91 \begin {gather*} - \frac {32 x}{27} - \frac {- 502254 x^{3} - 1235675 x^{2} - 979978 x - 247043}{1458 x^{4} + 4860 x^{3} + 5994 x^{2} + 3240 x + 648} + \frac {29375 \log {\left (x + \frac {2}{3} \right )}}{27} - 1085 \log {\left (x + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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