Optimal. Leaf size=50 \[ -\frac {8 x^2}{9}-\frac {12083 x+11597}{81 \left (3 x^2+5 x+2\right )}+\frac {112 x}{27}+83 \log (x+1)-\frac {1625}{27} \log (3 x+2) \]
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Rubi [A] time = 0.06, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {8 x^2}{9}-\frac {12083 x+11597}{81 \left (3 x^2+5 x+2\right )}+\frac {112 x}{27}+83 \log (x+1)-\frac {1625}{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)^4}{\left (2+5 x+3 x^2\right )^2} \, dx &=\int \frac {\frac {13}{2} (3+2 x)^4-\frac {1}{2} (3+2 x)^5}{\left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac {11597+12083 x}{81 \left (2+5 x+3 x^2\right )}-\int \frac {\frac {169}{27}-\frac {2312 x}{27}-\frac {32 x^2}{9}+\frac {16 x^3}{3}}{2+5 x+3 x^2} \, dx\\ &=-\frac {11597+12083 x}{81 \left (2+5 x+3 x^2\right )}-\int \left (-\frac {112}{27}+\frac {16 x}{9}+\frac {131-616 x}{9 \left (2+5 x+3 x^2\right )}\right ) \, dx\\ &=\frac {112 x}{27}-\frac {8 x^2}{9}-\frac {11597+12083 x}{81 \left (2+5 x+3 x^2\right )}-\frac {1}{9} \int \frac {131-616 x}{2+5 x+3 x^2} \, dx\\ &=\frac {112 x}{27}-\frac {8 x^2}{9}-\frac {11597+12083 x}{81 \left (2+5 x+3 x^2\right )}-\frac {1625}{9} \int \frac {1}{2+3 x} \, dx+249 \int \frac {1}{3+3 x} \, dx\\ &=\frac {112 x}{27}-\frac {8 x^2}{9}-\frac {11597+12083 x}{81 \left (2+5 x+3 x^2\right )}+83 \log (1+x)-\frac {1625}{27} \log (2+3 x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 56, normalized size = 1.12 \begin {gather*} \frac {1}{81} \left (-\frac {12083 x+11597}{3 x^2+5 x+2}-18 (2 x+3)^2+276 (2 x+3)-4875 \log (-6 x-4)+6723 \log (-2 (x+1))\right ) \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)^4}{\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 = 0.39, size = 68, normalized size = 1.36 \begin {gather*} -\frac {216 \, x^{4} - 648 \, x^{3} - 1536 \, x^{2} + 4875 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 6723 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (x + 1\right ) + 11411 \, x + 11597}{81 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 44, normalized size = 0.88 \begin {gather*} -\frac {8}{9} \, x^{2} + \frac {112}{27} \, x - \frac {12083 \, x + 11597}{81 \, {\left (3 \, x + 2\right )} {\left (x + 1\right )}} - \frac {1625}{27} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + 83 \, \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 = 40, normalized size = 0.80 \begin {gather*} -\frac {8 x^{2}}{9}+\frac {112 x}{27}-\frac {1625 \ln \left (3 x +2\right )}{27}+83 \ln \left (x +1\right )-\frac {10625}{81 \left (3 x +2\right )}-\frac {6}{x +1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 42, normalized size = 0.84 \begin {gather*} -\frac {8}{9} \, x^{2} + \frac {112}{27} \, x - \frac {12083 \, x + 11597}{81 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} - \frac {1625}{27} \, \log \left (3 \, x + 2\right ) + 83 \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 38, normalized size = 0.76 \begin {gather*} \frac {112\,x}{27}+83\,\ln \left (x+1\right )-\frac {1625\,\ln \left (x+\frac {2}{3}\right )}{27}-\frac {\frac {12083\,x}{243}+\frac {11597}{243}}{x^2+\frac {5\,x}{3}+\frac {2}{3}}-\frac {8\,x^2}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 42, normalized size = 0.84 \begin {gather*} - \frac {8 x^{2}}{9} + \frac {112 x}{27} - \frac {12083 x + 11597}{243 x^{2} + 405 x + 162} - \frac {1625 \log {\left (x + \frac {2}{3} \right )}}{27} + 83 \log {\left (x + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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