Optimal. Leaf size=87 \[ \frac {-2 x-3}{4 \left (x^2+3 x+2\right )^4}+\frac {35 (2 x+3)}{x^2+3 x+2}-\frac {35 (2 x+3)}{6 \left (x^2+3 x+2\right )^2}+\frac {7 (2 x+3)}{6 \left (x^2+3 x+2\right )^3}+70 \log (x+1)-70 \log (x+2) \]
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Rubi [A] time = 0.02, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {614, 616, 31} \[ \frac {35 (2 x+3)}{x^2+3 x+2}-\frac {35 (2 x+3)}{6 \left (x^2+3 x+2\right )^2}+\frac {7 (2 x+3)}{6 \left (x^2+3 x+2\right )^3}-\frac {2 x+3}{4 \left (x^2+3 x+2\right )^4}+70 \log (x+1)-70 \log (x+2) \]
Antiderivative was successfully verified.
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Rule 31
Rule 614
Rule 616
Rubi steps
\begin {align*} \int \frac {1}{\left (2+3 x+x^2\right )^5} \, dx &=-\frac {3+2 x}{4 \left (2+3 x+x^2\right )^4}-\frac {7}{2} \int \frac {1}{\left (2+3 x+x^2\right )^4} \, dx\\ &=-\frac {3+2 x}{4 \left (2+3 x+x^2\right )^4}+\frac {7 (3+2 x)}{6 \left (2+3 x+x^2\right )^3}+\frac {35}{3} \int \frac {1}{\left (2+3 x+x^2\right )^3} \, dx\\ &=-\frac {3+2 x}{4 \left (2+3 x+x^2\right )^4}+\frac {7 (3+2 x)}{6 \left (2+3 x+x^2\right )^3}-\frac {35 (3+2 x)}{6 \left (2+3 x+x^2\right )^2}-35 \int \frac {1}{\left (2+3 x+x^2\right )^2} \, dx\\ &=-\frac {3+2 x}{4 \left (2+3 x+x^2\right )^4}+\frac {7 (3+2 x)}{6 \left (2+3 x+x^2\right )^3}-\frac {35 (3+2 x)}{6 \left (2+3 x+x^2\right )^2}+\frac {35 (3+2 x)}{2+3 x+x^2}+70 \int \frac {1}{2+3 x+x^2} \, dx\\ &=-\frac {3+2 x}{4 \left (2+3 x+x^2\right )^4}+\frac {7 (3+2 x)}{6 \left (2+3 x+x^2\right )^3}-\frac {35 (3+2 x)}{6 \left (2+3 x+x^2\right )^2}+\frac {35 (3+2 x)}{2+3 x+x^2}+70 \int \frac {1}{1+x} \, dx-70 \int \frac {1}{2+x} \, dx\\ &=-\frac {3+2 x}{4 \left (2+3 x+x^2\right )^4}+\frac {7 (3+2 x)}{6 \left (2+3 x+x^2\right )^3}-\frac {35 (3+2 x)}{6 \left (2+3 x+x^2\right )^2}+\frac {35 (3+2 x)}{2+3 x+x^2}+70 \log (1+x)-70 \log (2+x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 87, normalized size = 1.00 \[ \frac {-2 x-3}{4 \left (x^2+3 x+2\right )^4}+\frac {35 (2 x+3)}{x^2+3 x+2}-\frac {35 (2 x+3)}{6 \left (x^2+3 x+2\right )^2}+\frac {7 (2 x+3)}{6 \left (x^2+3 x+2\right )^3}+70 \log (x+1)-70 \log (x+2) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 58, normalized size = 0.67 \[ \frac {(2 x+3) \left (420 x^6+3780 x^5+13790 x^4+26040 x^3+26824 x^2+14322 x+3105\right )}{12 \left (x^2+3 x+2\right )^4}-140 \tanh ^{-1}(2 x+3) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 165, normalized size = 1.90 \[ \frac {840 \, x^{7} + 8820 \, x^{6} + 38920 \, x^{5} + 93450 \, x^{4} + 131768 \, x^{3} + 109116 \, x^{2} - 840 \, {\left (x^{8} + 12 \, x^{7} + 62 \, x^{6} + 180 \, x^{5} + 321 \, x^{4} + 360 \, x^{3} + 248 \, x^{2} + 96 \, x + 16\right )} \log \left (x + 2\right ) + 840 \, {\left (x^{8} + 12 \, x^{7} + 62 \, x^{6} + 180 \, x^{5} + 321 \, x^{4} + 360 \, x^{3} + 248 \, x^{2} + 96 \, x + 16\right )} \log \left (x + 1\right ) + 49176 \, x + 9315}{12 \, {\left (x^{8} + 12 \, x^{7} + 62 \, x^{6} + 180 \, x^{5} + 321 \, x^{4} + 360 \, x^{3} + 248 \, x^{2} + 96 \, x + 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.90, size = 62, normalized size = 0.71 \[ \frac {840 \, x^{7} + 8820 \, x^{6} + 38920 \, x^{5} + 93450 \, x^{4} + 131768 \, x^{3} + 109116 \, x^{2} + 49176 \, x + 9315}{12 \, {\left (x^{2} + 3 \, x + 2\right )}^{4}} - 70 \, \log \left ({\left | x + 2 \right |}\right ) + 70 \, \log \left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 60, normalized size = 0.69
method | result | size |
norman | \(\frac {4098 x +\frac {9730}{3} x^{5}+\frac {32942}{3} x^{3}+70 x^{7}+735 x^{6}+9093 x^{2}+\frac {15575}{2} x^{4}+\frac {3105}{4}}{\left (x^{2}+3 x +2\right )^{4}}+70 \ln \left (1+x \right )-70 \ln \left (2+x \right )\) | \(60\) |
risch | \(\frac {4098 x +\frac {9730}{3} x^{5}+\frac {32942}{3} x^{3}+70 x^{7}+735 x^{6}+9093 x^{2}+\frac {15575}{2} x^{4}+\frac {3105}{4}}{\left (x^{2}+3 x +2\right )^{4}}+70 \ln \left (1+x \right )-70 \ln \left (2+x \right )\) | \(60\) |
default | \(\frac {1}{4 \left (2+x \right )^{4}}+\frac {5}{3 \left (2+x \right )^{3}}+\frac {15}{2 \left (2+x \right )^{2}}+\frac {35}{2+x}-70 \ln \left (2+x \right )-\frac {1}{4 \left (1+x \right )^{4}}+\frac {5}{3 \left (1+x \right )^{3}}-\frac {15}{2 \left (1+x \right )^{2}}+\frac {35}{1+x}+70 \ln \left (1+x \right )\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 90, normalized size = 1.03 \[ \frac {840 \, x^{7} + 8820 \, x^{6} + 38920 \, x^{5} + 93450 \, x^{4} + 131768 \, x^{3} + 109116 \, x^{2} + 49176 \, x + 9315}{12 \, {\left (x^{8} + 12 \, x^{7} + 62 \, x^{6} + 180 \, x^{5} + 321 \, x^{4} + 360 \, x^{3} + 248 \, x^{2} + 96 \, x + 16\right )}} - 70 \, \log \left (x + 2\right ) + 70 \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 65, normalized size = 0.75 \[ 70\,\ln \left (\frac {x+1}{x+2}\right )+70\,\left (x+\frac {3}{2}\right )\,\left (\frac {1}{x^2+3\,x+2}-\frac {1}{6\,{\left (x^2+3\,x+2\right )}^2}+\frac {1}{30\,{\left (x^2+3\,x+2\right )}^3}-\frac {1}{140\,{\left (x^2+3\,x+2\right )}^4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 88, normalized size = 1.01 \[ \frac {840 x^{7} + 8820 x^{6} + 38920 x^{5} + 93450 x^{4} + 131768 x^{3} + 109116 x^{2} + 49176 x + 9315}{12 x^{8} + 144 x^{7} + 744 x^{6} + 2160 x^{5} + 3852 x^{4} + 4320 x^{3} + 2976 x^{2} + 1152 x + 192} + 70 \log {\left (x + 1 \right )} - 70 \log {\left (x + 2 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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