Integrand size = 16, antiderivative size = 19 \[ \int \frac {3+3 x+2 x^2}{(1+x)^3} \, dx=-\frac {1}{(1+x)^2}+\frac {1}{1+x}+2 \log (1+x) \]
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Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {712} \[ \int \frac {3+3 x+2 x^2}{(1+x)^3} \, dx=\frac {1}{x+1}-\frac {1}{(x+1)^2}+2 \log (x+1) \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2}{(1+x)^3}-\frac {1}{(1+x)^2}+\frac {2}{1+x}\right ) \, dx \\ & = -\frac {1}{(1+x)^2}+\frac {1}{1+x}+2 \log (1+x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {3+3 x+2 x^2}{(1+x)^3} \, dx=-\frac {1}{(1+x)^2}+\frac {1}{1+x}+2 \log (1+x) \]
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Time = 20.70 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79
method | result | size |
norman | \(\frac {x}{\left (1+x \right )^{2}}+2 \ln \left (1+x \right )\) | \(15\) |
risch | \(\frac {x}{\left (1+x \right )^{2}}+2 \ln \left (1+x \right )\) | \(15\) |
default | \(-\frac {1}{\left (1+x \right )^{2}}+\frac {1}{1+x}+2 \ln \left (1+x \right )\) | \(20\) |
parallelrisch | \(\frac {2 \ln \left (1+x \right ) x^{2}+4 \ln \left (1+x \right ) x +2 \ln \left (1+x \right )+x}{\left (1+x \right )^{2}}\) | \(31\) |
meijerg | \(\frac {3 x \left (2+x \right )}{2 \left (1+x \right )^{2}}-\frac {x \left (9 x +6\right )}{3 \left (1+x \right )^{2}}+2 \ln \left (1+x \right )+\frac {3 x^{2}}{2 \left (1+x \right )^{2}}\) | \(42\) |
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none
Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {3+3 x+2 x^2}{(1+x)^3} \, dx=\frac {2 \, {\left (x^{2} + 2 \, x + 1\right )} \log \left (x + 1\right ) + x}{x^{2} + 2 \, x + 1} \]
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Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {3+3 x+2 x^2}{(1+x)^3} \, dx=\frac {x}{x^{2} + 2 x + 1} + 2 \log {\left (x + 1 \right )} \]
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none
Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {3+3 x+2 x^2}{(1+x)^3} \, dx=\frac {x}{x^{2} + 2 \, x + 1} + 2 \, \log \left (x + 1\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {3+3 x+2 x^2}{(1+x)^3} \, dx=\frac {x}{{\left (x + 1\right )}^{2}} + 2 \, \log \left ({\left | x + 1 \right |}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {3+3 x+2 x^2}{(1+x)^3} \, dx=2\,\ln \left (x+1\right )+\frac {x}{{\left (x+1\right )}^2} \]
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