Integrand size = 33, antiderivative size = 14 \[ \int \frac {3 x+3 x^2+x^3}{1+4 x+6 x^2+4 x^3+x^4} \, dx=\frac {1}{3 (1+x)^3}+\log (1+x) \]
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Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1608, 1694, 14} \[ \int \frac {3 x+3 x^2+x^3}{1+4 x+6 x^2+4 x^3+x^4} \, dx=\frac {1}{3 (x+1)^3}+\log (x+1) \]
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Rule 14
Rule 1608
Rule 1694
Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (3+3 x+x^2\right )}{1+4 x+6 x^2+4 x^3+x^4} \, dx \\ & = \text {Subst}\left (\int \frac {-1+x^3}{x^4} \, dx,x,1+x\right ) \\ & = \text {Subst}\left (\int \left (-\frac {1}{x^4}+\frac {1}{x}\right ) \, dx,x,1+x\right ) \\ & = \frac {1}{3 (1+x)^3}+\log (1+x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {3 x+3 x^2+x^3}{1+4 x+6 x^2+4 x^3+x^4} \, dx=\frac {1}{3 (1+x)^3}+\log (1+x) \]
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Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {1}{3 \left (x +1\right )^{3}}+\ln \left (x +1\right )\) | \(13\) |
norman | \(\frac {1}{3 \left (x +1\right )^{3}}+\ln \left (x +1\right )\) | \(13\) |
risch | \(\frac {1}{3 x^{3}+9 x^{2}+9 x +3}+\ln \left (x +1\right )\) | \(23\) |
parallelrisch | \(\frac {3 \ln \left (x +1\right ) x^{3}+1+9 \ln \left (x +1\right ) x^{2}+9 \ln \left (x +1\right ) x +3 \ln \left (x +1\right )}{3 x^{3}+9 x^{2}+9 x +3}\) | \(51\) |
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Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (12) = 24\).
Time = 0.30 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.71 \[ \int \frac {3 x+3 x^2+x^3}{1+4 x+6 x^2+4 x^3+x^4} \, dx=\frac {3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \left (x + 1\right ) + 1}{3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} \]
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Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int \frac {3 x+3 x^2+x^3}{1+4 x+6 x^2+4 x^3+x^4} \, dx=\log {\left (x + 1 \right )} + \frac {1}{3 x^{3} + 9 x^{2} + 9 x + 3} \]
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none
Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int \frac {3 x+3 x^2+x^3}{1+4 x+6 x^2+4 x^3+x^4} \, dx=\frac {1}{3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} + \log \left (x + 1\right ) \]
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none
Time = 0.30 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {3 x+3 x^2+x^3}{1+4 x+6 x^2+4 x^3+x^4} \, dx=\frac {1}{3 \, {\left (x + 1\right )}^{3}} + \log \left ({\left | x + 1 \right |}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {3 x+3 x^2+x^3}{1+4 x+6 x^2+4 x^3+x^4} \, dx=\ln \left (x+1\right )+\frac {1}{3\,{\left (x+1\right )}^3} \]
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