Integrand size = 39, antiderivative size = 17 \[ \int \frac {-46-117 x-72 x^2-2 x^3+\left (-46-46 x-x^2\right ) \log (x)}{529+46 x+x^2} \, dx=3-\frac {x (2+x) (x+\log (x))}{23+x} \]
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Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 6874, 45, 2404, 2332, 2351, 31} \[ \int \frac {-46-117 x-72 x^2-2 x^3+\left (-46-46 x-x^2\right ) \log (x)}{529+46 x+x^2} \, dx=-x^2+21 x+\frac {11109}{x+23}+\frac {21 x \log (x)}{x+23}-x \log (x) \]
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Rule 27
Rule 31
Rule 45
Rule 2332
Rule 2351
Rule 2404
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-46-117 x-72 x^2-2 x^3+\left (-46-46 x-x^2\right ) \log (x)}{(23+x)^2} \, dx \\ & = \int \left (-\frac {46}{(23+x)^2}-\frac {117 x}{(23+x)^2}-\frac {72 x^2}{(23+x)^2}-\frac {2 x^3}{(23+x)^2}-\frac {\left (46+46 x+x^2\right ) \log (x)}{(23+x)^2}\right ) \, dx \\ & = \frac {46}{23+x}-2 \int \frac {x^3}{(23+x)^2} \, dx-72 \int \frac {x^2}{(23+x)^2} \, dx-117 \int \frac {x}{(23+x)^2} \, dx-\int \frac {\left (46+46 x+x^2\right ) \log (x)}{(23+x)^2} \, dx \\ & = \frac {46}{23+x}-2 \int \left (-46+x-\frac {12167}{(23+x)^2}+\frac {1587}{23+x}\right ) \, dx-72 \int \left (1+\frac {529}{(23+x)^2}-\frac {46}{23+x}\right ) \, dx-117 \int \left (-\frac {23}{(23+x)^2}+\frac {1}{23+x}\right ) \, dx-\int \left (\log (x)-\frac {483 \log (x)}{(23+x)^2}\right ) \, dx \\ & = 20 x-x^2+\frac {11109}{23+x}+21 \log (23+x)+483 \int \frac {\log (x)}{(23+x)^2} \, dx-\int \log (x) \, dx \\ & = 21 x-x^2+\frac {11109}{23+x}-x \log (x)+\frac {21 x \log (x)}{23+x}+21 \log (23+x)-21 \int \frac {1}{23+x} \, dx \\ & = 21 x-x^2+\frac {11109}{23+x}-x \log (x)+\frac {21 x \log (x)}{23+x} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82 \[ \int \frac {-46-117 x-72 x^2-2 x^3+\left (-46-46 x-x^2\right ) \log (x)}{529+46 x+x^2} \, dx=21 x-x^2+\frac {483 (23-\log (x))}{23+x}+21 \log (x)-x \log (x) \]
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Time = 0.93 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.76
method | result | size |
norman | \(\frac {-2 x^{2}-x^{3}-2 x \ln \left (x \right )-x^{2} \ln \left (x \right )}{x +23}\) | \(30\) |
parallelrisch | \(\frac {-2 x^{2}-x^{3}-2 x \ln \left (x \right )-x^{2} \ln \left (x \right )}{x +23}\) | \(30\) |
default | \(-x^{2}+21 x +\frac {11109}{x +23}-x \ln \left (x \right )+\frac {21 \ln \left (x \right ) x}{x +23}\) | \(32\) |
parts | \(-x^{2}+21 x +\frac {11109}{x +23}-x \ln \left (x \right )+\frac {21 \ln \left (x \right ) x}{x +23}\) | \(32\) |
risch | \(-\frac {\left (x^{2}+23 x +483\right ) \ln \left (x \right )}{x +23}+\frac {-x^{3}+21 x \ln \left (x \right )-2 x^{2}+483 \ln \left (x \right )+483 x +11109}{x +23}\) | \(49\) |
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Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.76 \[ \int \frac {-46-117 x-72 x^2-2 x^3+\left (-46-46 x-x^2\right ) \log (x)}{529+46 x+x^2} \, dx=-\frac {x^{3} + 2 \, x^{2} + {\left (x^{2} + 2 \, x\right )} \log \left (x\right ) - 483 \, x - 11109}{x + 23} \]
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Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (14) = 28\).
Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.88 \[ \int \frac {-46-117 x-72 x^2-2 x^3+\left (-46-46 x-x^2\right ) \log (x)}{529+46 x+x^2} \, dx=- x^{2} + 21 x + 21 \log {\left (x \right )} + \frac {\left (- x^{2} - 23 x - 483\right ) \log {\left (x \right )}}{x + 23} + \frac {11109}{x + 23} \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (17) = 34\).
Time = 0.23 (sec) , antiderivative size = 54, normalized size of antiderivative = 3.18 \[ \int \frac {-46-117 x-72 x^2-2 x^3+\left (-46-46 x-x^2\right ) \log (x)}{529+46 x+x^2} \, dx=-x^{2} + 20 \, x + \frac {x^{2} - {\left (x^{2} + 23 \, x + 529\right )} \log \left (x\right ) + 23 \, x}{x + 23} + \frac {46 \, \log \left (x\right )}{x + 23} + \frac {11109}{x + 23} + 21 \, \log \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.94 \[ \int \frac {-46-117 x-72 x^2-2 x^3+\left (-46-46 x-x^2\right ) \log (x)}{529+46 x+x^2} \, dx=-x^{2} - {\left (x + \frac {483}{x + 23}\right )} \log \left (x\right ) + 21 \, x + \frac {11109}{x + 23} + 21 \, \log \left (x\right ) \]
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Time = 14.64 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {-46-117 x-72 x^2-2 x^3+\left (-46-46 x-x^2\right ) \log (x)}{529+46 x+x^2} \, dx=-\frac {x\,\left (x+\ln \left (x\right )\right )\,\left (x+2\right )}{x+23} \]
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