Integrand size = 47, antiderivative size = 20 \[ \int \frac {-9+250 x+54 x^2+3 x^3+e^x \left (8 x+x^2\right )+x \log (x)}{243 x+54 x^2+3 x^3} \, dx=x+\frac {-8+e^x-\log (x)}{3 (9+x)} \]
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Leaf count is larger than twice the leaf count of optimal. \(41\) vs. \(2(20)=40\).
Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.05, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.191, Rules used = {1608, 27, 12, 6874, 46, 45, 2228, 2351, 31} \[ \int \frac {-9+250 x+54 x^2+3 x^3+e^x \left (8 x+x^2\right )+x \log (x)}{243 x+54 x^2+3 x^3} \, dx=x+\frac {e^x}{3 (x+9)}-\frac {8}{3 (x+9)}+\frac {x \log (x)}{27 (x+9)}-\frac {\log (x)}{27} \]
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Rule 12
Rule 27
Rule 31
Rule 45
Rule 46
Rule 1608
Rule 2228
Rule 2351
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-9+250 x+54 x^2+3 x^3+e^x \left (8 x+x^2\right )+x \log (x)}{x \left (243+54 x+3 x^2\right )} \, dx \\ & = \int \frac {-9+250 x+54 x^2+3 x^3+e^x \left (8 x+x^2\right )+x \log (x)}{3 x (9+x)^2} \, dx \\ & = \frac {1}{3} \int \frac {-9+250 x+54 x^2+3 x^3+e^x \left (8 x+x^2\right )+x \log (x)}{x (9+x)^2} \, dx \\ & = \frac {1}{3} \int \left (\frac {250}{(9+x)^2}-\frac {9}{x (9+x)^2}+\frac {54 x}{(9+x)^2}+\frac {3 x^2}{(9+x)^2}+\frac {e^x (8+x)}{(9+x)^2}+\frac {\log (x)}{(9+x)^2}\right ) \, dx \\ & = -\frac {250}{3 (9+x)}+\frac {1}{3} \int \frac {e^x (8+x)}{(9+x)^2} \, dx+\frac {1}{3} \int \frac {\log (x)}{(9+x)^2} \, dx-3 \int \frac {1}{x (9+x)^2} \, dx+18 \int \frac {x}{(9+x)^2} \, dx+\int \frac {x^2}{(9+x)^2} \, dx \\ & = -\frac {250}{3 (9+x)}+\frac {e^x}{3 (9+x)}+\frac {x \log (x)}{27 (9+x)}-\frac {1}{27} \int \frac {1}{9+x} \, dx-3 \int \left (\frac {1}{81 x}-\frac {1}{9 (9+x)^2}-\frac {1}{81 (9+x)}\right ) \, dx+18 \int \left (-\frac {9}{(9+x)^2}+\frac {1}{9+x}\right ) \, dx+\int \left (1+\frac {81}{(9+x)^2}-\frac {18}{9+x}\right ) \, dx \\ & = x-\frac {8}{3 (9+x)}+\frac {e^x}{3 (9+x)}-\frac {\log (x)}{27}+\frac {x \log (x)}{27 (9+x)} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {-9+250 x+54 x^2+3 x^3+e^x \left (8 x+x^2\right )+x \log (x)}{243 x+54 x^2+3 x^3} \, dx=\frac {-8+e^x+27 x+3 x^2-\log (x)}{3 (9+x)} \]
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Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00
method | result | size |
norman | \(\frac {x^{2}-\frac {\ln \left (x \right )}{3}-\frac {251}{3}+\frac {{\mathrm e}^{x}}{3}}{x +9}\) | \(20\) |
parallelrisch | \(\frac {3 x^{2}-251-\ln \left (x \right )+{\mathrm e}^{x}}{3 x +27}\) | \(21\) |
risch | \(-\frac {\ln \left (x \right )}{3 \left (x +9\right )}+\frac {3 x^{2}+27 x +{\mathrm e}^{x}-8}{3 x +27}\) | \(30\) |
default | \(\frac {x \ln \left (x \right )}{27 x +243}+\frac {{\mathrm e}^{x}}{3 x +27}+x -\frac {\ln \left (x \right )}{27}-\frac {8}{3 \left (x +9\right )}\) | \(33\) |
parts | \(\frac {x \ln \left (x \right )}{27 x +243}+\frac {{\mathrm e}^{x}}{3 x +27}+x -\frac {\ln \left (x \right )}{27}-\frac {8}{3 \left (x +9\right )}\) | \(33\) |
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Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {-9+250 x+54 x^2+3 x^3+e^x \left (8 x+x^2\right )+x \log (x)}{243 x+54 x^2+3 x^3} \, dx=\frac {3 \, x^{2} + 27 \, x + e^{x} - \log \left (x\right ) - 8}{3 \, {\left (x + 9\right )}} \]
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Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {-9+250 x+54 x^2+3 x^3+e^x \left (8 x+x^2\right )+x \log (x)}{243 x+54 x^2+3 x^3} \, dx=x + \frac {e^{x}}{3 x + 27} - \frac {\log {\left (x \right )}}{3 x + 27} - \frac {8}{3 x + 27} \]
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\[ \int \frac {-9+250 x+54 x^2+3 x^3+e^x \left (8 x+x^2\right )+x \log (x)}{243 x+54 x^2+3 x^3} \, dx=\int { \frac {3 \, x^{3} + 54 \, x^{2} + {\left (x^{2} + 8 \, x\right )} e^{x} + x \log \left (x\right ) + 250 \, x - 9}{3 \, {\left (x^{3} + 18 \, x^{2} + 81 \, x\right )}} \,d x } \]
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Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {-9+250 x+54 x^2+3 x^3+e^x \left (8 x+x^2\right )+x \log (x)}{243 x+54 x^2+3 x^3} \, dx=\frac {3 \, x^{2} + 27 \, x + e^{x} - \log \left (x\right ) - 8}{3 \, {\left (x + 9\right )}} \]
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Time = 9.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {-9+250 x+54 x^2+3 x^3+e^x \left (8 x+x^2\right )+x \log (x)}{243 x+54 x^2+3 x^3} \, dx=x-\frac {\frac {\ln \left (x\right )}{3}-\frac {{\mathrm {e}}^x}{3}+\frac {8}{3}}{x+9} \]
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