Integrand size = 30, antiderivative size = 27 \[ \int \frac {e^x \left (-5+5 x+x^2\right )-x^2 \log \left (\frac {5+x}{3}\right )}{x^2} \, dx=\frac {13}{2}+x+(5+x) \left (\frac {e^x}{x}-\log \left (\frac {5+x}{3}\right )\right ) \]
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Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {14, 2230, 2225, 2208, 2209, 2436, 2332} \[ \int \frac {e^x \left (-5+5 x+x^2\right )-x^2 \log \left (\frac {5+x}{3}\right )}{x^2} \, dx=x+e^x+\frac {5 e^x}{x}+x \log (3)-(x+5) \log (x+5) \]
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Rule 14
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rule 2332
Rule 2436
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^x \left (-5+5 x+x^2\right )}{x^2}+\log (3)-\log (5+x)\right ) \, dx \\ & = x \log (3)+\int \frac {e^x \left (-5+5 x+x^2\right )}{x^2} \, dx-\int \log (5+x) \, dx \\ & = x \log (3)+\int \left (e^x-\frac {5 e^x}{x^2}+\frac {5 e^x}{x}\right ) \, dx-\text {Subst}(\int \log (x) \, dx,x,5+x) \\ & = x+x \log (3)-(5+x) \log (5+x)-5 \int \frac {e^x}{x^2} \, dx+5 \int \frac {e^x}{x} \, dx+\int e^x \, dx \\ & = e^x+\frac {5 e^x}{x}+x+5 \text {Ei}(x)+x \log (3)-(5+x) \log (5+x)-5 \int \frac {e^x}{x} \, dx \\ & = e^x+\frac {5 e^x}{x}+x+x \log (3)-(5+x) \log (5+x) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {e^x \left (-5+5 x+x^2\right )-x^2 \log \left (\frac {5+x}{3}\right )}{x^2} \, dx=e^x+\frac {5 e^x}{x}+x+x \log (3)-(5+x) \log (5+x) \]
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Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {5 \,{\mathrm e}^{x}}{x}+{\mathrm e}^{x}-3 \ln \left (\frac {5}{3}+\frac {x}{3}\right ) \left (\frac {5}{3}+\frac {x}{3}\right )+5+x\) | \(26\) |
parts | \(\frac {5 \,{\mathrm e}^{x}}{x}+{\mathrm e}^{x}-3 \ln \left (\frac {5}{3}+\frac {x}{3}\right ) \left (\frac {5}{3}+\frac {x}{3}\right )+5+x\) | \(26\) |
norman | \(\frac {x^{2}+{\mathrm e}^{x} x -5 x \ln \left (\frac {5}{3}+\frac {x}{3}\right )-x^{2} \ln \left (\frac {5}{3}+\frac {x}{3}\right )+5 \,{\mathrm e}^{x}}{x}\) | \(37\) |
risch | \(-x \ln \left (\frac {5}{3}+\frac {x}{3}\right )-\frac {5 x \ln \left (5+x \right )-x^{2}-{\mathrm e}^{x} x -5 \,{\mathrm e}^{x}}{x}\) | \(38\) |
parallelrisch | \(\frac {-x^{2} \ln \left (\frac {5}{3}+\frac {x}{3}\right )+x^{2}+{\mathrm e}^{x} x -5 x \ln \left (\frac {5}{3}+\frac {x}{3}\right )-5 x +5 \,{\mathrm e}^{x}}{x}\) | \(40\) |
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none
Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {e^x \left (-5+5 x+x^2\right )-x^2 \log \left (\frac {5+x}{3}\right )}{x^2} \, dx=\frac {x^{2} + {\left (x + 5\right )} e^{x} - {\left (x^{2} + 5 \, x\right )} \log \left (\frac {1}{3} \, x + \frac {5}{3}\right )}{x} \]
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Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {e^x \left (-5+5 x+x^2\right )-x^2 \log \left (\frac {5+x}{3}\right )}{x^2} \, dx=- x \log {\left (\frac {x}{3} + \frac {5}{3} \right )} + x - 5 \log {\left (x + 5 \right )} + \frac {\left (x + 5\right ) e^{x}}{x} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (-5+5 x+x^2\right )-x^2 \log \left (\frac {5+x}{3}\right )}{x^2} \, dx=-{\left (x + 5\right )} \log \left (\frac {1}{3} \, x + \frac {5}{3}\right ) + x + 5 \, {\rm Ei}\left (x\right ) + e^{x} - 5 \, \Gamma \left (-1, -x\right ) + 5 \]
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Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (22) = 44\).
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.52 \[ \int \frac {e^x \left (-5+5 x+x^2\right )-x^2 \log \left (\frac {5+x}{3}\right )}{x^2} \, dx=-\frac {{\left (x + 5\right )}^{2} e^{5} \log \left (\frac {1}{3} \, x + \frac {5}{3}\right ) - {\left (x + 5\right )}^{2} e^{5} - 5 \, {\left (x + 5\right )} e^{5} \log \left (\frac {1}{3} \, x + \frac {5}{3}\right ) + 5 \, {\left (x + 5\right )} e^{5} - {\left (x + 5\right )} e^{\left (x + 5\right )}}{{\left (x + 5\right )} e^{5} - 5 \, e^{5}} \]
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Time = 12.52 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {e^x \left (-5+5 x+x^2\right )-x^2 \log \left (\frac {5+x}{3}\right )}{x^2} \, dx=x-5\,\ln \left (x+5\right )+{\mathrm {e}}^x+\frac {5\,{\mathrm {e}}^x}{x}-x\,\ln \left (\frac {x}{3}+\frac {5}{3}\right ) \]
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