Integrand size = 34, antiderivative size = 26 \[ \int \frac {-x+x^2+e^{-x+x^2} \left (-1-x+2 x^2\right )}{5 x^2} \, dx=4+\frac {1}{5} \left (11+\frac {e^{-x+x^2}}{x}+x-\log (x)\right ) \]
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Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {12, 14, 2327, 45} \[ \int \frac {-x+x^2+e^{-x+x^2} \left (-1-x+2 x^2\right )}{5 x^2} \, dx=\frac {e^{x^2-x}}{5 x}+\frac {x}{5}-\frac {\log (x)}{5} \]
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Rule 12
Rule 14
Rule 45
Rule 2327
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \int \frac {-x+x^2+e^{-x+x^2} \left (-1-x+2 x^2\right )}{x^2} \, dx \\ & = \frac {1}{5} \int \left (\frac {e^{-x+x^2} (-1-2 x) (1-x)}{x^2}+\frac {-1+x}{x}\right ) \, dx \\ & = \frac {1}{5} \int \frac {e^{-x+x^2} (-1-2 x) (1-x)}{x^2} \, dx+\frac {1}{5} \int \frac {-1+x}{x} \, dx \\ & = \frac {e^{-x+x^2}}{5 x}+\frac {1}{5} \int \left (1-\frac {1}{x}\right ) \, dx \\ & = \frac {e^{-x+x^2}}{5 x}+\frac {x}{5}-\frac {\log (x)}{5} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int \frac {-x+x^2+e^{-x+x^2} \left (-1-x+2 x^2\right )}{5 x^2} \, dx=\frac {1}{5} \left (\frac {e^{(-1+x) x}}{x}+x-\log (x)\right ) \]
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Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77
method | result | size |
risch | \(\frac {x}{5}-\frac {\ln \left (x \right )}{5}+\frac {{\mathrm e}^{x \left (-1+x \right )}}{5 x}\) | \(20\) |
parts | \(\frac {x}{5}-\frac {\ln \left (x \right )}{5}+\frac {{\mathrm e}^{x^{2}-x}}{5 x}\) | \(22\) |
norman | \(\frac {\frac {x^{2}}{5}+\frac {{\mathrm e}^{x^{2}-x}}{5}}{x}-\frac {\ln \left (x \right )}{5}\) | \(26\) |
parallelrisch | \(-\frac {x \ln \left (x \right )-x^{2}-{\mathrm e}^{x^{2}-x}}{5 x}\) | \(26\) |
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Time = 0.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {-x+x^2+e^{-x+x^2} \left (-1-x+2 x^2\right )}{5 x^2} \, dx=\frac {x^{2} - x \log \left (x\right ) + e^{\left (x^{2} - x\right )}}{5 \, x} \]
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Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int \frac {-x+x^2+e^{-x+x^2} \left (-1-x+2 x^2\right )}{5 x^2} \, dx=\frac {x}{5} - \frac {\log {\left (x \right )}}{5} + \frac {e^{x^{2} - x}}{5 x} \]
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\[ \int \frac {-x+x^2+e^{-x+x^2} \left (-1-x+2 x^2\right )}{5 x^2} \, dx=\int { \frac {x^{2} + {\left (2 \, x^{2} - x - 1\right )} e^{\left (x^{2} - x\right )} - x}{5 \, x^{2}} \,d x } \]
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Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {-x+x^2+e^{-x+x^2} \left (-1-x+2 x^2\right )}{5 x^2} \, dx=\frac {x^{2} - x \log \left (x\right ) + e^{\left (x^{2} - x\right )}}{5 \, x} \]
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Time = 11.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {-x+x^2+e^{-x+x^2} \left (-1-x+2 x^2\right )}{5 x^2} \, dx=\frac {{\mathrm {e}}^{x^2-x}+x^2}{5\,x}-\frac {\ln \left (x\right )}{5} \]
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