Integrand size = 40, antiderivative size = 31 \[ \int \frac {5 x^2+8 x^3+e^{x^2} \left (-5+10 x^2+e \left (-1+2 x^2\right )\right )}{5 x^2} \, dx=\frac {\frac {1}{5} e^{x^2} (5+e)-x+x \left (x+\frac {4 x^2}{5}\right )}{x} \]
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Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {12, 14, 2326} \[ \int \frac {5 x^2+8 x^3+e^{x^2} \left (-5+10 x^2+e \left (-1+2 x^2\right )\right )}{5 x^2} \, dx=\frac {4 x^2}{5}+\frac {(5+e) e^{x^2}}{5 x}+x \]
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Rule 12
Rule 14
Rule 2326
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \int \frac {5 x^2+8 x^3+e^{x^2} \left (-5+10 x^2+e \left (-1+2 x^2\right )\right )}{x^2} \, dx \\ & = \frac {1}{5} \int \left (5+8 x+\frac {e^{x^2} (5+e) \left (-1+2 x^2\right )}{x^2}\right ) \, dx \\ & = x+\frac {4 x^2}{5}+\frac {1}{5} (5+e) \int \frac {e^{x^2} \left (-1+2 x^2\right )}{x^2} \, dx \\ & = \frac {e^{x^2} (5+e)}{5 x}+x+\frac {4 x^2}{5} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {5 x^2+8 x^3+e^{x^2} \left (-5+10 x^2+e \left (-1+2 x^2\right )\right )}{5 x^2} \, dx=\frac {1}{5} \left (\frac {5 e^{x^2}}{x}+\frac {e^{1+x^2}}{x}+5 x+4 x^2\right ) \]
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Time = 0.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {4 x^{2}}{5}+x +\frac {\left ({\mathrm e}+5\right ) {\mathrm e}^{x^{2}}}{5 x}\) | \(21\) |
norman | \(\frac {x^{2}+\left (\frac {{\mathrm e}}{5}+1\right ) {\mathrm e}^{x^{2}}+\frac {4 x^{3}}{5}}{x}\) | \(25\) |
parallelrisch | \(\frac {4 x^{3}+{\mathrm e} \,{\mathrm e}^{x^{2}}+5 x^{2}+5 \,{\mathrm e}^{x^{2}}}{5 x}\) | \(30\) |
default | \(\frac {4 x^{2}}{5}+x +\frac {{\mathrm e} \sqrt {\pi }\, \operatorname {erfi}\left (x \right )}{5}+\frac {{\mathrm e}^{x^{2}}}{x}-\frac {{\mathrm e} \left (-\frac {{\mathrm e}^{x^{2}}}{x}+\sqrt {\pi }\, \operatorname {erfi}\left (x \right )\right )}{5}\) | \(45\) |
parts | \(\frac {4 x^{2}}{5}+x +\frac {{\mathrm e} \sqrt {\pi }\, \operatorname {erfi}\left (x \right )}{5}+\frac {{\mathrm e}^{x^{2}}}{x}-\frac {{\mathrm e} \left (-\frac {{\mathrm e}^{x^{2}}}{x}+\sqrt {\pi }\, \operatorname {erfi}\left (x \right )\right )}{5}\) | \(45\) |
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Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \frac {5 x^2+8 x^3+e^{x^2} \left (-5+10 x^2+e \left (-1+2 x^2\right )\right )}{5 x^2} \, dx=\frac {4 \, x^{3} + 5 \, x^{2} + {\left (e + 5\right )} e^{\left (x^{2}\right )}}{5 \, x} \]
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Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {5 x^2+8 x^3+e^{x^2} \left (-5+10 x^2+e \left (-1+2 x^2\right )\right )}{5 x^2} \, dx=\frac {4 x^{2}}{5} + x + \frac {\left (e + 5\right ) e^{x^{2}}}{5 x} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.16 \[ \int \frac {5 x^2+8 x^3+e^{x^2} \left (-5+10 x^2+e \left (-1+2 x^2\right )\right )}{5 x^2} \, dx=-\frac {1}{5} i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e + \frac {4}{5} \, x^{2} - i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) + \frac {\sqrt {-x^{2}} e \Gamma \left (-\frac {1}{2}, -x^{2}\right )}{10 \, x} + x + \frac {\sqrt {-x^{2}} \Gamma \left (-\frac {1}{2}, -x^{2}\right )}{2 \, x} \]
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Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {5 x^2+8 x^3+e^{x^2} \left (-5+10 x^2+e \left (-1+2 x^2\right )\right )}{5 x^2} \, dx=\frac {4 \, x^{3} + 5 \, x^{2} + e^{\left (x^{2} + 1\right )} + 5 \, e^{\left (x^{2}\right )}}{5 \, x} \]
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Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {5 x^2+8 x^3+e^{x^2} \left (-5+10 x^2+e \left (-1+2 x^2\right )\right )}{5 x^2} \, dx=x+\frac {4\,x^2}{5}+\frac {{\mathrm {e}}^{x^2}\,\left (\mathrm {e}+5\right )}{5\,x} \]
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