Optimal. Leaf size=31 \[ \frac {\frac {1}{5} e^{x^2} (5+e)-x+x \left (x+\frac {4 x^2}{5}\right )}{x} \]
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Rubi [A] time = 0.05, antiderivative size = 24, normalized size of antiderivative = 0.77, number of steps used = 4, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {12, 14, 2288} \begin {gather*} \frac {4 x^2}{5}+\frac {(5+e) e^{x^2}}{5 x}+x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2288
Rubi steps
\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 34, normalized size = 1.10 \begin {gather*} \frac {1}{5} \left (\frac {5 e^{x^2}}{x}+\frac {e^{1+x^2}}{x}+5 x+4 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 25, normalized size = 0.81 \begin {gather*} \frac {4 \, x^{3} + 5 \, x^{2} + {\left (e + 5\right )} e^{\left (x^{2}\right )}}{5 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 28, normalized size = 0.90 \begin {gather*} \frac {4 \, x^{3} + 5 \, x^{2} + e^{\left (x^{2} + 1\right )} + 5 \, e^{\left (x^{2}\right )}}{5 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 21, normalized size = 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 | \(\frac {x^{2}+\left (\frac {{\mathrm e}}{5}+1\right ) {\mathrm e}^{x^{2}}+\frac {4 x^{3}}{5}}{x}\) | \(25\) |
default | \(\frac {4 x^{2}}{5}+x +\frac {{\mathrm e} \sqrt {\pi }\, \erfi \relax (x )}{5}+\frac {{\mathrm e}^{x^{2}}}{x}-\frac {{\mathrm e} \left (-\frac {{\mathrm e}^{x^{2}}}{x}+\sqrt {\pi }\, \erfi \relax (x )\right )}{5}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.41, size = 67, normalized size = 2.16 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 20, normalized size = 0.65 \begin {gather*} x+\frac {4\,x^2}{5}+\frac {{\mathrm {e}}^{x^2}\,\left (\mathrm {e}+5\right )}{5\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 20, normalized size = 0.65 \begin {gather*} \frac {4 x^{2}}{5} + x + \frac {\left (e + 5\right ) e^{x^{2}}}{5 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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