Optimal. Leaf size=23 \[ \frac {x}{e^{5 e (2-x)^2 x}+\frac {81}{x}} \]
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Rubi [F] time = 1.93, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {162 x+e^{e \left (20 x-20 x^2+5 x^3\right )} \left (x^2+e \left (-20 x^3+40 x^4-15 x^5\right )\right )}{6561+162 e^{e \left (20 x-20 x^2+5 x^3\right )} x+e^{2 e \left (20 x-20 x^2+5 x^3\right )} x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (162+e^{5 e (-2+x)^2 x} x-5 e^{1+5 e (-2+x)^2 x} x^2 \left (4-8 x+3 x^2\right )\right )}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2} \, dx\\ &=\int \left (-\frac {x \left (-1+20 e x-40 e x^2+15 e x^3\right )}{81+e^{5 e (-2+x)^2 x} x}+\frac {81 x \left (1+20 e x-40 e x^2+15 e x^3\right )}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2}\right ) \, dx\\ &=81 \int \frac {x \left (1+20 e x-40 e x^2+15 e x^3\right )}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2} \, dx-\int \frac {x \left (-1+20 e x-40 e x^2+15 e x^3\right )}{81+e^{5 e (-2+x)^2 x} x} \, dx\\ &=81 \int \left (\frac {x}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2}+\frac {20 e x^2}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2}-\frac {40 e x^3}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2}+\frac {15 e x^4}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2}\right ) \, dx-\int \left (-\frac {x}{81+e^{5 e (-2+x)^2 x} x}+\frac {20 e x^2}{81+e^{5 e (-2+x)^2 x} x}-\frac {40 e x^3}{81+e^{5 e (-2+x)^2 x} x}+\frac {15 e x^4}{81+e^{5 e (-2+x)^2 x} x}\right ) \, dx\\ &=81 \int \frac {x}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2} \, dx-(15 e) \int \frac {x^4}{81+e^{5 e (-2+x)^2 x} x} \, dx-(20 e) \int \frac {x^2}{81+e^{5 e (-2+x)^2 x} x} \, dx+(40 e) \int \frac {x^3}{81+e^{5 e (-2+x)^2 x} x} \, dx+(1215 e) \int \frac {x^4}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2} \, dx+(1620 e) \int \frac {x^2}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2} \, dx-(3240 e) \int \frac {x^3}{\left (81+e^{5 e (-2+x)^2 x} x\right )^2} \, dx+\int \frac {x}{81+e^{5 e (-2+x)^2 x} x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.50, size = 21, normalized size = 0.91 \begin {gather*} \frac {x^2}{81+e^{5 e (-2+x)^2 x} x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.92, size = 27, normalized size = 1.17 \begin {gather*} \frac {x^{2}}{x e^{\left (5 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e\right )} + 81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.84, size = 31, normalized size = 1.35 \begin {gather*} \frac {x^{2}}{x e^{\left (5 \, x^{3} e - 20 \, x^{2} e + 20 \, x e\right )} + 81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 22, normalized size = 0.96
method | result | size |
risch | \(\frac {x^{2}}{x \,{\mathrm e}^{5 x \left (x -2\right )^{2} {\mathrm e}}+81}\) | \(22\) |
norman | \(\frac {x^{2}}{x \,{\mathrm e}^{\left (5 x^{3}-20 x^{2}+20 x \right ) {\mathrm e}}+81}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 41, normalized size = 1.78 \begin {gather*} \frac {x^{2} e^{\left (20 \, x^{2} e\right )}}{x e^{\left (5 \, x^{3} e + 20 \, x e\right )} + 81 \, e^{\left (20 \, x^{2} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 32, normalized size = 1.39 \begin {gather*} \frac {x^2}{x\,{\mathrm {e}}^{5\,x^3\,\mathrm {e}}\,{\mathrm {e}}^{-20\,x^2\,\mathrm {e}}\,{\mathrm {e}}^{20\,x\,\mathrm {e}}+81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 24, normalized size = 1.04 \begin {gather*} \frac {x^{2}}{x e^{e \left (5 x^{3} - 20 x^{2} + 20 x\right )} + 81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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