Optimal. Leaf size=28 \[ \frac {3 x}{x+\frac {-e^x-\frac {5}{3 x}+x}{2 x}} \]
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Rubi [F] time = 1.85, 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 {-270 x^2+54 x^4+e^x \left (-108 x^3+54 x^4\right )}{25-30 x^2+9 e^{2 x} x^2-60 x^3+9 x^4+36 x^5+36 x^6+e^x \left (30 x-18 x^3-36 x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {54 x^2 \left (-5+e^x (-2+x) x+x^2\right )}{\left (5+3 e^x x-3 x^2-6 x^3\right )^2} \, dx\\ &=54 \int \frac {x^2 \left (-5+e^x (-2+x) x+x^2\right )}{\left (5+3 e^x x-3 x^2-6 x^3\right )^2} \, dx\\ &=54 \int \left (-\frac {(-2+x) x^2}{3 \left (-5-3 e^x x+3 x^2+6 x^3\right )}+\frac {x^2 \left (-5-5 x-3 x^2-9 x^3+6 x^4\right )}{3 \left (-5-3 e^x x+3 x^2+6 x^3\right )^2}\right ) \, dx\\ &=-\left (18 \int \frac {(-2+x) x^2}{-5-3 e^x x+3 x^2+6 x^3} \, dx\right )+18 \int \frac {x^2 \left (-5-5 x-3 x^2-9 x^3+6 x^4\right )}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2} \, dx\\ &=18 \int \left (-\frac {5 x^2}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2}-\frac {5 x^3}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2}-\frac {3 x^4}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2}-\frac {9 x^5}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2}+\frac {6 x^6}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2}\right ) \, dx-18 \int \left (-\frac {2 x^2}{-5-3 e^x x+3 x^2+6 x^3}+\frac {x^3}{-5-3 e^x x+3 x^2+6 x^3}\right ) \, dx\\ &=-\left (18 \int \frac {x^3}{-5-3 e^x x+3 x^2+6 x^3} \, dx\right )+36 \int \frac {x^2}{-5-3 e^x x+3 x^2+6 x^3} \, dx-54 \int \frac {x^4}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2} \, dx-90 \int \frac {x^2}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2} \, dx-90 \int \frac {x^3}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2} \, dx+108 \int \frac {x^6}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2} \, dx-162 \int \frac {x^5}{\left (-5-3 e^x x+3 x^2+6 x^3\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.39, size = 25, normalized size = 0.89 \begin {gather*} -\frac {18 x^3}{5+3 e^x x-3 x^2-6 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 24, normalized size = 0.86 \begin {gather*} \frac {18 \, x^{3}}{6 \, x^{3} + 3 \, x^{2} - 3 \, x e^{x} - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 24, normalized size = 0.86 \begin {gather*} \frac {18 \, x^{3}}{6 \, x^{3} + 3 \, x^{2} - 3 \, x e^{x} - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 25, normalized size = 0.89
method | result | size |
risch | \(\frac {18 x^{3}}{6 x^{3}+3 x^{2}-3 \,{\mathrm e}^{x} x -5}\) | \(25\) |
norman | \(\frac {-9 x^{2}+9 \,{\mathrm e}^{x} x +15}{6 x^{3}+3 x^{2}-3 \,{\mathrm e}^{x} x -5}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 24, normalized size = 0.86 \begin {gather*} \frac {18 \, x^{3}}{6 \, x^{3} + 3 \, x^{2} - 3 \, x e^{x} - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.21, size = 24, normalized size = 0.86 \begin {gather*} -\frac {18\,x^3}{3\,x\,{\mathrm {e}}^x-3\,x^2-6\,x^3+5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 24, normalized size = 0.86 \begin {gather*} - \frac {18 x^{3}}{- 6 x^{3} - 3 x^{2} + 3 x e^{x} + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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