Optimal. Leaf size=22 \[ 2+x+\frac {x}{e^x-5 x}-\frac {10 x}{-4+x} \]
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Rubi [F] time = 0.77, 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 {1400 x^2-200 x^3+25 x^4+e^{2 x} \left (56-8 x+x^2\right )+e^x \left (16-584 x+89 x^2-11 x^3\right )}{400 x^2-200 x^3+25 x^4+e^{2 x} \left (16-8 x+x^2\right )+e^x \left (-160 x+80 x^2-10 x^3\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 {e^{2 x} \left (56-8 x+x^2\right )+25 x^2 \left (56-8 x+x^2\right )+e^x \left (16-584 x+89 x^2-11 x^3\right )}{\left (e^x-5 x\right )^2 (4-x)^2} \, dx\\ &=\int \left (-\frac {-1+x}{e^x-5 x}-\frac {5 (-1+x) x}{\left (e^x-5 x\right )^2}+\frac {56-8 x+x^2}{(-4+x)^2}\right ) \, dx\\ &=-\left (5 \int \frac {(-1+x) x}{\left (e^x-5 x\right )^2} \, dx\right )-\int \frac {-1+x}{e^x-5 x} \, dx+\int \frac {56-8 x+x^2}{(-4+x)^2} \, dx\\ &=-\left (5 \int \left (-\frac {x}{\left (e^x-5 x\right )^2}+\frac {x^2}{\left (e^x-5 x\right )^2}\right ) \, dx\right )+\int \left (1+\frac {40}{(-4+x)^2}\right ) \, dx-\int \left (-\frac {1}{e^x-5 x}+\frac {x}{e^x-5 x}\right ) \, dx\\ &=\frac {40}{4-x}+x+5 \int \frac {x}{\left (e^x-5 x\right )^2} \, dx-5 \int \frac {x^2}{\left (e^x-5 x\right )^2} \, dx+\int \frac {1}{e^x-5 x} \, dx-\int \frac {x}{e^x-5 x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.16, size = 20, normalized size = 0.91 \begin {gather*} -\frac {40}{-4+x}+x+\frac {x}{e^x-5 x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 45, normalized size = 2.05 \begin {gather*} \frac {5 \, x^{3} - 21 \, x^{2} - {\left (x^{2} - 4 \, x - 40\right )} e^{x} - 196 \, x}{5 \, x^{2} - {\left (x - 4\right )} e^{x} - 20 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 51, normalized size = 2.32 \begin {gather*} \frac {5 \, x^{3} - x^{2} e^{x} - 21 \, x^{2} + 4 \, x e^{x} - 196 \, x + 40 \, e^{x}}{5 \, x^{2} - x e^{x} - 20 \, x + 4 \, e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 23, normalized size = 1.05
method | result | size |
risch | \(x -\frac {40}{x -4}-\frac {x}{5 x -{\mathrm e}^{x}}\) | \(23\) |
norman | \(\frac {-276 x -x^{2}+56 \,{\mathrm e}^{x}+5 x^{3}-{\mathrm e}^{x} x^{2}}{5 x^{2}-{\mathrm e}^{x} x -20 x +4 \,{\mathrm e}^{x}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 45, normalized size = 2.05 \begin {gather*} \frac {5 \, x^{3} - 21 \, x^{2} - {\left (x^{2} - 4 \, x - 40\right )} e^{x} - 196 \, x}{5 \, x^{2} - {\left (x - 4\right )} e^{x} - 20 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.65, size = 22, normalized size = 1.00 \begin {gather*} x-\frac {40}{x-4}-\frac {x}{5\,x-{\mathrm {e}}^x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 14, normalized size = 0.64 \begin {gather*} x + \frac {x}{- 5 x + e^{x}} - \frac {40}{x - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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