Optimal. Leaf size=30 \[ x+\frac {x^2-3 \left (e^2+x+\frac {1}{5 \left (-\frac {4}{x}+x\right )}\right )}{x} \]
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Rubi [A] time = 0.09, antiderivative size = 25, normalized size of antiderivative = 0.83, number of steps used = 6, number of rules used = 5, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {1594, 28, 1805, 1253, 14} \begin {gather*} \frac {3}{5 \left (4-x^2\right )}+2 x-\frac {3 e^2}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 28
Rule 1253
Rule 1594
Rule 1805
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {160 x^2+6 x^3-80 x^4+10 x^6+e^2 \left (240-120 x^2+15 x^4\right )}{x^2 \left (80-40 x^2+5 x^4\right )} \, dx\\ &=5 \int \frac {160 x^2+6 x^3-80 x^4+10 x^6+e^2 \left (240-120 x^2+15 x^4\right )}{x^2 \left (-20+5 x^2\right )^2} \, dx\\ &=\frac {3}{5 \left (4-x^2\right )}+\frac {1}{8} \int \frac {-480 e^2-40 \left (8-3 e^2\right ) x^2+80 x^4}{x^2 \left (-20+5 x^2\right )} \, dx\\ &=\frac {3}{5 \left (4-x^2\right )}+\frac {1}{8} \int \frac {24 e^2+16 x^2}{x^2} \, dx\\ &=\frac {3}{5 \left (4-x^2\right )}+\frac {1}{8} \int \left (16+\frac {24 e^2}{x^2}\right ) \, dx\\ &=-\frac {3 e^2}{x}+2 x+\frac {3}{5 \left (4-x^2\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 23, normalized size = 0.77 \begin {gather*} -\frac {3 e^2}{x}+2 x-\frac {3}{5 \left (-4+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 34, normalized size = 1.13 \begin {gather*} \frac {10 \, x^{4} - 40 \, x^{2} - 15 \, {\left (x^{2} - 4\right )} e^{2} - 3 \, x}{5 \, {\left (x^{3} - 4 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 28, normalized size = 0.93 \begin {gather*} 2 \, x - \frac {3 \, {\left (5 \, x^{2} e^{2} + x - 20 \, e^{2}\right )}}{5 \, {\left (x^{3} - 4 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 26, normalized size = 0.87
method | result | size |
default | \(2 x -\frac {3 \,{\mathrm e}^{2}}{x}-\frac {3}{20 \left (x -2\right )}+\frac {3}{20 \left (2+x \right )}\) | \(26\) |
risch | \(2 x +\frac {-3 x^{2} {\mathrm e}^{2}+12 \,{\mathrm e}^{2}-\frac {3 x}{5}}{x \left (x^{2}-4\right )}\) | \(31\) |
norman | \(\frac {-\frac {3 x}{5}+\left (-3 \,{\mathrm e}^{2}-8\right ) x^{2}+2 x^{4}+12 \,{\mathrm e}^{2}}{x \left (x^{2}-4\right )}\) | \(35\) |
gosper | \(-\frac {-10 x^{4}+15 x^{2} {\mathrm e}^{2}+40 x^{2}-60 \,{\mathrm e}^{2}+3 x}{5 x \left (x^{2}-4\right )}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 28, normalized size = 0.93 \begin {gather*} 2 \, x - \frac {3 \, {\left (5 \, x^{2} e^{2} + x - 20 \, e^{2}\right )}}{5 \, {\left (x^{3} - 4 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 31, normalized size = 1.03 \begin {gather*} 2\,x-\frac {3\,{\mathrm {e}}^2\,x^2+\frac {3\,x}{5}-12\,{\mathrm {e}}^2}{x\,\left (x^2-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 27, normalized size = 0.90 \begin {gather*} 2 x + \frac {- 15 x^{2} e^{2} - 3 x + 60 e^{2}}{5 x^{3} - 20 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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