Optimal. Leaf size=23 \[ \frac {\left (\frac {4}{x}+x\right )^2}{e^2 (5-x) x} \]
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Rubi [A] time = 0.07, antiderivative size = 45, normalized size of antiderivative = 1.96, number of steps used = 5, number of rules used = 4, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {12, 1585, 27, 1620} \begin {gather*} \frac {16}{5 e^2 x^3}+\frac {16}{25 e^2 x^2}+\frac {216}{125 e^2 x}+\frac {841}{125 e^2 (5-x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 1585
Rule 1620
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-240+64 x-40 x^2+16 x^3+5 x^4}{x^2 \left (25 x^2-10 x^3+x^4\right )} \, dx}{e^2}\\ &=\frac {\int \frac {-240+64 x-40 x^2+16 x^3+5 x^4}{x^4 \left (25-10 x+x^2\right )} \, dx}{e^2}\\ &=\frac {\int \frac {-240+64 x-40 x^2+16 x^3+5 x^4}{(-5+x)^2 x^4} \, dx}{e^2}\\ &=\frac {\int \left (\frac {841}{125 (-5+x)^2}-\frac {48}{5 x^4}-\frac {32}{25 x^3}-\frac {216}{125 x^2}\right ) \, dx}{e^2}\\ &=\frac {841}{125 e^2 (5-x)}+\frac {16}{5 e^2 x^3}+\frac {16}{25 e^2 x^2}+\frac {216}{125 e^2 x}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.09 \begin {gather*} -\frac {16+8 x^2+5 x^3}{e^2 (-5+x) x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 27, normalized size = 1.17 \begin {gather*} -\frac {{\left (5 \, x^{3} + 8 \, x^{2} + 16\right )} e^{\left (-2\right )}}{x^{4} - 5 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 27, normalized size = 1.17 \begin {gather*} -\frac {841 \, e^{\left (-2\right )}}{125 \, {\left (x - 5\right )}} + \frac {8 \, {\left (27 \, x^{2} + 10 \, x + 50\right )} e^{\left (-2\right )}}{125 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 24, normalized size = 1.04
method | result | size |
risch | \(\frac {{\mathrm e}^{-2} \left (-5 x^{3}-8 x^{2}-16\right )}{x^{3} \left (x -5\right )}\) | \(24\) |
default | \({\mathrm e}^{-3} {\mathrm e}^{5} {\mathrm e}^{-4} \left (\frac {16}{5 x^{3}}+\frac {16}{25 x^{2}}+\frac {216}{125 x}-\frac {841}{125 \left (x -5\right )}\right )\) | \(33\) |
gosper | \(-\frac {{\mathrm e}^{5} \left (5 x^{3}+8 x^{2}+16\right ) {\mathrm e}^{-4} {\mathrm e}^{-3}}{x^{3} \left (x -5\right )}\) | \(36\) |
norman | \(\frac {\left (-5 \,{\mathrm e}^{-2} {\mathrm e}^{-3} {\mathrm e}^{5} x^{3}-16 \,{\mathrm e}^{-2} {\mathrm e}^{-3} {\mathrm e}^{5}-8 \,{\mathrm e}^{-2} {\mathrm e}^{-3} {\mathrm e}^{5} x^{2}\right ) {\mathrm e}^{-2}}{x^{3} \left (x -5\right )}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 27, normalized size = 1.17 \begin {gather*} -\frac {{\left (5 \, x^{3} + 8 \, x^{2} + 16\right )} e^{\left (-2\right )}}{x^{4} - 5 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 19, normalized size = 0.83 \begin {gather*} -\frac {{\mathrm {e}}^{-2}\,{\left (x^2+4\right )}^2}{x^3\,\left (x-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, size = 27, normalized size = 1.17 \begin {gather*} \frac {- 5 x^{3} - 8 x^{2} - 16}{x^{4} e^{2} - 5 x^{3} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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