Optimal. Leaf size=26 \[ 2-\frac {4}{x^2}-x+\frac {1}{2} \left (-3+e-\frac {5}{x^2}-x\right ) x \]
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Rubi [A] time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.15, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6, 12, 14} \begin {gather*} -\frac {x^2}{2}-\frac {4}{x^2}-\frac {1}{2} (5-e) x-\frac {5}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 14
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {16+5 x+(-5+e) x^3-2 x^4}{2 x^3} \, dx\\ &=\frac {1}{2} \int \frac {16+5 x+(-5+e) x^3-2 x^4}{x^3} \, dx\\ &=\frac {1}{2} \int \left (-5 \left (1-\frac {e}{5}\right )+\frac {16}{x^3}+\frac {5}{x^2}-2 x\right ) \, dx\\ &=-\frac {4}{x^2}-\frac {5}{2 x}-\frac {1}{2} (5-e) x-\frac {x^2}{2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.00, size = 26, normalized size = 1.00 \begin {gather*} \frac {1}{2} \left (-\frac {8}{x^2}-\frac {5}{x}-5 x+e x-x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 25, normalized size = 0.96 \begin {gather*} -\frac {x^{4} - x^{3} e + 5 \, x^{3} + 5 \, x + 8}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 24, normalized size = 0.92 \begin {gather*} -\frac {1}{2} \, x^{2} + \frac {1}{2} \, x e - \frac {5}{2} \, x - \frac {5 \, x + 8}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 25, normalized size = 0.96
method | result | size |
default | \(-\frac {x^{2}}{2}+\frac {x \,{\mathrm e}}{2}-\frac {5 x}{2}-\frac {4}{x^{2}}-\frac {5}{2 x}\) | \(25\) |
norman | \(\frac {-4+\left (-\frac {5}{2}+\frac {{\mathrm e}}{2}\right ) x^{3}-\frac {5 x}{2}-\frac {x^{4}}{2}}{x^{2}}\) | \(25\) |
risch | \(\frac {x \,{\mathrm e}}{2}-\frac {x^{2}}{2}-\frac {5 x}{2}+\frac {-5 x -8}{2 x^{2}}\) | \(25\) |
gosper | \(\frac {x^{3} {\mathrm e}-x^{4}-5 x^{3}-5 x -8}{2 x^{2}}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 23, normalized size = 0.88 \begin {gather*} -\frac {1}{2} \, x^{2} + \frac {1}{2} \, x {\left (e - 5\right )} - \frac {5 \, x + 8}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 24, normalized size = 0.92 \begin {gather*} x\,\left (\frac {\mathrm {e}}{2}-\frac {5}{2}\right )-\frac {x^2}{2}-\frac {\frac {5\,x}{2}+4}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 24, normalized size = 0.92 \begin {gather*} - \frac {x^{2}}{2} - \frac {x \left (5 - e\right )}{2} - \frac {5 x + 8}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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