Optimal. Leaf size=33 \[ \frac {1}{2} x \left (-3+x-\frac {x^2}{5}\right )+\frac {e^4+x-\frac {2 x}{3+x}}{x} \]
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Rubi [A] time = 0.09, antiderivative size = 34, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1594, 27, 12, 1620} \begin {gather*} -\frac {x^3}{10}+\frac {x^2}{2}-\frac {3 x}{2}-\frac {2}{x+3}+\frac {e^4}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 1594
Rule 1620
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-115 x^2+18 x^4-8 x^5-3 x^6+e^4 \left (-90-60 x-10 x^2\right )}{x^2 \left (90+60 x+10 x^2\right )} \, dx\\ &=\int \frac {-115 x^2+18 x^4-8 x^5-3 x^6+e^4 \left (-90-60 x-10 x^2\right )}{10 x^2 (3+x)^2} \, dx\\ &=\frac {1}{10} \int \frac {-115 x^2+18 x^4-8 x^5-3 x^6+e^4 \left (-90-60 x-10 x^2\right )}{x^2 (3+x)^2} \, dx\\ &=\frac {1}{10} \int \left (-15-\frac {10 e^4}{x^2}+10 x-3 x^2+\frac {20}{(3+x)^2}\right ) \, dx\\ &=\frac {e^4}{x}-\frac {3 x}{2}+\frac {x^2}{2}-\frac {x^3}{10}-\frac {2}{3+x}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 1.21 \begin {gather*} \frac {e^4}{x}-\frac {2}{3+x}-\frac {36 (3+x)}{5}+\frac {7}{5} (3+x)^2-\frac {1}{10} (3+x)^3 \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 35, normalized size = 1.06 \begin {gather*} -\frac {x^{5} - 2 \, x^{4} + 45 \, x^{2} - 10 \, {\left (x + 3\right )} e^{4} + 20 \, x}{10 \, {\left (x^{2} + 3 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 36, normalized size = 1.09 \begin {gather*} -\frac {1}{10} \, x^{3} + \frac {1}{2} \, x^{2} - \frac {3}{2} \, x + \frac {x e^{4} - 2 \, x + 3 \, e^{4}}{x^{2} + 3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 28, normalized size = 0.85
method | result | size |
default | \(-\frac {x^{3}}{10}+\frac {x^{2}}{2}-\frac {3 x}{2}-\frac {2}{3+x}+\frac {{\mathrm e}^{4}}{x}\) | \(28\) |
norman | \(\frac {\left (\frac {23}{2}+{\mathrm e}^{4}\right ) x +\frac {x^{4}}{5}-\frac {x^{5}}{10}+3 \,{\mathrm e}^{4}}{\left (3+x \right ) x}\) | \(31\) |
gosper | \(\frac {-x^{5}+2 x^{4}+10 x \,{\mathrm e}^{4}+30 \,{\mathrm e}^{4}+115 x}{10 x \left (3+x \right )}\) | \(34\) |
risch | \(-\frac {x^{3}}{10}+\frac {x^{2}}{2}-\frac {3 x}{2}+\frac {\frac {\left (10 \,{\mathrm e}^{4}-20\right ) x}{10}+3 \,{\mathrm e}^{4}}{\left (3+x \right ) x}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 35, normalized size = 1.06 \begin {gather*} -\frac {1}{10} \, x^{3} + \frac {1}{2} \, x^{2} - \frac {3}{2} \, x + \frac {x {\left (e^{4} - 2\right )} + 3 \, e^{4}}{x^{2} + 3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.12, size = 35, normalized size = 1.06 \begin {gather*} \frac {3\,{\mathrm {e}}^4+x\,\left ({\mathrm {e}}^4-2\right )}{x^2+3\,x}-\frac {3\,x}{2}+\frac {x^2}{2}-\frac {x^3}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 32, normalized size = 0.97 \begin {gather*} - \frac {x^{3}}{10} + \frac {x^{2}}{2} - \frac {3 x}{2} - \frac {x \left (2 - e^{4}\right ) - 3 e^{4}}{x^{2} + 3 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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