Optimal. Leaf size=24 \[ x+\frac {x}{-5+\frac {3 \left (\frac {e^e}{x}+2 x\right )}{x}} \]
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Rubi [A] time = 0.02, antiderivative size = 21, normalized size of antiderivative = 0.88, number of steps used = 4, number of rules used = 4, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {28, 1157, 21, 8} \begin {gather*} 2 x-\frac {3 e^e x}{x^2+3 e^e} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 21
Rule 28
Rule 1157
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9 e^{2 e}+15 e^e x^2+2 x^4}{\left (3 e^e+x^2\right )^2} \, dx\\ &=-\frac {3 e^e x}{3 e^e+x^2}-\frac {1}{6} e^{-e} \int \frac {-36 e^{2 e}-12 e^e x^2}{3 e^e+x^2} \, dx\\ &=-\frac {3 e^e x}{3 e^e+x^2}+2 \int 1 \, dx\\ &=2 x-\frac {3 e^e x}{3 e^e+x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 21, normalized size = 0.88 \begin {gather*} 2 x-\frac {3 e^e x}{3 e^e+x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 24, normalized size = 1.00 \begin {gather*} \frac {2 \, x^{3} + 3 \, x e^{e}}{x^{2} + 3 \, e^{e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 22, normalized size = 0.92
method | result | size |
risch | \(2 x -\frac {x \,{\mathrm e}^{{\mathrm e}}}{\frac {x^{2}}{3}+{\mathrm e}^{{\mathrm e}}}\) | \(22\) |
gosper | \(\frac {x \left (2 x^{2}+3 \,{\mathrm e}^{{\mathrm e}}\right )}{x^{2}+3 \,{\mathrm e}^{{\mathrm e}}}\) | \(25\) |
norman | \(\frac {2 x^{3}+3 x \,{\mathrm e}^{{\mathrm e}}}{x^{2}+3 \,{\mathrm e}^{{\mathrm e}}}\) | \(25\) |
default | \(2 x -\frac {{\mathrm e}^{2 \,{\mathrm e}} {\mathrm e}^{-{\mathrm e}} x}{\frac {x^{2}}{3}+{\mathrm e}^{{\mathrm e}}}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 21, normalized size = 0.88 \begin {gather*} 2 \, x - \frac {3 \, x e^{e}}{x^{2} + 3 \, e^{e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 24, normalized size = 1.00 \begin {gather*} \frac {x\,\left (2\,x^2+3\,{\mathrm {e}}^{\mathrm {e}}\right )}{x^2+3\,{\mathrm {e}}^{\mathrm {e}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 20, normalized size = 0.83 \begin {gather*} 2 x - \frac {3 x e^{e}}{x^{2} + 3 e^{e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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