Optimal. Leaf size=26 \[ x+\left (-x+\frac {2 e}{x \left (x-\frac {4 x}{1+x}\right )}\right )^2 \]
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Rubi [B] time = 0.13, antiderivative size = 79, normalized size of antiderivative = 3.04, number of steps used = 2, number of rules used = 1, integrand size = 90, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {2074} \begin {gather*} \frac {4 e^2}{9 x^4}+\frac {32 e^2}{27 x^3}+x^2+\frac {32 e^2}{27 x^2}+x+\frac {16 e (81+10 e)}{243 (3-x)}+\frac {64 e^2}{81 (3-x)^2}+\frac {4 e (81+40 e)}{243 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2074
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-\frac {128 e^2}{81 (-3+x)^3}+\frac {16 e (81+10 e)}{243 (-3+x)^2}-\frac {16 e^2}{9 x^5}-\frac {32 e^2}{9 x^4}-\frac {64 e^2}{27 x^3}-\frac {4 e (81+40 e)}{243 x^2}+2 x\right ) \, dx\\ &=\frac {64 e^2}{81 (3-x)^2}+\frac {16 e (81+10 e)}{243 (3-x)}+\frac {4 e^2}{9 x^4}+\frac {32 e^2}{27 x^3}+\frac {32 e^2}{27 x^2}+\frac {4 e (81+40 e)}{243 x}+x+x^2\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 39, normalized size = 1.50 \begin {gather*} \frac {(1+x) \left (-4 e (-3+x) x^3+(-3+x)^2 x^5+4 e^2 (1+x)\right )}{(-3+x)^2 x^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 66, normalized size = 2.54 \begin {gather*} \frac {x^{8} - 5 \, x^{7} + 3 \, x^{6} + 9 \, x^{5} + 4 \, {\left (x^{2} + 2 \, x + 1\right )} e^{2} - 4 \, {\left (x^{5} - 2 \, x^{4} - 3 \, x^{3}\right )} e}{x^{6} - 6 \, x^{5} + 9 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 67, normalized size = 2.58 \begin {gather*} x^{2} + x - \frac {16 \, {\left (10 \, x e^{2} + 81 \, x e - 42 \, e^{2} - 243 \, e\right )}}{243 \, {\left (x - 3\right )}^{2}} + \frac {4 \, {\left (40 \, x^{3} e^{2} + 81 \, x^{3} e + 72 \, x^{2} e^{2} + 72 \, x e^{2} + 27 \, e^{2}\right )}}{243 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 58, normalized size = 2.23
method | result | size |
risch | \(x^{2}+x +\frac {-4 x^{5} {\mathrm e}+8 x^{4} {\mathrm e}+4 x^{2} {\mathrm e}^{2}+12 x^{3} {\mathrm e}+8 \,{\mathrm e}^{2} x +4 \,{\mathrm e}^{2}}{x^{4} \left (x^{2}-6 x +9\right )}\) | \(58\) |
default | \(x^{2}+x -\frac {-\frac {160 \,{\mathrm e}^{2}}{243}-\frac {4 \,{\mathrm e}}{3}}{x}+\frac {4 \,{\mathrm e}^{2}}{9 x^{4}}+\frac {32 \,{\mathrm e}^{2}}{27 x^{3}}+\frac {32 \,{\mathrm e}^{2}}{27 x^{2}}-\frac {\frac {160 \,{\mathrm e}^{2}}{243}+\frac {16 \,{\mathrm e}}{3}}{x -3}+\frac {64 \,{\mathrm e}^{2}}{81 \left (x -3\right )^{2}}\) | \(66\) |
norman | \(\frac {x^{8}+\left (8 \,{\mathrm e}-27\right ) x^{4}+\left (-4 \,{\mathrm e}+27\right ) x^{5}-5 x^{7}+4 \,{\mathrm e}^{2}+8 \,{\mathrm e}^{2} x +4 x^{2} {\mathrm e}^{2}+12 x^{3} {\mathrm e}}{x^{4} \left (x -3\right )^{2}}\) | \(68\) |
gosper | \(\frac {x^{8}-5 x^{7}-4 x^{5} {\mathrm e}+8 x^{4} {\mathrm e}+27 x^{5}+4 x^{2} {\mathrm e}^{2}+12 x^{3} {\mathrm e}-27 x^{4}+8 \,{\mathrm e}^{2} x +4 \,{\mathrm e}^{2}}{x^{4} \left (x^{2}-6 x +9\right )}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 60, normalized size = 2.31 \begin {gather*} x^{2} + x - \frac {4 \, {\left (x^{5} e - 2 \, x^{4} e - 3 \, x^{3} e - x^{2} e^{2} - 2 \, x e^{2} - e^{2}\right )}}{x^{6} - 6 \, x^{5} + 9 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.42, size = 49, normalized size = 1.88 \begin {gather*} \frac {\left (x+1\right )\,\left (x^7-6\,x^6+9\,x^5-4\,\mathrm {e}\,x^4+12\,\mathrm {e}\,x^3+4\,{\mathrm {e}}^2\,x+4\,{\mathrm {e}}^2\right )}{x^4\,{\left (x-3\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.44, size = 63, normalized size = 2.42 \begin {gather*} x^{2} + x + \frac {- 4 e x^{5} + 8 e x^{4} + 12 e x^{3} + 4 x^{2} e^{2} + 8 x e^{2} + 4 e^{2}}{x^{6} - 6 x^{5} + 9 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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