Integrand size = 59, antiderivative size = 25 \[ \int \frac {-18+\sqrt {e} (6-4 x)+12 x+4 x^2}{9+e-6 x+7 x^2-2 x^3+x^4+\sqrt {e} \left (-6+2 x-2 x^2\right )} \, dx=-2+\frac {2 (-3+x) x}{3-\sqrt {e}-x+x^2} \]
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Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(25)=50\).
Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.44, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {1694, 12, 1828, 8} \[ \int \frac {-18+\sqrt {e} (6-4 x)+12 x+4 x^2}{9+e-6 x+7 x^2-2 x^3+x^4+\sqrt {e} \left (-6+2 x-2 x^2\right )} \, dx=-\frac {8 \left (2 \left (11-4 \sqrt {e}\right ) \left (x-\frac {1}{2}\right )+4 e-27 \sqrt {e}+44\right )}{\left (11-4 \sqrt {e}\right ) \left (4 \left (x-\frac {1}{2}\right )^2-4 \sqrt {e}+11\right )} \]
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Rule 8
Rule 12
Rule 1694
Rule 1828
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {16 \left (-11+4 \sqrt {e}+4 \left (4-\sqrt {e}\right ) x+4 x^2\right )}{\left (11-4 \sqrt {e}+4 x^2\right )^2} \, dx,x,-\frac {1}{2}+x\right ) \\ & = 16 \text {Subst}\left (\int \frac {-11+4 \sqrt {e}+4 \left (4-\sqrt {e}\right ) x+4 x^2}{\left (11-4 \sqrt {e}+4 x^2\right )^2} \, dx,x,-\frac {1}{2}+x\right ) \\ & = -\frac {8 \left (44-27 \sqrt {e}+4 e-\left (11-4 \sqrt {e}\right ) (1-2 x)\right )}{\left (11-4 \sqrt {e}\right ) \left (11-4 \sqrt {e}+(-1+2 x)^2\right )}-\frac {8 \text {Subst}\left (\int 0 \, dx,x,-\frac {1}{2}+x\right )}{11-4 \sqrt {e}} \\ & = -\frac {8 \left (44-27 \sqrt {e}+4 e-\left (11-4 \sqrt {e}\right ) (1-2 x)\right )}{\left (11-4 \sqrt {e}\right ) \left (11-4 \sqrt {e}+(-1+2 x)^2\right )} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {-18+\sqrt {e} (6-4 x)+12 x+4 x^2}{9+e-6 x+7 x^2-2 x^3+x^4+\sqrt {e} \left (-6+2 x-2 x^2\right )} \, dx=\frac {2 \left (-3+\sqrt {e}-2 x\right )}{3-\sqrt {e}-x+x^2} \]
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Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
method | result | size |
gosper | \(-\frac {2 \left (-3-2 x +{\mathrm e}^{\frac {1}{2}}\right )}{-x^{2}+{\mathrm e}^{\frac {1}{2}}+x -3}\) | \(22\) |
norman | \(\frac {4 x -2 \,{\mathrm e}^{\frac {1}{2}}+6}{-x^{2}+{\mathrm e}^{\frac {1}{2}}+x -3}\) | \(23\) |
risch | \(\frac {4 x -2 \,{\mathrm e}^{\frac {1}{2}}+6}{-x^{2}+{\mathrm e}^{\frac {1}{2}}+x -3}\) | \(23\) |
parallelrisch | \(-\frac {2 \,{\mathrm e}^{\frac {1}{2}}-4 x -6}{-x^{2}+{\mathrm e}^{\frac {1}{2}}+x -3}\) | \(24\) |
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Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-18+\sqrt {e} (6-4 x)+12 x+4 x^2}{9+e-6 x+7 x^2-2 x^3+x^4+\sqrt {e} \left (-6+2 x-2 x^2\right )} \, dx=-\frac {2 \, {\left (2 \, x - e^{\frac {1}{2}} + 3\right )}}{x^{2} - x - e^{\frac {1}{2}} + 3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (19) = 38\).
Time = 0.45 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.48 \[ \int \frac {-18+\sqrt {e} (6-4 x)+12 x+4 x^2}{9+e-6 x+7 x^2-2 x^3+x^4+\sqrt {e} \left (-6+2 x-2 x^2\right )} \, dx=\frac {x \left (-484 - 64 e + 352 e^{\frac {1}{2}}\right ) - 272 e - 726 + 32 e^{\frac {3}{2}} + 770 e^{\frac {1}{2}}}{x^{2} \left (- 88 e^{\frac {1}{2}} + 16 e + 121\right ) + x \left (-121 - 16 e + 88 e^{\frac {1}{2}}\right ) - 385 e^{\frac {1}{2}} - 16 e^{\frac {3}{2}} + 363 + 136 e} \]
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\[ \int \frac {-18+\sqrt {e} (6-4 x)+12 x+4 x^2}{9+e-6 x+7 x^2-2 x^3+x^4+\sqrt {e} \left (-6+2 x-2 x^2\right )} \, dx=\int { \frac {2 \, {\left (2 \, x^{2} - {\left (2 \, x - 3\right )} e^{\frac {1}{2}} + 6 \, x - 9\right )}}{x^{4} - 2 \, x^{3} + 7 \, x^{2} - 2 \, {\left (x^{2} - x + 3\right )} e^{\frac {1}{2}} - 6 \, x + e + 9} \,d x } \]
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Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-18+\sqrt {e} (6-4 x)+12 x+4 x^2}{9+e-6 x+7 x^2-2 x^3+x^4+\sqrt {e} \left (-6+2 x-2 x^2\right )} \, dx=-\frac {2 \, {\left (2 \, x - e^{\frac {1}{2}} + 3\right )}}{x^{2} - x - e^{\frac {1}{2}} + 3} \]
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Time = 0.13 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {-18+\sqrt {e} (6-4 x)+12 x+4 x^2}{9+e-6 x+7 x^2-2 x^3+x^4+\sqrt {e} \left (-6+2 x-2 x^2\right )} \, dx=\frac {4\,x-2\,\sqrt {\mathrm {e}}+6}{-x^2+x+\sqrt {\mathrm {e}}-3} \]
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