Integrand size = 45, antiderivative size = 26 \[ \int \frac {12 x-18 x^2+e^2 \left (3-6 x+3 x^2\right )}{e^2 \left (25 x^4-50 x^5+25 x^6\right )} \, dx=\frac {\frac {6}{-1+x}-\frac {e^2}{x}}{25 e^2 x^2} \]
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Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {12, 1608, 27, 1834} \[ \int \frac {12 x-18 x^2+e^2 \left (3-6 x+3 x^2\right )}{e^2 \left (25 x^4-50 x^5+25 x^6\right )} \, dx=-\frac {1}{25 x^3}-\frac {6}{25 e^2 x^2}-\frac {6}{25 e^2 x}-\frac {6}{25 e^2 (1-x)} \]
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Rule 12
Rule 27
Rule 1608
Rule 1834
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {12 x-18 x^2+e^2 \left (3-6 x+3 x^2\right )}{25 x^4-50 x^5+25 x^6} \, dx}{e^2} \\ & = \frac {\int \frac {12 x-18 x^2+e^2 \left (3-6 x+3 x^2\right )}{x^4 \left (25-50 x+25 x^2\right )} \, dx}{e^2} \\ & = \frac {\int \frac {12 x-18 x^2+e^2 \left (3-6 x+3 x^2\right )}{25 (-1+x)^2 x^4} \, dx}{e^2} \\ & = \frac {\int \frac {12 x-18 x^2+e^2 \left (3-6 x+3 x^2\right )}{(-1+x)^2 x^4} \, dx}{25 e^2} \\ & = \frac {\int \left (-\frac {6}{(-1+x)^2}+\frac {3 e^2}{x^4}+\frac {12}{x^3}+\frac {6}{x^2}\right ) \, dx}{25 e^2} \\ & = -\frac {6}{25 e^2 (1-x)}-\frac {1}{25 x^3}-\frac {6}{25 e^2 x^2}-\frac {6}{25 e^2 x} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {12 x-18 x^2+e^2 \left (3-6 x+3 x^2\right )}{e^2 \left (25 x^4-50 x^5+25 x^6\right )} \, dx=\frac {-1+\frac {6 x}{e^2 (-1+x)}}{25 x^3} \]
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Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96
method | result | size |
risch | \(\frac {{\mathrm e}^{-2} \left (\left (-\frac {{\mathrm e}^{2}}{25}+\frac {6}{25}\right ) x +\frac {{\mathrm e}^{2}}{25}\right )}{x^{3} \left (-1+x \right )}\) | \(25\) |
gosper | \(-\frac {\left ({\mathrm e}^{2} x -{\mathrm e}^{2}-6 x \right ) {\mathrm e}^{-2}}{25 x^{3} \left (-1+x \right )}\) | \(27\) |
parallelrisch | \(-\frac {\left ({\mathrm e}^{2} x -{\mathrm e}^{2}-6 x \right ) {\mathrm e}^{-2}}{25 x^{3} \left (-1+x \right )}\) | \(27\) |
default | \(\frac {{\mathrm e}^{-2} \left (-\frac {{\mathrm e}^{2}}{x^{3}}-\frac {6}{x^{2}}-\frac {6}{x}+\frac {6}{-1+x}\right )}{25}\) | \(32\) |
norman | \(\frac {\left (\frac {{\mathrm e}^{-1} {\mathrm e}^{2}}{25}-\frac {\left ({\mathrm e}^{2}-6\right ) {\mathrm e}^{-1} x}{25}\right ) {\mathrm e}^{-1}}{x^{3} \left (-1+x \right )}\) | \(34\) |
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Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {12 x-18 x^2+e^2 \left (3-6 x+3 x^2\right )}{e^2 \left (25 x^4-50 x^5+25 x^6\right )} \, dx=-\frac {{\left ({\left (x - 1\right )} e^{2} - 6 \, x\right )} e^{\left (-2\right )}}{25 \, {\left (x^{4} - x^{3}\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {12 x-18 x^2+e^2 \left (3-6 x+3 x^2\right )}{e^2 \left (25 x^4-50 x^5+25 x^6\right )} \, dx=- \frac {x \left (-6 + e^{2}\right ) - e^{2}}{25 x^{4} e^{2} - 25 x^{3} e^{2}} \]
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Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {12 x-18 x^2+e^2 \left (3-6 x+3 x^2\right )}{e^2 \left (25 x^4-50 x^5+25 x^6\right )} \, dx=-\frac {{\left (x {\left (e^{2} - 6\right )} - e^{2}\right )} e^{\left (-2\right )}}{25 \, {\left (x^{4} - x^{3}\right )}} \]
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Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {12 x-18 x^2+e^2 \left (3-6 x+3 x^2\right )}{e^2 \left (25 x^4-50 x^5+25 x^6\right )} \, dx=\frac {1}{25} \, {\left (\frac {6}{x - 1} - \frac {6 \, x^{2} + 6 \, x + e^{2}}{x^{3}}\right )} e^{\left (-2\right )} \]
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Time = 0.13 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {12 x-18 x^2+e^2 \left (3-6 x+3 x^2\right )}{e^2 \left (25 x^4-50 x^5+25 x^6\right )} \, dx=-\frac {{\mathrm {e}}^2-x\,\left ({\mathrm {e}}^2-6\right )}{25\,x^3\,{\mathrm {e}}^2-25\,x^4\,{\mathrm {e}}^2} \]
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