Integrand size = 36, antiderivative size = 20 \[ \int \frac {e^4 (225-150 x)+1125 e^2 x^2}{e^4-30 e^2 x+225 x^2} \, dx=\frac {5 (3-x) x}{\frac {1}{15}-\frac {x}{e^2}} \]
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Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.50, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {27, 1864} \[ \int \frac {e^4 (225-150 x)+1125 e^2 x^2}{e^4-30 e^2 x+225 x^2} \, dx=5 e^2 x+\frac {e^4 \left (45-e^2\right )}{3 \left (e^2-15 x\right )} \]
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Rule 27
Rule 1864
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^4 (225-150 x)+1125 e^2 x^2}{\left (e^2-15 x\right )^2} \, dx \\ & = \int \left (5 e^2-\frac {5 e^4 \left (-45+e^2\right )}{\left (e^2-15 x\right )^2}\right ) \, dx \\ & = \frac {e^4 \left (45-e^2\right )}{3 \left (e^2-15 x\right )}+5 e^2 x \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.85 \[ \int \frac {e^4 (225-150 x)+1125 e^2 x^2}{e^4-30 e^2 x+225 x^2} \, dx=-\frac {2 e^6+225 e^2 x^2-15 e^4 (3+2 x)}{3 \left (e^2-15 x\right )} \]
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Time = 0.33 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05
method | result | size |
gosper | \(\frac {15 \left (-5 x^{2}+{\mathrm e}^{2}\right ) {\mathrm e}^{2}}{{\mathrm e}^{2}-15 x}\) | \(21\) |
norman | \(\frac {-75 x^{2} {\mathrm e}^{2}+15 \,{\mathrm e}^{4}}{{\mathrm e}^{2}-15 x}\) | \(24\) |
parallelrisch | \(\frac {-1125 x^{2} {\mathrm e}^{2}+225 \,{\mathrm e}^{4}}{15 \,{\mathrm e}^{2}-225 x}\) | \(25\) |
risch | \(5 \,{\mathrm e}^{2} x -\frac {{\mathrm e}^{6}}{3 \left ({\mathrm e}^{2}-15 x \right )}+\frac {15 \,{\mathrm e}^{4}}{{\mathrm e}^{2}-15 x}\) | \(31\) |
meijerg | \(\frac {225 x}{1-15 x \,{\mathrm e}^{-2}}-\frac {2 \,{\mathrm e}^{4} \left (\frac {15 x \,{\mathrm e}^{-2}}{1-15 x \,{\mathrm e}^{-2}}+\ln \left (1-15 x \,{\mathrm e}^{-2}\right )\right )}{3}-\frac {{\mathrm e}^{4} \left (-\frac {5 x \,{\mathrm e}^{-2} \left (-45 x \,{\mathrm e}^{-2}+6\right )}{1-15 x \,{\mathrm e}^{-2}}-2 \ln \left (1-15 x \,{\mathrm e}^{-2}\right )\right )}{3}\) | \(77\) |
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Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {e^4 (225-150 x)+1125 e^2 x^2}{e^4-30 e^2 x+225 x^2} \, dx=\frac {225 \, x^{2} e^{2} - 15 \, {\left (x + 3\right )} e^{4} + e^{6}}{3 \, {\left (15 \, x - e^{2}\right )}} \]
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Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {e^4 (225-150 x)+1125 e^2 x^2}{e^4-30 e^2 x+225 x^2} \, dx=5 x e^{2} + \frac {- 45 e^{4} + e^{6}}{45 x - 3 e^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {e^4 (225-150 x)+1125 e^2 x^2}{e^4-30 e^2 x+225 x^2} \, dx=5 \, x e^{2} + \frac {e^{6} - 45 \, e^{4}}{3 \, {\left (15 \, x - e^{2}\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {e^4 (225-150 x)+1125 e^2 x^2}{e^4-30 e^2 x+225 x^2} \, dx=5 \, x e^{2} + \frac {e^{6} - 45 \, e^{4}}{3 \, {\left (15 \, x - e^{2}\right )}} \]
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Time = 9.96 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {e^4 (225-150 x)+1125 e^2 x^2}{e^4-30 e^2 x+225 x^2} \, dx=5\,x\,{\mathrm {e}}^2-\frac {15\,{\mathrm {e}}^4-\frac {{\mathrm {e}}^6}{3}}{15\,x-{\mathrm {e}}^2} \]
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