Integrand size = 62, antiderivative size = 28 \[ \int \frac {-100+200 e^3-100 e^6+e^{2 x} \left (-18 x^2+18 x^3\right )+e^x \left (90 x-30 x^2+e^3 \left (-90 x+30 x^2\right )\right )}{225 x^5} \, dx=5+\frac {\left (\frac {e^x}{5}+\frac {-1+e^3}{3 x}\right )^2}{x^2} \]
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Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {12, 14, 2228} \[ \int \frac {-100+200 e^3-100 e^6+e^{2 x} \left (-18 x^2+18 x^3\right )+e^x \left (90 x-30 x^2+e^3 \left (-90 x+30 x^2\right )\right )}{225 x^5} \, dx=\frac {\left (1-e^3\right )^2}{9 x^4}-\frac {2 \left (1-e^3\right ) e^x}{15 x^3}+\frac {e^{2 x}}{25 x^2} \]
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Rule 12
Rule 14
Rule 2228
Rubi steps \begin{align*} \text {integral}& = \frac {1}{225} \int \frac {-100+200 e^3-100 e^6+e^{2 x} \left (-18 x^2+18 x^3\right )+e^x \left (90 x-30 x^2+e^3 \left (-90 x+30 x^2\right )\right )}{x^5} \, dx \\ & = \frac {1}{225} \int \left (-\frac {100 \left (-1+e^3\right )^2}{x^5}+\frac {30 (-1+e) e^x \left (1+e+e^2\right ) (-3+x)}{x^4}+\frac {18 e^{2 x} (-1+x)}{x^3}\right ) \, dx \\ & = \frac {\left (1-e^3\right )^2}{9 x^4}+\frac {2}{25} \int \frac {e^{2 x} (-1+x)}{x^3} \, dx+\frac {1}{15} \left (2 (-1+e) \left (1+e+e^2\right )\right ) \int \frac {e^x (-3+x)}{x^4} \, dx \\ & = \frac {\left (1-e^3\right )^2}{9 x^4}-\frac {2 (1-e) e^x \left (1+e+e^2\right )}{15 x^3}+\frac {e^{2 x}}{25 x^2} \\ \end{align*}
Time = 1.37 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {-100+200 e^3-100 e^6+e^{2 x} \left (-18 x^2+18 x^3\right )+e^x \left (90 x-30 x^2+e^3 \left (-90 x+30 x^2\right )\right )}{225 x^5} \, dx=\frac {\left (-5+5 e^3+3 e^x x\right )^2}{225 x^4} \]
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Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29
method | result | size |
norman | \(\frac {\left (\frac {2 \,{\mathrm e}^{3}}{15}-\frac {2}{15}\right ) x \,{\mathrm e}^{x}+\frac {{\mathrm e}^{2 x} x^{2}}{25}+\frac {{\mathrm e}^{6}}{9}-\frac {2 \,{\mathrm e}^{3}}{9}+\frac {1}{9}}{x^{4}}\) | \(36\) |
risch | \(\frac {30 \,{\mathrm e}^{3+x} x +9 \,{\mathrm e}^{2 x} x^{2}-30 \,{\mathrm e}^{x} x -50 \,{\mathrm e}^{3}+25 \,{\mathrm e}^{6}+25}{225 x^{4}}\) | \(37\) |
parallelrisch | \(\frac {30 x \,{\mathrm e}^{3} {\mathrm e}^{x}+9 \,{\mathrm e}^{2 x} x^{2}-30 \,{\mathrm e}^{x} x -50 \,{\mathrm e}^{3}+25 \,{\mathrm e}^{6}+25}{225 x^{4}}\) | \(39\) |
parts | \(-\frac {-\frac {4 \,{\mathrm e}^{6}}{9}+\frac {8 \,{\mathrm e}^{3}}{9}-\frac {4}{9}}{4 x^{4}}+\frac {2 \,{\mathrm e}^{3} \left (-\frac {{\mathrm e}^{x}}{2 x^{2}}-\frac {{\mathrm e}^{x}}{2 x}-\frac {\operatorname {Ei}_{1}\left (-x \right )}{2}\right )}{15}-\frac {2 \,{\mathrm e}^{x}}{15 x^{3}}-\frac {2 \,{\mathrm e}^{3} \left (-\frac {{\mathrm e}^{x}}{3 x^{3}}-\frac {{\mathrm e}^{x}}{6 x^{2}}-\frac {{\mathrm e}^{x}}{6 x}-\frac {\operatorname {Ei}_{1}\left (-x \right )}{6}\right )}{5}+\frac {{\mathrm e}^{2 x}}{25 x^{2}}\) | \(94\) |
default | \(\frac {1}{9 x^{4}}-\frac {2 \,{\mathrm e}^{3}}{9 x^{4}}+\frac {{\mathrm e}^{6}}{9 x^{4}}+\frac {{\mathrm e}^{2 x}}{25 x^{2}}-\frac {2 \,{\mathrm e}^{x}}{15 x^{3}}-\frac {2 \,{\mathrm e}^{3} \left (-\frac {{\mathrm e}^{x}}{3 x^{3}}-\frac {{\mathrm e}^{x}}{6 x^{2}}-\frac {{\mathrm e}^{x}}{6 x}-\frac {\operatorname {Ei}_{1}\left (-x \right )}{6}\right )}{5}+\frac {2 \,{\mathrm e}^{3} \left (-\frac {{\mathrm e}^{x}}{2 x^{2}}-\frac {{\mathrm e}^{x}}{2 x}-\frac {\operatorname {Ei}_{1}\left (-x \right )}{2}\right )}{15}\) | \(98\) |
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Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {-100+200 e^3-100 e^6+e^{2 x} \left (-18 x^2+18 x^3\right )+e^x \left (90 x-30 x^2+e^3 \left (-90 x+30 x^2\right )\right )}{225 x^5} \, dx=\frac {9 \, x^{2} e^{\left (2 \, x\right )} + 30 \, {\left (x e^{3} - x\right )} e^{x} + 25 \, e^{6} - 50 \, e^{3} + 25}{225 \, x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (20) = 40\).
Time = 0.10 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {-100+200 e^3-100 e^6+e^{2 x} \left (-18 x^2+18 x^3\right )+e^x \left (90 x-30 x^2+e^3 \left (-90 x+30 x^2\right )\right )}{225 x^5} \, dx=- \frac {- \frac {4 e^{6}}{9} - \frac {4}{9} + \frac {8 e^{3}}{9}}{4 x^{4}} + \frac {15 x^{3} e^{2 x} + \left (- 50 x^{2} + 50 x^{2} e^{3}\right ) e^{x}}{375 x^{5}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.23 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.36 \[ \int \frac {-100+200 e^3-100 e^6+e^{2 x} \left (-18 x^2+18 x^3\right )+e^x \left (90 x-30 x^2+e^3 \left (-90 x+30 x^2\right )\right )}{225 x^5} \, dx=-\frac {2}{15} \, e^{3} \Gamma \left (-2, -x\right ) - \frac {2}{5} \, e^{3} \Gamma \left (-3, -x\right ) + \frac {e^{6}}{9 \, x^{4}} - \frac {2 \, e^{3}}{9 \, x^{4}} + \frac {1}{9 \, x^{4}} + \frac {4}{25} \, \Gamma \left (-1, -2 \, x\right ) + \frac {2}{15} \, \Gamma \left (-2, -x\right ) + \frac {8}{25} \, \Gamma \left (-2, -2 \, x\right ) + \frac {2}{5} \, \Gamma \left (-3, -x\right ) \]
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Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {-100+200 e^3-100 e^6+e^{2 x} \left (-18 x^2+18 x^3\right )+e^x \left (90 x-30 x^2+e^3 \left (-90 x+30 x^2\right )\right )}{225 x^5} \, dx=\frac {9 \, x^{2} e^{\left (2 \, x\right )} + 30 \, x e^{\left (x + 3\right )} - 30 \, x e^{x} + 25 \, e^{6} - 50 \, e^{3} + 25}{225 \, x^{4}} \]
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Time = 12.77 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {-100+200 e^3-100 e^6+e^{2 x} \left (-18 x^2+18 x^3\right )+e^x \left (90 x-30 x^2+e^3 \left (-90 x+30 x^2\right )\right )}{225 x^5} \, dx=\frac {25\,{\left ({\mathrm {e}}^3-1\right )}^2+9\,x^2\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^x\,\left (30\,{\mathrm {e}}^3-30\right )}{225\,x^4} \]
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