Integrand size = 27, antiderivative size = 22 \[ \int \frac {1}{5} e^{-6 x/5} \left (-75-18 x+e^{2 x} (25+4 x)\right ) \, dx=e^{-x/5} \left (3 e^{-x}+e^x\right ) (5+x) \]
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Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(22)=44\).
Time = 0.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.14, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {12, 6874, 2225, 2207} \[ \int \frac {1}{5} e^{-6 x/5} \left (-75-18 x+e^{2 x} (25+4 x)\right ) \, dx=3 e^{-6 x/5} x+15 e^{-6 x/5}-\frac {5}{4} e^{4 x/5}+\frac {1}{4} e^{4 x/5} (4 x+25) \]
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Rule 12
Rule 2207
Rule 2225
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \int e^{-6 x/5} \left (-75-18 x+e^{2 x} (25+4 x)\right ) \, dx \\ & = \frac {1}{5} \int \left (-75 e^{-6 x/5}-18 e^{-6 x/5} x+e^{4 x/5} (25+4 x)\right ) \, dx \\ & = \frac {1}{5} \int e^{4 x/5} (25+4 x) \, dx-\frac {18}{5} \int e^{-6 x/5} x \, dx-15 \int e^{-6 x/5} \, dx \\ & = \frac {25}{2} e^{-6 x/5}+3 e^{-6 x/5} x+\frac {1}{4} e^{4 x/5} (25+4 x)-3 \int e^{-6 x/5} \, dx-\int e^{4 x/5} \, dx \\ & = 15 e^{-6 x/5}-\frac {5}{4} e^{4 x/5}+3 e^{-6 x/5} x+\frac {1}{4} e^{4 x/5} (25+4 x) \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{5} e^{-6 x/5} \left (-75-18 x+e^{2 x} (25+4 x)\right ) \, dx=e^{-6 x/5} \left (3+e^{2 x}\right ) (5+x) \]
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Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09
method | result | size |
risch | \(\frac {\left (25+5 x \right ) {\mathrm e}^{\frac {4 x}{5}}}{5}+\frac {\left (15 x +75\right ) {\mathrm e}^{-\frac {6 x}{5}}}{5}\) | \(24\) |
default | \(15 \,{\mathrm e}^{-\frac {6 x}{5}}+5 \,{\mathrm e}^{\frac {4 x}{5}}+3 x \,{\mathrm e}^{-\frac {6 x}{5}}+x \,{\mathrm e}^{\frac {4 x}{5}}\) | \(27\) |
parts | \(15 \,{\mathrm e}^{-\frac {6 x}{5}}+5 \,{\mathrm e}^{\frac {4 x}{5}}+3 x \,{\mathrm e}^{-\frac {6 x}{5}}+x \,{\mathrm e}^{\frac {4 x}{5}}\) | \(27\) |
norman | \(\left (15+x \,{\mathrm e}^{2 x}+3 x +5 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{-\frac {6 x}{5}}\) | \(29\) |
parallelrisch | \(\frac {\left (375+25 x \,{\mathrm e}^{2 x}+125 \,{\mathrm e}^{2 x}+75 x \right ) {\mathrm e}^{-x} {\mathrm e}^{-\frac {x}{5}}}{25}\) | \(31\) |
meijerg | \(-15+\frac {25 \,{\mathrm e}^{-\frac {6 x}{5}}}{2}+\frac {5 \,{\mathrm e}^{2 x} {\mathrm e}^{-2 x} \left (1-\frac {\left (2-\frac {8 x}{5}\right ) {\mathrm e}^{\frac {4 x}{5}}}{2}\right )}{4}-\frac {25 \,{\mathrm e}^{2 x} {\mathrm e}^{-2 x} \left (1-{\mathrm e}^{\frac {4 x}{5}}\right )}{4}+\frac {5 \left (2+\frac {12 x}{5}\right ) {\mathrm e}^{-\frac {6 x}{5}}}{4}\) | \(61\) |
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Time = 0.33 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {1}{5} e^{-6 x/5} \left (-75-18 x+e^{2 x} (25+4 x)\right ) \, dx={\left (3 \, {\left (x + 5\right )} e^{\left (-2 \, x\right )} + x + 5\right )} e^{\left (\frac {4}{5} \, x\right )} \]
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Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{5} e^{-6 x/5} \left (-75-18 x+e^{2 x} (25+4 x)\right ) \, dx=\left (x + 5\right ) e^{\frac {4 x}{5}} + \left (3 x + 15\right ) e^{- \frac {6 x}{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (17) = 34\).
Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {1}{5} e^{-6 x/5} \left (-75-18 x+e^{2 x} (25+4 x)\right ) \, dx=\frac {1}{4} \, {\left (4 \, x - 5\right )} e^{\left (\frac {4}{5} \, x\right )} + \frac {1}{2} \, {\left (6 \, x + 5\right )} e^{\left (-\frac {6}{5} \, x\right )} + \frac {25}{4} \, e^{\left (\frac {4}{5} \, x\right )} + \frac {25}{2} \, e^{\left (-\frac {6}{5} \, x\right )} \]
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Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {1}{5} e^{-6 x/5} \left (-75-18 x+e^{2 x} (25+4 x)\right ) \, dx=x e^{\left (\frac {4}{5} \, x\right )} + 3 \, x e^{\left (-\frac {6}{5} \, x\right )} + 5 \, e^{\left (\frac {4}{5} \, x\right )} + 15 \, e^{\left (-\frac {6}{5} \, x\right )} \]
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Time = 18.50 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int \frac {1}{5} e^{-6 x/5} \left (-75-18 x+e^{2 x} (25+4 x)\right ) \, dx={\mathrm {e}}^{-\frac {6\,x}{5}}\,\left ({\mathrm {e}}^{2\,x}+3\right )\,\left (x+5\right ) \]
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