Integrand size = 21, antiderivative size = 22 \[ \int \frac {-8+4 e^{6+2 x} x^2}{3 x^2} \, dx=-5+\frac {2 \left (4+\left (6+e^{6+2 x}\right ) x\right )}{3 x} \]
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Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 14, 2225} \[ \int \frac {-8+4 e^{6+2 x} x^2}{3 x^2} \, dx=\frac {2}{3} e^{2 x+6}+\frac {8}{3 x} \]
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Rule 12
Rule 14
Rule 2225
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \frac {-8+4 e^{6+2 x} x^2}{x^2} \, dx \\ & = \frac {1}{3} \int \left (4 e^{6+2 x}-\frac {8}{x^2}\right ) \, dx \\ & = \frac {8}{3 x}+\frac {4}{3} \int e^{6+2 x} \, dx \\ & = \frac {2}{3} e^{6+2 x}+\frac {8}{3 x} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-8+4 e^{6+2 x} x^2}{3 x^2} \, dx=\frac {4}{3} \left (\frac {1}{2} e^{6+2 x}+\frac {2}{x}\right ) \]
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Time = 0.12 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {2 \,{\mathrm e}^{2 x +6}}{3}+\frac {8}{3 x}\) | \(15\) |
default | \(\frac {2 \,{\mathrm e}^{6} {\mathrm e}^{2 x}}{3}+\frac {8}{3 x}\) | \(17\) |
parts | \(\frac {2 \,{\mathrm e}^{6} {\mathrm e}^{2 x}}{3}+\frac {8}{3 x}\) | \(17\) |
norman | \(\frac {\frac {8}{3}+\frac {2 x \,{\mathrm e}^{6} {\mathrm e}^{2 x}}{3}}{x}\) | \(18\) |
parallelrisch | \(\frac {8+2 x \,{\mathrm e}^{6} {\mathrm e}^{2 x}}{3 x}\) | \(19\) |
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Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {-8+4 e^{6+2 x} x^2}{3 x^2} \, dx=\frac {2 \, {\left (x e^{\left (2 \, x + 6\right )} + 4\right )}}{3 \, x} \]
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Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {-8+4 e^{6+2 x} x^2}{3 x^2} \, dx=\frac {2 e^{6} e^{2 x}}{3} + \frac {8}{3 x} \]
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Time = 0.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int \frac {-8+4 e^{6+2 x} x^2}{3 x^2} \, dx=\frac {8}{3 \, x} + \frac {2}{3} \, e^{\left (2 \, x + 6\right )} \]
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Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {-8+4 e^{6+2 x} x^2}{3 x^2} \, dx=\frac {2 \, {\left (x e^{\left (2 \, x + 6\right )} + 4\right )}}{3 \, x} \]
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Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int \frac {-8+4 e^{6+2 x} x^2}{3 x^2} \, dx=\frac {2\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^6}{3}+\frac {8}{3\,x} \]
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