Integrand size = 24, antiderivative size = 28 \[ \int \frac {\frac {48}{e}+12 e^3 x+9 x^5}{4 x^3} \, dx=3 \left (-\frac {e^3}{x}+\frac {-\frac {2}{e}+\frac {x^5}{4}}{x^2}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {12, 14} \[ \int \frac {\frac {48}{e}+12 e^3 x+9 x^5}{4 x^3} \, dx=\frac {3 x^3}{4}-\frac {6}{e x^2}-\frac {3 e^3}{x} \]
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Rule 12
Rule 14
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \int \frac {\frac {48}{e}+12 e^3 x+9 x^5}{x^3} \, dx \\ & = \frac {1}{4} \int \left (\frac {48}{e x^3}+\frac {12 e^3}{x^2}+9 x^2\right ) \, dx \\ & = -\frac {6}{e x^2}-\frac {3 e^3}{x}+\frac {3 x^3}{4} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {\frac {48}{e}+12 e^3 x+9 x^5}{4 x^3} \, dx=\frac {3 \left (-\frac {8}{x^2}-\frac {4 e^4}{x}+e x^3\right )}{4 e} \]
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Time = 0.18 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75
method | result | size |
risch | \(-\frac {3 \left (-x^{5}+8 \,{\mathrm e}^{-1}+4 x \,{\mathrm e}^{3}\right )}{4 x^{2}}\) | \(21\) |
norman | \(\frac {\frac {3 x^{5}}{4}-6 \,{\mathrm e}^{-1}-3 x \,{\mathrm e}^{3}}{x^{2}}\) | \(22\) |
gosper | \(-\frac {3 \left (-x^{5}+4 x \,{\mathrm e}^{3}+4 \,{\mathrm e}^{\ln \left (2\right )-1}\right )}{4 x^{2}}\) | \(24\) |
default | \(\frac {3 x^{3}}{4}-\frac {3 \,{\mathrm e}^{3}}{x}-\frac {3 \,{\mathrm e}^{\ln \left (2\right )-1}}{x^{2}}\) | \(24\) |
parallelrisch | \(-\frac {-3 x^{5}+12 x \,{\mathrm e}^{3}+12 \,{\mathrm e}^{\ln \left (2\right )-1}}{4 x^{2}}\) | \(24\) |
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Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {\frac {48}{e}+12 e^3 x+9 x^5}{4 x^3} \, dx=\frac {3 \, {\left (x^{5} e^{\left (3 \, \log \left (2\right ) - 3\right )} - 32 \, x - 4 \, e^{\left (4 \, \log \left (2\right ) - 4\right )}\right )} e^{\left (-3 \, \log \left (2\right ) + 3\right )}}{4 \, x^{2}} \]
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Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {\frac {48}{e}+12 e^3 x+9 x^5}{4 x^3} \, dx=\frac {3 e x^{3} + \frac {- 12 x e^{4} - 24}{x^{2}}}{4 e} \]
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Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {\frac {48}{e}+12 e^3 x+9 x^5}{4 x^3} \, dx=\frac {3}{4} \, x^{3} - \frac {3 \, {\left (x e^{4} + 2\right )} e^{\left (-1\right )}}{x^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {\frac {48}{e}+12 e^3 x+9 x^5}{4 x^3} \, dx=\frac {3}{4} \, x^{3} - \frac {3 \, {\left (x e^{3} + e^{\left (\log \left (2\right ) - 1\right )}\right )}}{x^{2}} \]
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Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {\frac {48}{e}+12 e^3 x+9 x^5}{4 x^3} \, dx=\frac {3\,x^3}{4}-\frac {{\mathrm {e}}^{-1}\,\left (3\,x\,{\mathrm {e}}^4+6\right )}{x^2} \]
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