Integrand size = 31, antiderivative size = 30 \[ \int \frac {16 e^{4/x}+2 x^2+e^4 x^2+8 x^3}{x^2} \, dx=2 x+e^4 \left (e^{4/3}+x\right )-4 \left (e^{4/x}-x^2\right ) \]
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Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {6, 14, 2240} \[ \int \frac {16 e^{4/x}+2 x^2+e^4 x^2+8 x^3}{x^2} \, dx=4 x^2+\left (2+e^4\right ) x-4 e^{4/x} \]
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Rule 6
Rule 14
Rule 2240
Rubi steps \begin{align*} \text {integral}& = \int \frac {16 e^{4/x}+\left (2+e^4\right ) x^2+8 x^3}{x^2} \, dx \\ & = \int \left (2 \left (1+\frac {e^4}{2}\right )+\frac {16 e^{4/x}}{x^2}+8 x\right ) \, dx \\ & = \left (2+e^4\right ) x+4 x^2+16 \int \frac {e^{4/x}}{x^2} \, dx \\ & = -4 e^{4/x}+\left (2+e^4\right ) x+4 x^2 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.77 \[ \int \frac {16 e^{4/x}+2 x^2+e^4 x^2+8 x^3}{x^2} \, dx=-4 e^{4/x}+2 x+e^4 x+4 x^2 \]
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Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(4 x^{2}+2 x +x \,{\mathrm e}^{4}-4 \,{\mathrm e}^{\frac {4}{x}}\) | \(22\) |
default | \(4 x^{2}+2 x +x \,{\mathrm e}^{4}-4 \,{\mathrm e}^{\frac {4}{x}}\) | \(22\) |
risch | \(4 x^{2}+2 x +x \,{\mathrm e}^{4}-4 \,{\mathrm e}^{\frac {4}{x}}\) | \(22\) |
parallelrisch | \(4 x^{2}+2 x +x \,{\mathrm e}^{4}-4 \,{\mathrm e}^{\frac {4}{x}}\) | \(22\) |
parts | \(4 x^{2}+2 x +x \,{\mathrm e}^{4}-4 \,{\mathrm e}^{\frac {4}{x}}\) | \(22\) |
norman | \(\frac {\left (2+{\mathrm e}^{4}\right ) x^{2}+4 x^{3}-4 x \,{\mathrm e}^{\frac {4}{x}}}{x}\) | \(28\) |
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Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.70 \[ \int \frac {16 e^{4/x}+2 x^2+e^4 x^2+8 x^3}{x^2} \, dx=4 \, x^{2} + x e^{4} + 2 \, x - 4 \, e^{\frac {4}{x}} \]
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Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.57 \[ \int \frac {16 e^{4/x}+2 x^2+e^4 x^2+8 x^3}{x^2} \, dx=4 x^{2} + x \left (2 + e^{4}\right ) - 4 e^{\frac {4}{x}} \]
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Time = 0.18 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.70 \[ \int \frac {16 e^{4/x}+2 x^2+e^4 x^2+8 x^3}{x^2} \, dx=4 \, x^{2} + x e^{4} + 2 \, x - 4 \, e^{\frac {4}{x}} \]
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Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \frac {16 e^{4/x}+2 x^2+e^4 x^2+8 x^3}{x^2} \, dx=x^{2} {\left (\frac {e^{4}}{x} + \frac {2}{x} - \frac {4 \, e^{\frac {4}{x}}}{x^{2}} + 4\right )} \]
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Time = 8.61 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.70 \[ \int \frac {16 e^{4/x}+2 x^2+e^4 x^2+8 x^3}{x^2} \, dx=2\,x-4\,{\mathrm {e}}^{4/x}+x\,{\mathrm {e}}^4+4\,x^2 \]
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