Integrand size = 21, antiderivative size = 16 \[ \int \frac {-5+x^3+e^x x^3+2 x^4}{x^3} \, dx=2+e^x+\frac {5}{2 x^2}+x+x^2 \]
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Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {14, 2225} \[ \int \frac {-5+x^3+e^x x^3+2 x^4}{x^3} \, dx=x^2+\frac {5}{2 x^2}+x+e^x \]
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Rule 14
Rule 2225
Rubi steps \begin{align*} \text {integral}& = \int \left (e^x+\frac {-5+x^3+2 x^4}{x^3}\right ) \, dx \\ & = \int e^x \, dx+\int \frac {-5+x^3+2 x^4}{x^3} \, dx \\ & = e^x+\int \left (1-\frac {5}{x^3}+2 x\right ) \, dx \\ & = e^x+\frac {5}{2 x^2}+x+x^2 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {-5+x^3+e^x x^3+2 x^4}{x^3} \, dx=e^x+\frac {5}{2 x^2}+x+x^2 \]
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Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
default | \(x^{2}+x +\frac {5}{2 x^{2}}+{\mathrm e}^{x}\) | \(13\) |
risch | \(x^{2}+x +\frac {5}{2 x^{2}}+{\mathrm e}^{x}\) | \(13\) |
parts | \(x^{2}+x +\frac {5}{2 x^{2}}+{\mathrm e}^{x}\) | \(13\) |
norman | \(\frac {\frac {5}{2}+{\mathrm e}^{x} x^{2}+x^{4}+x^{3}}{x^{2}}\) | \(19\) |
parallelrisch | \(\frac {2 x^{4}+2 \,{\mathrm e}^{x} x^{2}+2 x^{3}+5}{2 x^{2}}\) | \(25\) |
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Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50 \[ \int \frac {-5+x^3+e^x x^3+2 x^4}{x^3} \, dx=\frac {2 \, x^{4} + 2 \, x^{3} + 2 \, x^{2} e^{x} + 5}{2 \, x^{2}} \]
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Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {-5+x^3+e^x x^3+2 x^4}{x^3} \, dx=x^{2} + x + e^{x} + \frac {5}{2 x^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {-5+x^3+e^x x^3+2 x^4}{x^3} \, dx=x^{2} + x + \frac {5}{2 \, x^{2}} + e^{x} \]
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Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50 \[ \int \frac {-5+x^3+e^x x^3+2 x^4}{x^3} \, dx=\frac {2 \, x^{4} + 2 \, x^{3} + 2 \, x^{2} e^{x} + 5}{2 \, x^{2}} \]
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Time = 13.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {-5+x^3+e^x x^3+2 x^4}{x^3} \, dx=x+{\mathrm {e}}^x+\frac {5}{2\,x^2}+x^2 \]
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