Integrand size = 28, antiderivative size = 30 \[ \int \frac {-3 e^{\frac {3+4 x}{x}}+5 x^2+2 x^3}{x^2} \, dx=1+e^{4+\frac {3}{x}}+x+5 x^2+4 \left (x-x^2\right )-\log (4) \]
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Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.53, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {14, 2240} \[ \int \frac {-3 e^{\frac {3+4 x}{x}}+5 x^2+2 x^3}{x^2} \, dx=x^2+5 x+e^{\frac {3}{x}+4} \]
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Rule 14
Rule 2240
Rubi steps \begin{align*} \text {integral}& = \int \left (5-\frac {3 e^{4+\frac {3}{x}}}{x^2}+2 x\right ) \, dx \\ & = 5 x+x^2-3 \int \frac {e^{4+\frac {3}{x}}}{x^2} \, dx \\ & = e^{4+\frac {3}{x}}+5 x+x^2 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.53 \[ \int \frac {-3 e^{\frac {3+4 x}{x}}+5 x^2+2 x^3}{x^2} \, dx=e^{4+\frac {3}{x}}+5 x+x^2 \]
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Time = 0.11 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.53
method | result | size |
derivativedivides | \(x^{2}+5 x +{\mathrm e}^{4+\frac {3}{x}}\) | \(16\) |
default | \(x^{2}+5 x +{\mathrm e}^{4+\frac {3}{x}}\) | \(16\) |
risch | \(x^{2}+5 x +{\mathrm e}^{\frac {3+4 x}{x}}\) | \(18\) |
parallelrisch | \(x^{2}+5 x +{\mathrm e}^{\frac {3+4 x}{x}}\) | \(18\) |
parts | \(x^{2}+5 x +{\mathrm e}^{\frac {3+4 x}{x}}\) | \(18\) |
norman | \(\frac {x^{3}+{\mathrm e}^{\frac {3+4 x}{x}} x +5 x^{2}}{x}\) | \(26\) |
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Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.57 \[ \int \frac {-3 e^{\frac {3+4 x}{x}}+5 x^2+2 x^3}{x^2} \, dx=x^{2} + 5 \, x + e^{\left (\frac {4 \, x + 3}{x}\right )} \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.47 \[ \int \frac {-3 e^{\frac {3+4 x}{x}}+5 x^2+2 x^3}{x^2} \, dx=x^{2} + 5 x + e^{\frac {4 x + 3}{x}} \]
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Time = 0.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.50 \[ \int \frac {-3 e^{\frac {3+4 x}{x}}+5 x^2+2 x^3}{x^2} \, dx=x^{2} + 5 \, x + e^{\left (\frac {3}{x} + 4\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (20) = 40\).
Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.03 \[ \int \frac {-3 e^{\frac {3+4 x}{x}}+5 x^2+2 x^3}{x^2} \, dx=\frac {\frac {{\left (4 \, x + 3\right )}^{2} e^{\left (\frac {4 \, x + 3}{x}\right )}}{x^{2}} - \frac {8 \, {\left (4 \, x + 3\right )} e^{\left (\frac {4 \, x + 3}{x}\right )}}{x} + \frac {15 \, {\left (4 \, x + 3\right )}}{x} + 16 \, e^{\left (\frac {4 \, x + 3}{x}\right )} - 51}{\frac {{\left (4 \, x + 3\right )}^{2}}{x^{2}} - \frac {8 \, {\left (4 \, x + 3\right )}}{x} + 16} \]
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Time = 14.89 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.53 \[ \int \frac {-3 e^{\frac {3+4 x}{x}}+5 x^2+2 x^3}{x^2} \, dx=5\,x+{\mathrm {e}}^4\,{\mathrm {e}}^{3/x}+x^2 \]
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