Integrand size = 69, antiderivative size = 27 \[ \int \frac {x^2+e^{\frac {-48+12 x^2-12 x^3+e^5 \left (36 x^2+12 x^3\right )}{x}} \left (48+12 x^2-24 x^3+e^5 \left (36 x^2+24 x^3\right )\right )}{x^2} \, dx=-1+e^{12 \left (-\frac {4}{x}+x-x^2+e^5 x (3+x)\right )}+x \]
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Time = 0.32 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {14, 6838} \[ \int \frac {x^2+e^{\frac {-48+12 x^2-12 x^3+e^5 \left (36 x^2+12 x^3\right )}{x}} \left (48+12 x^2-24 x^3+e^5 \left (36 x^2+24 x^3\right )\right )}{x^2} \, dx=\exp \left (-12 \left (1-e^5\right ) x^2+12 \left (1+3 e^5\right ) x-\frac {48}{x}\right )+x \]
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Rule 14
Rule 6838
Rubi steps \begin{align*} \text {integral}& = \int \left (1+\frac {12 \exp \left (-\frac {48}{x}+12 \left (1+3 e^5\right ) x+12 \left (-1+e^5\right ) x^2\right ) \left (4+\left (1+3 e^5\right ) x^2-2 \left (1-e^5\right ) x^3\right )}{x^2}\right ) \, dx \\ & = x+12 \int \frac {\exp \left (-\frac {48}{x}+12 \left (1+3 e^5\right ) x+12 \left (-1+e^5\right ) x^2\right ) \left (4+\left (1+3 e^5\right ) x^2-2 \left (1-e^5\right ) x^3\right )}{x^2} \, dx \\ & = e^{-\frac {48}{x}+12 \left (1+3 e^5\right ) x-12 \left (1-e^5\right ) x^2}+x \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {x^2+e^{\frac {-48+12 x^2-12 x^3+e^5 \left (36 x^2+12 x^3\right )}{x}} \left (48+12 x^2-24 x^3+e^5 \left (36 x^2+24 x^3\right )\right )}{x^2} \, dx=e^{-\frac {48}{x}+12 \left (1+3 e^5\right ) x+12 \left (-1+e^5\right ) x^2}+x \]
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Time = 0.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19
method | result | size |
risch | \(x +{\mathrm e}^{\frac {12 x^{3} {\mathrm e}^{5}+36 x^{2} {\mathrm e}^{5}-12 x^{3}+12 x^{2}-48}{x}}\) | \(32\) |
parallelrisch | \(x +{\mathrm e}^{\frac {\left (12 x^{3}+36 x^{2}\right ) {\mathrm e}^{5}-12 x^{3}+12 x^{2}-48}{x}}\) | \(34\) |
parts | \(x +{\mathrm e}^{\frac {\left (12 x^{3}+36 x^{2}\right ) {\mathrm e}^{5}-12 x^{3}+12 x^{2}-48}{x}}\) | \(34\) |
norman | \(\frac {x^{2}+x \,{\mathrm e}^{\frac {\left (12 x^{3}+36 x^{2}\right ) {\mathrm e}^{5}-12 x^{3}+12 x^{2}-48}{x}}}{x}\) | \(42\) |
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Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {x^2+e^{\frac {-48+12 x^2-12 x^3+e^5 \left (36 x^2+12 x^3\right )}{x}} \left (48+12 x^2-24 x^3+e^5 \left (36 x^2+24 x^3\right )\right )}{x^2} \, dx=x + e^{\left (-\frac {12 \, {\left (x^{3} - x^{2} - {\left (x^{3} + 3 \, x^{2}\right )} e^{5} + 4\right )}}{x}\right )} \]
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Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {x^2+e^{\frac {-48+12 x^2-12 x^3+e^5 \left (36 x^2+12 x^3\right )}{x}} \left (48+12 x^2-24 x^3+e^5 \left (36 x^2+24 x^3\right )\right )}{x^2} \, dx=x + e^{\frac {- 12 x^{3} + 12 x^{2} + \left (12 x^{3} + 36 x^{2}\right ) e^{5} - 48}{x}} \]
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Time = 0.39 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {x^2+e^{\frac {-48+12 x^2-12 x^3+e^5 \left (36 x^2+12 x^3\right )}{x}} \left (48+12 x^2-24 x^3+e^5 \left (36 x^2+24 x^3\right )\right )}{x^2} \, dx=x + e^{\left (12 \, x^{2} e^{5} - 12 \, x^{2} + 36 \, x e^{5} + 12 \, x - \frac {48}{x}\right )} \]
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Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {x^2+e^{\frac {-48+12 x^2-12 x^3+e^5 \left (36 x^2+12 x^3\right )}{x}} \left (48+12 x^2-24 x^3+e^5 \left (36 x^2+24 x^3\right )\right )}{x^2} \, dx=x + e^{\left (\frac {12 \, {\left (x^{3} e^{5} - x^{3} + 3 \, x^{2} e^{5} + x^{2} - 4\right )}}{x}\right )} \]
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Time = 8.42 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {x^2+e^{\frac {-48+12 x^2-12 x^3+e^5 \left (36 x^2+12 x^3\right )}{x}} \left (48+12 x^2-24 x^3+e^5 \left (36 x^2+24 x^3\right )\right )}{x^2} \, dx=x+{\mathrm {e}}^{12\,x^2\,{\mathrm {e}}^5}\,{\mathrm {e}}^{12\,x}\,{\mathrm {e}}^{-12\,x^2}\,{\mathrm {e}}^{-\frac {48}{x}}\,{\mathrm {e}}^{36\,x\,{\mathrm {e}}^5} \]
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