Integrand size = 80, antiderivative size = 30 \[ \int \frac {-8-12 x-6 x^2-x^3-128 x^5-112 x^6-36 x^7-4 x^8+e^x \left (8 x^2+12 x^3+6 x^4+x^5\right )}{8 x^2+12 x^3+6 x^4+x^5} \, dx=e^x+\frac {1}{x}-x^2 \left (x+\frac {2 x^2}{2 x+x^2}\right )^2 \]
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Time = 0.48 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60, number of steps used = 17, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {6873, 6874, 2225, 46, 37, 45} \[ \int \frac {-8-12 x-6 x^2-x^3-128 x^5-112 x^6-36 x^7-4 x^8+e^x \left (8 x^2+12 x^3+6 x^4+x^5\right )}{8 x^2+12 x^3+6 x^4+x^5} \, dx=-x^4-4 x^3-\frac {x^2}{4 (x+2)^2}+4 x^2+e^x+\frac {63}{x+2}-\frac {63}{(x+2)^2}+\frac {1}{x} \]
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Rule 37
Rule 45
Rule 46
Rule 2225
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-8-12 x-6 x^2-x^3-128 x^5-112 x^6-36 x^7-4 x^8+e^x \left (8 x^2+12 x^3+6 x^4+x^5\right )}{x^2 (2+x)^3} \, dx \\ & = \int \left (e^x-\frac {6}{(2+x)^3}-\frac {8}{x^2 (2+x)^3}-\frac {12}{x (2+x)^3}-\frac {x}{(2+x)^3}-\frac {128 x^3}{(2+x)^3}-\frac {112 x^4}{(2+x)^3}-\frac {36 x^5}{(2+x)^3}-\frac {4 x^6}{(2+x)^3}\right ) \, dx \\ & = \frac {3}{(2+x)^2}-4 \int \frac {x^6}{(2+x)^3} \, dx-8 \int \frac {1}{x^2 (2+x)^3} \, dx-12 \int \frac {1}{x (2+x)^3} \, dx-36 \int \frac {x^5}{(2+x)^3} \, dx-112 \int \frac {x^4}{(2+x)^3} \, dx-128 \int \frac {x^3}{(2+x)^3} \, dx+\int e^x \, dx-\int \frac {x}{(2+x)^3} \, dx \\ & = e^x+\frac {3}{(2+x)^2}-\frac {x^2}{4 (2+x)^2}-4 \int \left (-80+24 x-6 x^2+x^3+\frac {64}{(2+x)^3}-\frac {192}{(2+x)^2}+\frac {240}{2+x}\right ) \, dx-8 \int \left (\frac {1}{8 x^2}-\frac {3}{16 x}+\frac {1}{4 (2+x)^3}+\frac {1}{4 (2+x)^2}+\frac {3}{16 (2+x)}\right ) \, dx-12 \int \left (\frac {1}{8 x}-\frac {1}{2 (2+x)^3}-\frac {1}{4 (2+x)^2}-\frac {1}{8 (2+x)}\right ) \, dx-36 \int \left (24-6 x+x^2-\frac {32}{(2+x)^3}+\frac {80}{(2+x)^2}-\frac {80}{2+x}\right ) \, dx-112 \int \left (-6+x+\frac {16}{(2+x)^3}-\frac {32}{(2+x)^2}+\frac {24}{2+x}\right ) \, dx-128 \int \left (1-\frac {8}{(2+x)^3}+\frac {12}{(2+x)^2}-\frac {6}{2+x}\right ) \, dx \\ & = e^x+\frac {1}{x}+4 x^2-4 x^3-x^4-\frac {63}{(2+x)^2}-\frac {x^2}{4 (2+x)^2}+\frac {63}{2+x} \\ \end{align*}
Time = 2.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60 \[ \int \frac {-8-12 x-6 x^2-x^3-128 x^5-112 x^6-36 x^7-4 x^8+e^x \left (8 x^2+12 x^3+6 x^4+x^5\right )}{8 x^2+12 x^3+6 x^4+x^5} \, dx=e^x+\frac {1}{x}+4 x^2-4 x^3-x^4-\frac {63}{(2+x)^2}-\frac {x^2}{4 (2+x)^2}+\frac {63}{2+x} \]
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Time = 0.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20
method | result | size |
default | \({\mathrm e}^{x}+\frac {1}{x}+\frac {64}{2+x}-\frac {64}{\left (2+x \right )^{2}}+4 x^{2}-4 x^{3}-x^{4}\) | \(36\) |
parts | \({\mathrm e}^{x}+\frac {1}{x}+\frac {64}{2+x}-\frac {64}{\left (2+x \right )^{2}}+4 x^{2}-4 x^{3}-x^{4}\) | \(36\) |
risch | \(-x^{4}-4 x^{3}+4 x^{2}+\frac {65 x^{2}+68 x +4}{x \left (x^{2}+4 x +4\right )}+{\mathrm e}^{x}\) | \(43\) |
norman | \(\frac {4+x^{2}+4 x +{\mathrm e}^{x} x^{3}-16 x^{5}-8 x^{6}-x^{7}+4 \,{\mathrm e}^{x} x +4 \,{\mathrm e}^{x} x^{2}}{x \left (2+x \right )^{2}}\) | \(51\) |
parallelrisch | \(-\frac {x^{7}+8 x^{6}+16 x^{5}-{\mathrm e}^{x} x^{3}-4-4 \,{\mathrm e}^{x} x^{2}-x^{2}-4 \,{\mathrm e}^{x} x -4 x}{x \left (x^{2}+4 x +4\right )}\) | \(58\) |
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (29) = 58\).
Time = 0.31 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.00 \[ \int \frac {-8-12 x-6 x^2-x^3-128 x^5-112 x^6-36 x^7-4 x^8+e^x \left (8 x^2+12 x^3+6 x^4+x^5\right )}{8 x^2+12 x^3+6 x^4+x^5} \, dx=-\frac {x^{7} + 8 \, x^{6} + 16 \, x^{5} - 16 \, x^{3} - 65 \, x^{2} - {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} e^{x} - 68 \, x - 4}{x^{3} + 4 \, x^{2} + 4 \, x} \]
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Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {-8-12 x-6 x^2-x^3-128 x^5-112 x^6-36 x^7-4 x^8+e^x \left (8 x^2+12 x^3+6 x^4+x^5\right )}{8 x^2+12 x^3+6 x^4+x^5} \, dx=- x^{4} - 4 x^{3} + 4 x^{2} - \frac {- 65 x^{2} - 68 x - 4}{x^{3} + 4 x^{2} + 4 x} + e^{x} \]
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\[ \int \frac {-8-12 x-6 x^2-x^3-128 x^5-112 x^6-36 x^7-4 x^8+e^x \left (8 x^2+12 x^3+6 x^4+x^5\right )}{8 x^2+12 x^3+6 x^4+x^5} \, dx=\int { -\frac {4 \, x^{8} + 36 \, x^{7} + 112 \, x^{6} + 128 \, x^{5} + x^{3} + 6 \, x^{2} - {\left (x^{5} + 6 \, x^{4} + 12 \, x^{3} + 8 \, x^{2}\right )} e^{x} + 12 \, x + 8}{x^{5} + 6 \, x^{4} + 12 \, x^{3} + 8 \, x^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (29) = 58\).
Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.10 \[ \int \frac {-8-12 x-6 x^2-x^3-128 x^5-112 x^6-36 x^7-4 x^8+e^x \left (8 x^2+12 x^3+6 x^4+x^5\right )}{8 x^2+12 x^3+6 x^4+x^5} \, dx=-\frac {x^{7} + 8 \, x^{6} + 16 \, x^{5} - x^{3} e^{x} - 16 \, x^{3} - 4 \, x^{2} e^{x} - 65 \, x^{2} - 4 \, x e^{x} - 68 \, x - 4}{x^{3} + 4 \, x^{2} + 4 \, x} \]
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Time = 0.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-8-12 x-6 x^2-x^3-128 x^5-112 x^6-36 x^7-4 x^8+e^x \left (8 x^2+12 x^3+6 x^4+x^5\right )}{8 x^2+12 x^3+6 x^4+x^5} \, dx={\mathrm {e}}^x+4\,x^2-4\,x^3-x^4+\frac {65\,x^2+68\,x+4}{x\,{\left (x+2\right )}^2} \]
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