Integrand size = 62, antiderivative size = 34 \[ \int \frac {-9 x^2-18 x^3-6 x^5-12 x^6-6 x^7+e^{12+x} \left (9+15 x+3 x^2-3 x^3\right )}{x^4+2 x^5+x^6} \, dx=\frac {3 \left (-\frac {e^{12+x}}{x^2}+x \left (-x^2+\frac {3}{x (1+x)}\right )\right )}{x} \]
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Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82, number of steps used = 15, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {1608, 27, 6874, 2228, 46, 45} \[ \int \frac {-9 x^2-18 x^3-6 x^5-12 x^6-6 x^7+e^{12+x} \left (9+15 x+3 x^2-3 x^3\right )}{x^4+2 x^5+x^6} \, dx=-\frac {3 e^{x+12}}{x^3}-3 x^2-\frac {9}{x+1}+\frac {9}{x} \]
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Rule 27
Rule 45
Rule 46
Rule 1608
Rule 2228
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-9 x^2-18 x^3-6 x^5-12 x^6-6 x^7+e^{12+x} \left (9+15 x+3 x^2-3 x^3\right )}{x^4 \left (1+2 x+x^2\right )} \, dx \\ & = \int \frac {-9 x^2-18 x^3-6 x^5-12 x^6-6 x^7+e^{12+x} \left (9+15 x+3 x^2-3 x^3\right )}{x^4 (1+x)^2} \, dx \\ & = \int \left (-\frac {3 e^{12+x} (-3+x)}{x^4}-\frac {9}{x^2 (1+x)^2}-\frac {18}{x (1+x)^2}-\frac {6 x}{(1+x)^2}-\frac {12 x^2}{(1+x)^2}-\frac {6 x^3}{(1+x)^2}\right ) \, dx \\ & = -\left (3 \int \frac {e^{12+x} (-3+x)}{x^4} \, dx\right )-6 \int \frac {x}{(1+x)^2} \, dx-6 \int \frac {x^3}{(1+x)^2} \, dx-9 \int \frac {1}{x^2 (1+x)^2} \, dx-12 \int \frac {x^2}{(1+x)^2} \, dx-18 \int \frac {1}{x (1+x)^2} \, dx \\ & = -\frac {3 e^{12+x}}{x^3}-6 \int \left (-\frac {1}{(1+x)^2}+\frac {1}{1+x}\right ) \, dx-6 \int \left (-2+x-\frac {1}{(1+x)^2}+\frac {3}{1+x}\right ) \, dx-9 \int \left (\frac {1}{x^2}-\frac {2}{x}+\frac {1}{(1+x)^2}+\frac {2}{1+x}\right ) \, dx-12 \int \left (1+\frac {1}{(1+x)^2}-\frac {2}{1+x}\right ) \, dx-18 \int \left (\frac {1}{-1-x}+\frac {1}{x}-\frac {1}{(1+x)^2}\right ) \, dx \\ & = -\frac {3 e^{12+x}}{x^3}+\frac {9}{x}-3 x^2-\frac {9}{1+x} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int \frac {-9 x^2-18 x^3-6 x^5-12 x^6-6 x^7+e^{12+x} \left (9+15 x+3 x^2-3 x^3\right )}{x^4+2 x^5+x^6} \, dx=-\frac {3 e^{12+x}}{x^3}+\frac {9}{x}-3 x^2-\frac {9}{1+x} \]
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Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76
method | result | size |
risch | \(-3 x^{2}+\frac {9}{x \left (1+x \right )}-\frac {3 \,{\mathrm e}^{x +12}}{x^{3}}\) | \(26\) |
parts | \(\frac {9}{x}-\frac {9}{1+x}-3 x^{2}-\frac {3 \,{\mathrm e}^{12} {\mathrm e}^{x}}{x^{3}}\) | \(28\) |
norman | \(\frac {9 x^{2}-3 x^{5}-3 x^{6}-3 \,{\mathrm e}^{12} {\mathrm e}^{x}-3 x \,{\mathrm e}^{12} {\mathrm e}^{x}}{x^{3} \left (1+x \right )}\) | \(39\) |
parallelrisch | \(-\frac {3 x^{6}+3 x^{5}+3 x \,{\mathrm e}^{12} {\mathrm e}^{x}+3 \,{\mathrm e}^{12} {\mathrm e}^{x}-9 x^{2}}{x^{3} \left (1+x \right )}\) | \(40\) |
default | \(\frac {9}{x}-\frac {9}{1+x}-3 x^{2}+9 \,{\mathrm e}^{12} \left (\frac {5 \,{\mathrm e}^{x}}{6 x^{2}}-\frac {13 \,{\mathrm e}^{x}}{6 x}+\frac {11 \,\operatorname {Ei}_{1}\left (-x \right )}{6}-\frac {{\mathrm e}^{x}}{3 x^{3}}-\frac {{\mathrm e}^{x}}{1+x}-5 \,{\mathrm e}^{-1} \operatorname {Ei}_{1}\left (-1-x \right )\right )+15 \,{\mathrm e}^{12} \left (-\frac {{\mathrm e}^{x}}{2 x^{2}}+\frac {3 \,{\mathrm e}^{x}}{2 x}-\frac {3 \,\operatorname {Ei}_{1}\left (-x \right )}{2}+\frac {{\mathrm e}^{x}}{1+x}+4 \,{\mathrm e}^{-1} \operatorname {Ei}_{1}\left (-1-x \right )\right )+3 \,{\mathrm e}^{12} \left (\operatorname {Ei}_{1}\left (-x \right )-\frac {{\mathrm e}^{x}}{1+x}-3 \,{\mathrm e}^{-1} \operatorname {Ei}_{1}\left (-1-x \right )-\frac {{\mathrm e}^{x}}{x}\right )-3 \,{\mathrm e}^{12} \left (-\operatorname {Ei}_{1}\left (-x \right )+\frac {{\mathrm e}^{x}}{1+x}+2 \,{\mathrm e}^{-1} \operatorname {Ei}_{1}\left (-1-x \right )\right )\) | \(185\) |
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Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {-9 x^2-18 x^3-6 x^5-12 x^6-6 x^7+e^{12+x} \left (9+15 x+3 x^2-3 x^3\right )}{x^4+2 x^5+x^6} \, dx=-\frac {3 \, {\left (x^{6} + x^{5} - 3 \, x^{2} + {\left (x + 1\right )} e^{\left (x + 12\right )}\right )}}{x^{4} + x^{3}} \]
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Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.65 \[ \int \frac {-9 x^2-18 x^3-6 x^5-12 x^6-6 x^7+e^{12+x} \left (9+15 x+3 x^2-3 x^3\right )}{x^4+2 x^5+x^6} \, dx=- 3 x^{2} + \frac {9}{x^{2} + x} - \frac {3 e^{12} e^{x}}{x^{3}} \]
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Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {-9 x^2-18 x^3-6 x^5-12 x^6-6 x^7+e^{12+x} \left (9+15 x+3 x^2-3 x^3\right )}{x^4+2 x^5+x^6} \, dx=-3 \, x^{2} + \frac {9 \, {\left (2 \, x + 1\right )}}{x^{2} + x} - \frac {18}{x + 1} - \frac {3 \, e^{\left (x + 12\right )}}{x^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \frac {-9 x^2-18 x^3-6 x^5-12 x^6-6 x^7+e^{12+x} \left (9+15 x+3 x^2-3 x^3\right )}{x^4+2 x^5+x^6} \, dx=-\frac {3 \, {\left (x^{6} + x^{5} - 3 \, x^{2} + x e^{\left (x + 12\right )} + e^{\left (x + 12\right )}\right )}}{x^{4} + x^{3}} \]
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Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int \frac {-9 x^2-18 x^3-6 x^5-12 x^6-6 x^7+e^{12+x} \left (9+15 x+3 x^2-3 x^3\right )}{x^4+2 x^5+x^6} \, dx=-3\,x^2-\frac {3\,{\mathrm {e}}^{x+12}+3\,x\,{\mathrm {e}}^{x+12}-9\,x^2}{x^3\,\left (x+1\right )} \]
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