Integrand size = 150, antiderivative size = 28 \[ \int \frac {-75-30 e^{10+x}-10 e^{10+2 x}-90 x-15 x^2}{81 e^{20}+12 e^{20+3 x}+e^{20+4 x}+225 x^2+270 x^3+111 x^4+18 x^5+x^6+e^{10} \left (270 x+162 x^2+18 x^3\right )+e^{2 x} \left (54 e^{20}+e^{10} \left (30 x+18 x^2+2 x^3\right )\right )+e^x \left (108 e^{20}+e^{10} \left (180 x+108 x^2+12 x^3\right )\right )} \, dx=\frac {5}{e^{10} \left (3+e^x\right )^2-x+x \left (x+(4+x)^2\right )} \]
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Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {6820, 12, 6818} \[ \int \frac {-75-30 e^{10+x}-10 e^{10+2 x}-90 x-15 x^2}{81 e^{20}+12 e^{20+3 x}+e^{20+4 x}+225 x^2+270 x^3+111 x^4+18 x^5+x^6+e^{10} \left (270 x+162 x^2+18 x^3\right )+e^{2 x} \left (54 e^{20}+e^{10} \left (30 x+18 x^2+2 x^3\right )\right )+e^x \left (108 e^{20}+e^{10} \left (180 x+108 x^2+12 x^3\right )\right )} \, dx=\frac {5}{x \left (x^2+9 x+15\right )+e^{2 (x+5)}+6 e^{x+10}+9 e^{10}} \]
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Rule 12
Rule 6818
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {5 \left (-2 e^{2 (5+x)}-6 e^{10+x}-3 \left (5+6 x+x^2\right )\right )}{\left (9 e^{10}+e^{2 (5+x)}+6 e^{10+x}+x \left (15+9 x+x^2\right )\right )^2} \, dx \\ & = 5 \int \frac {-2 e^{2 (5+x)}-6 e^{10+x}-3 \left (5+6 x+x^2\right )}{\left (9 e^{10}+e^{2 (5+x)}+6 e^{10+x}+x \left (15+9 x+x^2\right )\right )^2} \, dx \\ & = \frac {5}{9 e^{10}+e^{2 (5+x)}+6 e^{10+x}+x \left (15+9 x+x^2\right )} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {-75-30 e^{10+x}-10 e^{10+2 x}-90 x-15 x^2}{81 e^{20}+12 e^{20+3 x}+e^{20+4 x}+225 x^2+270 x^3+111 x^4+18 x^5+x^6+e^{10} \left (270 x+162 x^2+18 x^3\right )+e^{2 x} \left (54 e^{20}+e^{10} \left (30 x+18 x^2+2 x^3\right )\right )+e^x \left (108 e^{20}+e^{10} \left (180 x+108 x^2+12 x^3\right )\right )} \, dx=\frac {5}{9 e^{10}+e^{2 (5+x)}+6 e^{10+x}+x \left (15+9 x+x^2\right )} \]
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Time = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18
method | result | size |
risch | \(\frac {5}{{\mathrm e}^{2 x +10}+6 \,{\mathrm e}^{x +10}+x^{3}+9 \,{\mathrm e}^{10}+9 x^{2}+15 x}\) | \(33\) |
parallelrisch | \(\frac {5}{{\mathrm e}^{10} {\mathrm e}^{2 x}+6 \,{\mathrm e}^{10} {\mathrm e}^{x}+x^{3}+9 \,{\mathrm e}^{10}+9 x^{2}+15 x}\) | \(40\) |
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Time = 0.24 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {-75-30 e^{10+x}-10 e^{10+2 x}-90 x-15 x^2}{81 e^{20}+12 e^{20+3 x}+e^{20+4 x}+225 x^2+270 x^3+111 x^4+18 x^5+x^6+e^{10} \left (270 x+162 x^2+18 x^3\right )+e^{2 x} \left (54 e^{20}+e^{10} \left (30 x+18 x^2+2 x^3\right )\right )+e^x \left (108 e^{20}+e^{10} \left (180 x+108 x^2+12 x^3\right )\right )} \, dx=\frac {5 \, e^{10}}{{\left (x^{3} + 9 \, x^{2} + 15 \, x\right )} e^{10} + 9 \, e^{20} + e^{\left (2 \, x + 20\right )} + 6 \, e^{\left (x + 20\right )}} \]
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Time = 0.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {-75-30 e^{10+x}-10 e^{10+2 x}-90 x-15 x^2}{81 e^{20}+12 e^{20+3 x}+e^{20+4 x}+225 x^2+270 x^3+111 x^4+18 x^5+x^6+e^{10} \left (270 x+162 x^2+18 x^3\right )+e^{2 x} \left (54 e^{20}+e^{10} \left (30 x+18 x^2+2 x^3\right )\right )+e^x \left (108 e^{20}+e^{10} \left (180 x+108 x^2+12 x^3\right )\right )} \, dx=\frac {5}{x^{3} + 9 x^{2} + 15 x + e^{10} e^{2 x} + 6 e^{10} e^{x} + 9 e^{10}} \]
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Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {-75-30 e^{10+x}-10 e^{10+2 x}-90 x-15 x^2}{81 e^{20}+12 e^{20+3 x}+e^{20+4 x}+225 x^2+270 x^3+111 x^4+18 x^5+x^6+e^{10} \left (270 x+162 x^2+18 x^3\right )+e^{2 x} \left (54 e^{20}+e^{10} \left (30 x+18 x^2+2 x^3\right )\right )+e^x \left (108 e^{20}+e^{10} \left (180 x+108 x^2+12 x^3\right )\right )} \, dx=\frac {5}{x^{3} + 9 \, x^{2} + 15 \, x + 9 \, e^{10} + e^{\left (2 \, x + 10\right )} + 6 \, e^{\left (x + 10\right )}} \]
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Time = 0.58 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {-75-30 e^{10+x}-10 e^{10+2 x}-90 x-15 x^2}{81 e^{20}+12 e^{20+3 x}+e^{20+4 x}+225 x^2+270 x^3+111 x^4+18 x^5+x^6+e^{10} \left (270 x+162 x^2+18 x^3\right )+e^{2 x} \left (54 e^{20}+e^{10} \left (30 x+18 x^2+2 x^3\right )\right )+e^x \left (108 e^{20}+e^{10} \left (180 x+108 x^2+12 x^3\right )\right )} \, dx=\frac {10}{x^{3} + 9 \, x^{2} + 15 \, x + 9 \, e^{10} + e^{\left (2 \, x + 10\right )} + 6 \, e^{\left (x + 10\right )}} \]
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Timed out. \[ \int \frac {-75-30 e^{10+x}-10 e^{10+2 x}-90 x-15 x^2}{81 e^{20}+12 e^{20+3 x}+e^{20+4 x}+225 x^2+270 x^3+111 x^4+18 x^5+x^6+e^{10} \left (270 x+162 x^2+18 x^3\right )+e^{2 x} \left (54 e^{20}+e^{10} \left (30 x+18 x^2+2 x^3\right )\right )+e^x \left (108 e^{20}+e^{10} \left (180 x+108 x^2+12 x^3\right )\right )} \, dx=\int -\frac {90\,x+10\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{10}+30\,{\mathrm {e}}^{10}\,{\mathrm {e}}^x+15\,x^2+75}{81\,{\mathrm {e}}^{20}+12\,{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^{20}+{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{20}+{\mathrm {e}}^{2\,x}\,\left (54\,{\mathrm {e}}^{20}+{\mathrm {e}}^{10}\,\left (2\,x^3+18\,x^2+30\,x\right )\right )+{\mathrm {e}}^{10}\,\left (18\,x^3+162\,x^2+270\,x\right )+{\mathrm {e}}^x\,\left (108\,{\mathrm {e}}^{20}+{\mathrm {e}}^{10}\,\left (12\,x^3+108\,x^2+180\,x\right )\right )+225\,x^2+270\,x^3+111\,x^4+18\,x^5+x^6} \,d x \]
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