Integrand size = 53, antiderivative size = 26 \[ \int \frac {-24576+1152 x^2+96 x^3+e^{2 x} \left (8192 x+2560 x^2-2560 x^3+96 x^4+64 x^5\right )}{64+16 x+x^2} \, dx=16 (4-x)^2 \left (3+\frac {2 e^{2 x} x^2}{8+x}\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(57\) vs. \(2(26)=52\).
Time = 0.24 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.19, number of steps used = 18, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {27, 6874, 2230, 2225, 2207, 2208, 2209} \[ \int \frac {-24576+1152 x^2+96 x^3+e^{2 x} \left (8192 x+2560 x^2-2560 x^3+96 x^4+64 x^5\right )}{64+16 x+x^2} \, dx=32 e^{2 x} x^3-512 e^{2 x} x^2+4608 e^{2 x} x-36864 e^{2 x}+48 (4-x)^2+\frac {294912 e^{2 x}}{x+8} \]
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Rule 27
Rule 2207
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-24576+1152 x^2+96 x^3+e^{2 x} \left (8192 x+2560 x^2-2560 x^3+96 x^4+64 x^5\right )}{(8+x)^2} \, dx \\ & = \int \left (96 (-4+x)+\frac {32 e^{2 x} (-4+x) x \left (-64-36 x+11 x^2+2 x^3\right )}{(8+x)^2}\right ) \, dx \\ & = 48 (4-x)^2+32 \int \frac {e^{2 x} (-4+x) x \left (-64-36 x+11 x^2+2 x^3\right )}{(8+x)^2} \, dx \\ & = 48 (4-x)^2+32 \int \left (-2160 e^{2 x}+256 e^{2 x} x-29 e^{2 x} x^2+2 e^{2 x} x^3-\frac {9216 e^{2 x}}{(8+x)^2}+\frac {18432 e^{2 x}}{8+x}\right ) \, dx \\ & = 48 (4-x)^2+64 \int e^{2 x} x^3 \, dx-928 \int e^{2 x} x^2 \, dx+8192 \int e^{2 x} x \, dx-69120 \int e^{2 x} \, dx-294912 \int \frac {e^{2 x}}{(8+x)^2} \, dx+589824 \int \frac {e^{2 x}}{8+x} \, dx \\ & = -34560 e^{2 x}+48 (4-x)^2+4096 e^{2 x} x-464 e^{2 x} x^2+32 e^{2 x} x^3+\frac {294912 e^{2 x}}{8+x}+\frac {589824 \operatorname {ExpIntegralEi}(2 (8+x))}{e^{16}}-96 \int e^{2 x} x^2 \, dx+928 \int e^{2 x} x \, dx-4096 \int e^{2 x} \, dx-589824 \int \frac {e^{2 x}}{8+x} \, dx \\ & = -36608 e^{2 x}+48 (4-x)^2+4560 e^{2 x} x-512 e^{2 x} x^2+32 e^{2 x} x^3+\frac {294912 e^{2 x}}{8+x}+96 \int e^{2 x} x \, dx-464 \int e^{2 x} \, dx \\ & = -36840 e^{2 x}+48 (4-x)^2+4608 e^{2 x} x-512 e^{2 x} x^2+32 e^{2 x} x^3+\frac {294912 e^{2 x}}{8+x}-48 \int e^{2 x} \, dx \\ & = -36864 e^{2 x}+48 (4-x)^2+4608 e^{2 x} x-512 e^{2 x} x^2+32 e^{2 x} x^3+\frac {294912 e^{2 x}}{8+x} \\ \end{align*}
Time = 0.93 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-24576+1152 x^2+96 x^3+e^{2 x} \left (8192 x+2560 x^2-2560 x^3+96 x^4+64 x^5\right )}{64+16 x+x^2} \, dx=\frac {16 x \left (2 e^{2 x} (-4+x)^2 x+3 \left (-64+x^2\right )\right )}{8+x} \]
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Time = 0.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23
method | result | size |
risch | \(48 x^{2}-384 x +\frac {32 x^{2} \left (x^{2}-8 x +16\right ) {\mathrm e}^{2 x}}{x +8}\) | \(32\) |
norman | \(\frac {48 x^{3}+512 \,{\mathrm e}^{2 x} x^{2}-256 \,{\mathrm e}^{2 x} x^{3}+32 \,{\mathrm e}^{2 x} x^{4}+24576}{x +8}\) | \(41\) |
parallelrisch | \(\frac {48 x^{3}+512 \,{\mathrm e}^{2 x} x^{2}-256 \,{\mathrm e}^{2 x} x^{3}+32 \,{\mathrm e}^{2 x} x^{4}+24576}{x +8}\) | \(41\) |
derivativedivides | \(48 x^{2}-384 x +\frac {589824 \,{\mathrm e}^{2 x}}{2 x +16}-36864 \,{\mathrm e}^{2 x}+4608 x \,{\mathrm e}^{2 x}-512 \,{\mathrm e}^{2 x} x^{2}+32 \,{\mathrm e}^{2 x} x^{3}\) | \(54\) |
default | \(48 x^{2}-384 x +\frac {589824 \,{\mathrm e}^{2 x}}{2 x +16}-36864 \,{\mathrm e}^{2 x}+4608 x \,{\mathrm e}^{2 x}-512 \,{\mathrm e}^{2 x} x^{2}+32 \,{\mathrm e}^{2 x} x^{3}\) | \(54\) |
parts | \(48 x^{2}-384 x +\frac {589824 \,{\mathrm e}^{2 x}}{2 x +16}-36864 \,{\mathrm e}^{2 x}+4608 x \,{\mathrm e}^{2 x}-512 \,{\mathrm e}^{2 x} x^{2}+32 \,{\mathrm e}^{2 x} x^{3}\) | \(54\) |
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Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {-24576+1152 x^2+96 x^3+e^{2 x} \left (8192 x+2560 x^2-2560 x^3+96 x^4+64 x^5\right )}{64+16 x+x^2} \, dx=\frac {16 \, {\left (3 \, x^{3} + 2 \, {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )} e^{\left (2 \, x\right )} - 192 \, x\right )}}{x + 8} \]
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Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {-24576+1152 x^2+96 x^3+e^{2 x} \left (8192 x+2560 x^2-2560 x^3+96 x^4+64 x^5\right )}{64+16 x+x^2} \, dx=48 x^{2} - 384 x + \frac {\left (32 x^{4} - 256 x^{3} + 512 x^{2}\right ) e^{2 x}}{x + 8} \]
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Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {-24576+1152 x^2+96 x^3+e^{2 x} \left (8192 x+2560 x^2-2560 x^3+96 x^4+64 x^5\right )}{64+16 x+x^2} \, dx=48 \, x^{2} - 384 \, x + \frac {32 \, {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )} e^{\left (2 \, x\right )}}{x + 8} \]
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Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {-24576+1152 x^2+96 x^3+e^{2 x} \left (8192 x+2560 x^2-2560 x^3+96 x^4+64 x^5\right )}{64+16 x+x^2} \, dx=\frac {16 \, {\left (2 \, x^{4} e^{\left (2 \, x\right )} - 16 \, x^{3} e^{\left (2 \, x\right )} + 3 \, x^{3} + 32 \, x^{2} e^{\left (2 \, x\right )} - 192 \, x\right )}}{x + 8} \]
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Time = 14.84 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54 \[ \int \frac {-24576+1152 x^2+96 x^3+e^{2 x} \left (8192 x+2560 x^2-2560 x^3+96 x^4+64 x^5\right )}{64+16 x+x^2} \, dx=\frac {16\,x\,\left (32\,x\,{\mathrm {e}}^{2\,x}-16\,x^2\,{\mathrm {e}}^{2\,x}+2\,x^3\,{\mathrm {e}}^{2\,x}+3\,x^2-192\right )}{x+8} \]
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