Integrand size = 53, antiderivative size = 22 \[ \int \frac {2960 x+1865 x^2+300 x^3+15 x^4+e^x \left (640 x+480 x^2+90 x^3+5 x^4\right )}{64+16 x+x^2} \, dx=5 x^2 \left (4+e^x+x+\frac {5+2 x}{8+x}\right ) \]
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Time = 0.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32, number of steps used = 20, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.151, Rules used = {27, 6820, 12, 6874, 2227, 2207, 2225, 45} \[ \int \frac {2960 x+1865 x^2+300 x^3+15 x^4+e^x \left (640 x+480 x^2+90 x^3+5 x^4\right )}{64+16 x+x^2} \, dx=5 x^3+5 e^x x^2+30 x^2-55 x-\frac {3520}{x+8} \]
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Rule 12
Rule 27
Rule 45
Rule 2207
Rule 2225
Rule 2227
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {2960 x+1865 x^2+300 x^3+15 x^4+e^x \left (640 x+480 x^2+90 x^3+5 x^4\right )}{(8+x)^2} \, dx \\ & = \int \frac {5 x \left (592+373 x+60 x^2+3 x^3+e^x (2+x) (8+x)^2\right )}{(8+x)^2} \, dx \\ & = 5 \int \frac {x \left (592+373 x+60 x^2+3 x^3+e^x (2+x) (8+x)^2\right )}{(8+x)^2} \, dx \\ & = 5 \int \left (e^x x (2+x)+\frac {592 x}{(8+x)^2}+\frac {373 x^2}{(8+x)^2}+\frac {60 x^3}{(8+x)^2}+\frac {3 x^4}{(8+x)^2}\right ) \, dx \\ & = 5 \int e^x x (2+x) \, dx+15 \int \frac {x^4}{(8+x)^2} \, dx+300 \int \frac {x^3}{(8+x)^2} \, dx+1865 \int \frac {x^2}{(8+x)^2} \, dx+2960 \int \frac {x}{(8+x)^2} \, dx \\ & = 5 \int \left (2 e^x x+e^x x^2\right ) \, dx+15 \int \left (192-16 x+x^2+\frac {4096}{(8+x)^2}-\frac {2048}{8+x}\right ) \, dx+300 \int \left (-16+x-\frac {512}{(8+x)^2}+\frac {192}{8+x}\right ) \, dx+1865 \int \left (1+\frac {64}{(8+x)^2}-\frac {16}{8+x}\right ) \, dx+2960 \int \left (-\frac {8}{(8+x)^2}+\frac {1}{8+x}\right ) \, dx \\ & = -55 x+30 x^2+5 x^3-\frac {3520}{8+x}+5 \int e^x x^2 \, dx+10 \int e^x x \, dx \\ & = -55 x+10 e^x x+30 x^2+5 e^x x^2+5 x^3-\frac {3520}{8+x}-10 \int e^x \, dx-10 \int e^x x \, dx \\ & = -10 e^x-55 x+30 x^2+5 e^x x^2+5 x^3-\frac {3520}{8+x}+10 \int e^x \, dx \\ & = -55 x+30 x^2+5 e^x x^2+5 x^3-\frac {3520}{8+x} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {2960 x+1865 x^2+300 x^3+15 x^4+e^x \left (640 x+480 x^2+90 x^3+5 x^4\right )}{64+16 x+x^2} \, dx=5 \left (-11 x+\left (6+e^x\right ) x^2+x^3-\frac {704}{8+x}\right ) \]
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Time = 0.33 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32
method | result | size |
default | \(-\frac {3520}{x +8}-55 x +30 x^{2}+5 x^{3}+5 \,{\mathrm e}^{x} x^{2}\) | \(29\) |
risch | \(-\frac {3520}{x +8}-55 x +30 x^{2}+5 x^{3}+5 \,{\mathrm e}^{x} x^{2}\) | \(29\) |
parts | \(-\frac {3520}{x +8}-55 x +30 x^{2}+5 x^{3}+5 \,{\mathrm e}^{x} x^{2}\) | \(29\) |
norman | \(\frac {185 x^{2}+70 x^{3}+5 x^{4}+40 \,{\mathrm e}^{x} x^{2}+5 \,{\mathrm e}^{x} x^{3}}{x +8}\) | \(37\) |
parallelrisch | \(\frac {185 x^{2}+70 x^{3}+5 x^{4}+40 \,{\mathrm e}^{x} x^{2}+5 \,{\mathrm e}^{x} x^{3}}{x +8}\) | \(37\) |
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Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {2960 x+1865 x^2+300 x^3+15 x^4+e^x \left (640 x+480 x^2+90 x^3+5 x^4\right )}{64+16 x+x^2} \, dx=\frac {5 \, {\left (x^{4} + 14 \, x^{3} + 37 \, x^{2} + {\left (x^{3} + 8 \, x^{2}\right )} e^{x} - 88 \, x - 704\right )}}{x + 8} \]
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Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {2960 x+1865 x^2+300 x^3+15 x^4+e^x \left (640 x+480 x^2+90 x^3+5 x^4\right )}{64+16 x+x^2} \, dx=5 x^{3} + 5 x^{2} e^{x} + 30 x^{2} - 55 x - \frac {3520}{x + 8} \]
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Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {2960 x+1865 x^2+300 x^3+15 x^4+e^x \left (640 x+480 x^2+90 x^3+5 x^4\right )}{64+16 x+x^2} \, dx=5 \, x^{3} + 5 \, x^{2} e^{x} + 30 \, x^{2} - 55 \, x - \frac {3520}{x + 8} \]
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Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.73 \[ \int \frac {2960 x+1865 x^2+300 x^3+15 x^4+e^x \left (640 x+480 x^2+90 x^3+5 x^4\right )}{64+16 x+x^2} \, dx=\frac {5 \, {\left (x^{4} + x^{3} e^{x} + 14 \, x^{3} + 8 \, x^{2} e^{x} + 37 \, x^{2} - 88 \, x - 704\right )}}{x + 8} \]
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Time = 10.53 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {2960 x+1865 x^2+300 x^3+15 x^4+e^x \left (640 x+480 x^2+90 x^3+5 x^4\right )}{64+16 x+x^2} \, dx=x^2\,\left (5\,{\mathrm {e}}^x+30\right )-\frac {3520}{x+8}-55\,x+5\,x^3 \]
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