Integrand size = 70, antiderivative size = 24 \[ \int \frac {1600 x^4+1250 x^5+300 x^6+25 x^7+e^{\frac {6}{25 x^3}} \left (-1152-864 x-216 x^2-18 x^3\right )}{1600 x^4+1200 x^5+300 x^6+25 x^7} \, dx=1+e^{\frac {6}{25 x^3}}+x+\frac {x^2}{4 (4+x)^2} \]
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Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {6820, 2240, 1864} \[ \int \frac {1600 x^4+1250 x^5+300 x^6+25 x^7+e^{\frac {6}{25 x^3}} \left (-1152-864 x-216 x^2-18 x^3\right )}{1600 x^4+1200 x^5+300 x^6+25 x^7} \, dx=e^{\frac {6}{25 x^3}}+x-\frac {2}{x+4}+\frac {4}{(x+4)^2} \]
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Rule 1864
Rule 2240
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {18 e^{\frac {6}{25 x^3}}}{25 x^4}+\frac {64+50 x+12 x^2+x^3}{(4+x)^3}\right ) \, dx \\ & = -\left (\frac {18}{25} \int \frac {e^{\frac {6}{25 x^3}}}{x^4} \, dx\right )+\int \frac {64+50 x+12 x^2+x^3}{(4+x)^3} \, dx \\ & = e^{\frac {6}{25 x^3}}+\int \left (1-\frac {8}{(4+x)^3}+\frac {2}{(4+x)^2}\right ) \, dx \\ & = e^{\frac {6}{25 x^3}}+x+\frac {4}{(4+x)^2}-\frac {2}{4+x} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {1600 x^4+1250 x^5+300 x^6+25 x^7+e^{\frac {6}{25 x^3}} \left (-1152-864 x-216 x^2-18 x^3\right )}{1600 x^4+1200 x^5+300 x^6+25 x^7} \, dx=e^{\frac {6}{25 x^3}}+x+\frac {4}{(4+x)^2}-\frac {2}{4+x} \]
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Time = 1.63 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96
method | result | size |
parts | \(x -\frac {2}{4+x}+\frac {4}{\left (4+x \right )^{2}}+{\mathrm e}^{\frac {6}{25 x^{3}}}\) | \(23\) |
risch | \(x +\frac {-2 x -4}{x^{2}+8 x +16}+{\mathrm e}^{\frac {6}{25 x^{3}}}\) | \(25\) |
parallelrisch | \(\frac {200 x^{3}+200 \,{\mathrm e}^{\frac {6}{25 x^{3}}} x^{2}-6400+1250 x^{2}+1600 \,{\mathrm e}^{\frac {6}{25 x^{3}}} x +3200 \,{\mathrm e}^{\frac {6}{25 x^{3}}}}{200 x^{2}+1600 x +3200}\) | \(53\) |
norman | \(\frac {x^{6}-132 x^{3}-50 x^{4}+{\mathrm e}^{\frac {6}{25 x^{3}}} x^{5}+16 \,{\mathrm e}^{\frac {6}{25 x^{3}}} x^{3}+8 \,{\mathrm e}^{\frac {6}{25 x^{3}}} x^{4}}{x^{3} \left (4+x \right )^{2}}\) | \(56\) |
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (19) = 38\).
Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {1600 x^4+1250 x^5+300 x^6+25 x^7+e^{\frac {6}{25 x^3}} \left (-1152-864 x-216 x^2-18 x^3\right )}{1600 x^4+1200 x^5+300 x^6+25 x^7} \, dx=\frac {x^{3} + 8 \, x^{2} + {\left (x^{2} + 8 \, x + 16\right )} e^{\left (\frac {6}{25 \, x^{3}}\right )} + 14 \, x - 4}{x^{2} + 8 \, x + 16} \]
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Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1600 x^4+1250 x^5+300 x^6+25 x^7+e^{\frac {6}{25 x^3}} \left (-1152-864 x-216 x^2-18 x^3\right )}{1600 x^4+1200 x^5+300 x^6+25 x^7} \, dx=x + \frac {- 2 x - 4}{x^{2} + 8 x + 16} + e^{\frac {6}{25 x^{3}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (19) = 38\).
Time = 0.23 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.79 \[ \int \frac {1600 x^4+1250 x^5+300 x^6+25 x^7+e^{\frac {6}{25 x^3}} \left (-1152-864 x-216 x^2-18 x^3\right )}{1600 x^4+1200 x^5+300 x^6+25 x^7} \, dx=x - \frac {16 \, {\left (3 \, x + 10\right )}}{x^{2} + 8 \, x + 16} + \frac {96 \, {\left (x + 3\right )}}{x^{2} + 8 \, x + 16} - \frac {50 \, {\left (x + 2\right )}}{x^{2} + 8 \, x + 16} - \frac {32}{x^{2} + 8 \, x + 16} + e^{\left (\frac {6}{25 \, x^{3}}\right )} \]
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none
Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {1600 x^4+1250 x^5+300 x^6+25 x^7+e^{\frac {6}{25 x^3}} \left (-1152-864 x-216 x^2-18 x^3\right )}{1600 x^4+1200 x^5+300 x^6+25 x^7} \, dx=x - \frac {2 \, {\left (x + 2\right )}}{x^{2} + 8 \, x + 16} + e^{\left (\frac {6}{25 \, x^{3}}\right )} \]
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Time = 9.85 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {1600 x^4+1250 x^5+300 x^6+25 x^7+e^{\frac {6}{25 x^3}} \left (-1152-864 x-216 x^2-18 x^3\right )}{1600 x^4+1200 x^5+300 x^6+25 x^7} \, dx=x+{\mathrm {e}}^{\frac {6}{25\,x^3}}-\frac {2\,x+4}{{\left (x+4\right )}^2} \]
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