Integrand size = 75, antiderivative size = 28 \[ \int \frac {-10 3^{2/5}+6 \sqrt [5]{3} e^{x/5} x^3+1960 x^4+e^{x/5} \left (-840 x^4-84 x^5\right )+e^{2 x/5} \left (90 x^4+18 x^5\right )}{5 x^3} \, dx=\left (-5+3 \left (-3+e^{x/5}\right )+\frac {\sqrt [5]{3}}{x^2}\right )^2 x^2 \]
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Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89, number of steps used = 20, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {12, 14, 2227, 2207, 2225} \[ \int \frac {-10 3^{2/5}+6 \sqrt [5]{3} e^{x/5} x^3+1960 x^4+e^{x/5} \left (-840 x^4-84 x^5\right )+e^{2 x/5} \left (90 x^4+18 x^5\right )}{5 x^3} \, dx=-84 e^{x/5} x^2+9 e^{2 x/5} x^2+196 x^2+\frac {3^{2/5}}{x^2}+6 \sqrt [5]{3} e^{x/5} \]
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Rule 12
Rule 14
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \int \frac {-10 3^{2/5}+6 \sqrt [5]{3} e^{x/5} x^3+1960 x^4+e^{x/5} \left (-840 x^4-84 x^5\right )+e^{2 x/5} \left (90 x^4+18 x^5\right )}{x^3} \, dx \\ & = \frac {1}{5} \int \left (18 e^{2 x/5} x (5+x)+6 e^{x/5} \left (\sqrt [5]{3}-140 x-14 x^2\right )-\frac {10 \left (3^{2/5}-196 x^4\right )}{x^3}\right ) \, dx \\ & = \frac {6}{5} \int e^{x/5} \left (\sqrt [5]{3}-140 x-14 x^2\right ) \, dx-2 \int \frac {3^{2/5}-196 x^4}{x^3} \, dx+\frac {18}{5} \int e^{2 x/5} x (5+x) \, dx \\ & = \frac {6}{5} \int \left (\sqrt [5]{3} e^{x/5}-140 e^{x/5} x-14 e^{x/5} x^2\right ) \, dx-2 \int \left (\frac {3^{2/5}}{x^3}-196 x\right ) \, dx+\frac {18}{5} \int \left (5 e^{2 x/5} x+e^{2 x/5} x^2\right ) \, dx \\ & = \frac {3^{2/5}}{x^2}+196 x^2+\frac {18}{5} \int e^{2 x/5} x^2 \, dx-\frac {84}{5} \int e^{x/5} x^2 \, dx+18 \int e^{2 x/5} x \, dx-168 \int e^{x/5} x \, dx+\frac {1}{5} \left (6 \sqrt [5]{3}\right ) \int e^{x/5} \, dx \\ & = 6 \sqrt [5]{3} e^{x/5}+\frac {3^{2/5}}{x^2}-840 e^{x/5} x+45 e^{2 x/5} x+196 x^2-84 e^{x/5} x^2+9 e^{2 x/5} x^2-18 \int e^{2 x/5} x \, dx-45 \int e^{2 x/5} \, dx+168 \int e^{x/5} x \, dx+840 \int e^{x/5} \, dx \\ & = 4200 e^{x/5}+6 \sqrt [5]{3} e^{x/5}-\frac {225}{2} e^{2 x/5}+\frac {3^{2/5}}{x^2}+196 x^2-84 e^{x/5} x^2+9 e^{2 x/5} x^2+45 \int e^{2 x/5} \, dx-840 \int e^{x/5} \, dx \\ & = 6 \sqrt [5]{3} e^{x/5}+\frac {3^{2/5}}{x^2}+196 x^2-84 e^{x/5} x^2+9 e^{2 x/5} x^2 \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00 \[ \int \frac {-10 3^{2/5}+6 \sqrt [5]{3} e^{x/5} x^3+1960 x^4+e^{x/5} \left (-840 x^4-84 x^5\right )+e^{2 x/5} \left (90 x^4+18 x^5\right )}{5 x^3} \, dx=\frac {2}{5} \left (\frac {5\ 3^{2/5}}{2 x^2}+490 x^2+\frac {45}{2} e^{2 x/5} x^2+15 e^{x/5} \left (\sqrt [5]{3}-14 x^2\right )\right ) \]
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Time = 0.39 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43
method | result | size |
risch | \(196 x^{2}+\frac {3^{\frac {2}{5}}}{x^{2}}+9 x^{2} {\mathrm e}^{\frac {2 x}{5}}+\frac {\left (30 \,3^{\frac {1}{5}}-420 x^{2}\right ) {\mathrm e}^{\frac {x}{5}}}{5}\) | \(40\) |
derivativedivides | \(196 x^{2}+\frac {3^{\frac {2}{5}}}{x^{2}}-84 \,{\mathrm e}^{\frac {x}{5}} x^{2}+9 x^{2} {\mathrm e}^{\frac {2 x}{5}}+6 \,{\mathrm e}^{\frac {x}{5}} 3^{\frac {1}{5}}\) | \(43\) |
default | \(196 x^{2}+\frac {3^{\frac {2}{5}}}{x^{2}}-84 \,{\mathrm e}^{\frac {x}{5}} x^{2}+9 x^{2} {\mathrm e}^{\frac {2 x}{5}}+6 \,{\mathrm e}^{\frac {x}{5}} 3^{\frac {1}{5}}\) | \(43\) |
parts | \(196 x^{2}+\frac {3^{\frac {2}{5}}}{x^{2}}-84 \,{\mathrm e}^{\frac {x}{5}} x^{2}+9 x^{2} {\mathrm e}^{\frac {2 x}{5}}+6 \,{\mathrm e}^{\frac {x}{5}} 3^{\frac {1}{5}}\) | \(43\) |
norman | \(\frac {-84 x^{4} {\mathrm e}^{\frac {x}{5}}+9 x^{4} {\mathrm e}^{\frac {2 x}{5}}+6 x^{2} {\mathrm e}^{\frac {x}{5}} 3^{\frac {1}{5}}+196 x^{4}+3^{\frac {2}{5}}}{x^{2}}\) | \(46\) |
parallelrisch | \(\frac {30 x^{2} {\mathrm e}^{\frac {x}{5}} 3^{\frac {1}{5}}-420 x^{4} {\mathrm e}^{\frac {x}{5}}+45 x^{4} {\mathrm e}^{\frac {2 x}{5}}+5 \,3^{\frac {2}{5}}+980 x^{4}}{5 x^{2}}\) | \(49\) |
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Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {-10 3^{2/5}+6 \sqrt [5]{3} e^{x/5} x^3+1960 x^4+e^{x/5} \left (-840 x^4-84 x^5\right )+e^{2 x/5} \left (90 x^4+18 x^5\right )}{5 x^3} \, dx=\frac {9 \, x^{4} e^{\left (\frac {2}{5} \, x\right )} + 196 \, x^{4} - 6 \, {\left (14 \, x^{4} - 3^{\frac {1}{5}} x^{2}\right )} e^{\left (\frac {1}{5} \, x\right )} + 3^{\frac {2}{5}}}{x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).
Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {-10 3^{2/5}+6 \sqrt [5]{3} e^{x/5} x^3+1960 x^4+e^{x/5} \left (-840 x^4-84 x^5\right )+e^{2 x/5} \left (90 x^4+18 x^5\right )}{5 x^3} \, dx=9 x^{2} e^{\frac {2 x}{5}} + 196 x^{2} + \left (- 84 x^{2} + 6 \cdot \sqrt [5]{3}\right ) e^{\frac {x}{5}} + \frac {3^{\frac {2}{5}}}{x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (21) = 42\).
Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.57 \[ \int \frac {-10 3^{2/5}+6 \sqrt [5]{3} e^{x/5} x^3+1960 x^4+e^{x/5} \left (-840 x^4-84 x^5\right )+e^{2 x/5} \left (90 x^4+18 x^5\right )}{5 x^3} \, dx=196 \, x^{2} + \frac {9}{2} \, {\left (2 \, x^{2} - 10 \, x + 25\right )} e^{\left (\frac {2}{5} \, x\right )} + \frac {45}{2} \, {\left (2 \, x - 5\right )} e^{\left (\frac {2}{5} \, x\right )} - 84 \, {\left (x^{2} - 10 \, x + 50\right )} e^{\left (\frac {1}{5} \, x\right )} - 840 \, {\left (x - 5\right )} e^{\left (\frac {1}{5} \, x\right )} + 6 \cdot 3^{\frac {1}{5}} e^{\left (\frac {1}{5} \, x\right )} + \frac {3^{\frac {2}{5}}}{x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (21) = 42\).
Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {-10 3^{2/5}+6 \sqrt [5]{3} e^{x/5} x^3+1960 x^4+e^{x/5} \left (-840 x^4-84 x^5\right )+e^{2 x/5} \left (90 x^4+18 x^5\right )}{5 x^3} \, dx=\frac {9 \, x^{4} e^{\left (\frac {2}{5} \, x\right )} - 84 \, x^{4} e^{\left (\frac {1}{5} \, x\right )} + 196 \, x^{4} + 6 \cdot 3^{\frac {1}{5}} x^{2} e^{\left (\frac {1}{5} \, x\right )} + 3^{\frac {2}{5}}}{x^{2}} \]
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Time = 0.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {-10 3^{2/5}+6 \sqrt [5]{3} e^{x/5} x^3+1960 x^4+e^{x/5} \left (-840 x^4-84 x^5\right )+e^{2 x/5} \left (90 x^4+18 x^5\right )}{5 x^3} \, dx=\frac {3^{2/5}}{x^2}+x^2\,\left (9\,{\mathrm {e}}^{\frac {2\,x}{5}}-84\,{\mathrm {e}}^{x/5}+196\right )+6\,3^{1/5}\,{\mathrm {e}}^{x/5} \]
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