Optimal. Leaf size=29 \[ \frac {\left (5-\frac {7 x}{2}\right ) \left (e^{x/4}+x\right ) \left (\frac {2}{x}+x\right )^2}{x} \]
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Rubi [B] time = 0.29, antiderivative size = 95, normalized size of antiderivative = 3.28, number of steps used = 23, number of rules used = 7, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.101, Rules used = {12, 14, 2199, 2194, 2177, 2178, 2176} \begin {gather*} -\frac {7 x^3}{2}+\frac {20 e^{x/4}}{x^3}-\frac {7}{2} e^{x/4} x^2+5 x^2-\frac {14 e^{x/4}}{x^2}+\frac {20}{x^2}+5 e^{x/4} x-14 x-14 e^{x/4}+\frac {20 e^{x/4}}{x}-\frac {14}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2176
Rule 2177
Rule 2178
Rule 2194
Rule 2199
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{8} \int \frac {-320 x+112 x^2-112 x^4+80 x^5-84 x^6+e^{x/4} \left (-480+264 x-188 x^2+40 x^3+12 x^4-46 x^5-7 x^6\right )}{x^4} \, dx\\ &=\frac {1}{8} \int \left (-\frac {e^{x/4} \left (2+x^2\right ) \left (240-132 x-26 x^2+46 x^3+7 x^4\right )}{x^4}-\frac {4 \left (80-28 x+28 x^3-20 x^4+21 x^5\right )}{x^3}\right ) \, dx\\ &=-\left (\frac {1}{8} \int \frac {e^{x/4} \left (2+x^2\right ) \left (240-132 x-26 x^2+46 x^3+7 x^4\right )}{x^4} \, dx\right )-\frac {1}{2} \int \frac {80-28 x+28 x^3-20 x^4+21 x^5}{x^3} \, dx\\ &=-\left (\frac {1}{8} \int \left (-12 e^{x/4}+\frac {480 e^{x/4}}{x^4}-\frac {264 e^{x/4}}{x^3}+\frac {188 e^{x/4}}{x^2}-\frac {40 e^{x/4}}{x}+46 e^{x/4} x+7 e^{x/4} x^2\right ) \, dx\right )-\frac {1}{2} \int \left (28+\frac {80}{x^3}-\frac {28}{x^2}-20 x+21 x^2\right ) \, dx\\ &=\frac {20}{x^2}-\frac {14}{x}-14 x+5 x^2-\frac {7 x^3}{2}-\frac {7}{8} \int e^{x/4} x^2 \, dx+\frac {3}{2} \int e^{x/4} \, dx+5 \int \frac {e^{x/4}}{x} \, dx-\frac {23}{4} \int e^{x/4} x \, dx-\frac {47}{2} \int \frac {e^{x/4}}{x^2} \, dx+33 \int \frac {e^{x/4}}{x^3} \, dx-60 \int \frac {e^{x/4}}{x^4} \, dx\\ &=6 e^{x/4}+\frac {20 e^{x/4}}{x^3}+\frac {20}{x^2}-\frac {33 e^{x/4}}{2 x^2}-\frac {14}{x}+\frac {47 e^{x/4}}{2 x}-14 x-23 e^{x/4} x+5 x^2-\frac {7}{2} e^{x/4} x^2-\frac {7 x^3}{2}+5 \text {Ei}\left (\frac {x}{4}\right )+\frac {33}{8} \int \frac {e^{x/4}}{x^2} \, dx-5 \int \frac {e^{x/4}}{x^3} \, dx-\frac {47}{8} \int \frac {e^{x/4}}{x} \, dx+7 \int e^{x/4} x \, dx+23 \int e^{x/4} \, dx\\ &=98 e^{x/4}+\frac {20 e^{x/4}}{x^3}+\frac {20}{x^2}-\frac {14 e^{x/4}}{x^2}-\frac {14}{x}+\frac {155 e^{x/4}}{8 x}-14 x+5 e^{x/4} x+5 x^2-\frac {7}{2} e^{x/4} x^2-\frac {7 x^3}{2}-\frac {7 \text {Ei}\left (\frac {x}{4}\right )}{8}-\frac {5}{8} \int \frac {e^{x/4}}{x^2} \, dx+\frac {33}{32} \int \frac {e^{x/4}}{x} \, dx-28 \int e^{x/4} \, dx\\ &=-14 e^{x/4}+\frac {20 e^{x/4}}{x^3}+\frac {20}{x^2}-\frac {14 e^{x/4}}{x^2}-\frac {14}{x}+\frac {20 e^{x/4}}{x}-14 x+5 e^{x/4} x+5 x^2-\frac {7}{2} e^{x/4} x^2-\frac {7 x^3}{2}+\frac {5 \text {Ei}\left (\frac {x}{4}\right )}{32}-\frac {5}{32} \int \frac {e^{x/4}}{x} \, dx\\ &=-14 e^{x/4}+\frac {20 e^{x/4}}{x^3}+\frac {20}{x^2}-\frac {14 e^{x/4}}{x^2}-\frac {14}{x}+\frac {20 e^{x/4}}{x}-14 x+5 e^{x/4} x+5 x^2-\frac {7}{2} e^{x/4} x^2-\frac {7 x^3}{2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.19, size = 50, normalized size = 1.72 \begin {gather*} -\frac {e^{x/4} (-10+7 x) \left (2+x^2\right )^2+x \left (-40+28 x+28 x^3-10 x^4+7 x^5\right )}{2 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 59, normalized size = 2.03 \begin {gather*} -\frac {7 \, x^{6} - 10 \, x^{5} + 28 \, x^{4} + 28 \, x^{2} + {\left (7 \, x^{5} - 10 \, x^{4} + 28 \, x^{3} - 40 \, x^{2} + 28 \, x - 40\right )} e^{\left (\frac {1}{4} \, x\right )} - 40 \, x}{2 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 78, normalized size = 2.69 \begin {gather*} -\frac {7 \, x^{6} + 7 \, x^{5} e^{\left (\frac {1}{4} \, x\right )} - 10 \, x^{5} - 10 \, x^{4} e^{\left (\frac {1}{4} \, x\right )} + 28 \, x^{4} + 28 \, x^{3} e^{\left (\frac {1}{4} \, x\right )} - 40 \, x^{2} e^{\left (\frac {1}{4} \, x\right )} + 28 \, x^{2} + 28 \, x e^{\left (\frac {1}{4} \, x\right )} - 40 \, x - 40 \, e^{\left (\frac {1}{4} \, x\right )}}{2 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 59, normalized size = 2.03
method | result | size |
risch | \(-\frac {7 x^{3}}{2}+5 x^{2}-14 x +\frac {-112 x +160}{8 x^{2}}-\frac {\left (7 x^{5}-10 x^{4}+28 x^{3}-40 x^{2}+28 x -40\right ) {\mathrm e}^{\frac {x}{4}}}{2 x^{3}}\) | \(59\) |
derivativedivides | \(5 x^{2}-14 x +\frac {20}{x^{2}}-\frac {14}{x}-\frac {7 x^{3}}{2}+\frac {20 \,{\mathrm e}^{\frac {x}{4}}}{x^{3}}-\frac {14 \,{\mathrm e}^{\frac {x}{4}}}{x^{2}}+\frac {20 \,{\mathrm e}^{\frac {x}{4}}}{x}+5 x \,{\mathrm e}^{\frac {x}{4}}-14 \,{\mathrm e}^{\frac {x}{4}}-\frac {7 x^{2} {\mathrm e}^{\frac {x}{4}}}{2}\) | \(74\) |
default | \(5 x^{2}-14 x +\frac {20}{x^{2}}-\frac {14}{x}-\frac {7 x^{3}}{2}+\frac {20 \,{\mathrm e}^{\frac {x}{4}}}{x^{3}}-\frac {14 \,{\mathrm e}^{\frac {x}{4}}}{x^{2}}+\frac {20 \,{\mathrm e}^{\frac {x}{4}}}{x}+5 x \,{\mathrm e}^{\frac {x}{4}}-14 \,{\mathrm e}^{\frac {x}{4}}-\frac {7 x^{2} {\mathrm e}^{\frac {x}{4}}}{2}\) | \(74\) |
norman | \(\frac {20 x -14 x^{2}-14 x^{4}+5 x^{5}-\frac {7 x^{6}}{2}-14 x \,{\mathrm e}^{\frac {x}{4}}+20 x^{2} {\mathrm e}^{\frac {x}{4}}-14 \,{\mathrm e}^{\frac {x}{4}} x^{3}+5 \,{\mathrm e}^{\frac {x}{4}} x^{4}-\frac {7 \,{\mathrm e}^{\frac {x}{4}} x^{5}}{2}+20 \,{\mathrm e}^{\frac {x}{4}}}{x^{3}}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.63, size = 80, normalized size = 2.76 \begin {gather*} -\frac {7}{2} \, x^{3} + 5 \, x^{2} - \frac {7}{2} \, {\left (x^{2} - 8 \, x + 32\right )} e^{\left (\frac {1}{4} \, x\right )} - 23 \, {\left (x - 4\right )} e^{\left (\frac {1}{4} \, x\right )} - 14 \, x - \frac {14}{x} + \frac {20}{x^{2}} + 5 \, {\rm Ei}\left (\frac {1}{4} \, x\right ) + 6 \, e^{\left (\frac {1}{4} \, x\right )} - \frac {47}{8} \, \Gamma \left (-1, -\frac {1}{4} \, x\right ) - \frac {33}{16} \, \Gamma \left (-2, -\frac {1}{4} \, x\right ) - \frac {15}{16} \, \Gamma \left (-3, -\frac {1}{4} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 69, normalized size = 2.38 \begin {gather*} \frac {20\,{\mathrm {e}}^{x/4}+x^2\,\left (20\,{\mathrm {e}}^{x/4}-14\right )-x\,\left (14\,{\mathrm {e}}^{x/4}-20\right )}{x^3}-x^2\,\left (\frac {7\,{\mathrm {e}}^{x/4}}{2}-5\right )-14\,{\mathrm {e}}^{x/4}+x\,\left (5\,{\mathrm {e}}^{x/4}-14\right )-\frac {7\,x^3}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.15, size = 60, normalized size = 2.07 \begin {gather*} - \frac {7 x^{3}}{2} + 5 x^{2} - 14 x - \frac {28 x - 40}{2 x^{2}} + \frac {\left (- 7 x^{5} + 10 x^{4} - 28 x^{3} + 40 x^{2} - 28 x + 40\right ) e^{\frac {x}{4}}}{2 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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