Optimal. Leaf size=37 \[ 3-\frac {x^4}{4+x+\frac {x}{-e^{e^4+\frac {3}{2 x}}+5 x^2}} \]
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Rubi [F] time = 6.22, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-50 x^6-800 x^7-150 x^8+e^{\frac {3+2 e^4 x}{x}} \left (-32 x^3-6 x^4\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-3 x^3+6 x^4+320 x^5+60 x^6\right )}{2 x^2+80 x^3+820 x^4+400 x^5+50 x^6+e^{\frac {3+2 e^4 x}{x}} \left (32+16 x+2 x^2\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-16 x-324 x^2-160 x^3-20 x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^3 \left (-e^{2 e^4+\frac {3}{x}} (32+6 x)-50 x^3 \left (1+16 x+3 x^2\right )+e^{e^4+\frac {3}{2 x}} \left (-3+6 x+320 x^2+60 x^3\right )\right )}{2 \left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {x^3 \left (-e^{2 e^4+\frac {3}{x}} (32+6 x)-50 x^3 \left (1+16 x+3 x^2\right )+e^{e^4+\frac {3}{2 x}} \left (-3+6 x+320 x^2+60 x^3\right )\right )}{\left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {2 x^3 (16+3 x)}{(4+x)^2}+\frac {x^3 \left (12+43 x+6 x^2\right )}{(4+x)^2 \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )}-\frac {x^4 \left (12+251 x+440 x^2+175 x^3+20 x^4\right )}{(4+x)^2 \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2}\right ) \, dx\\ &=\frac {1}{2} \int \frac {x^3 \left (12+43 x+6 x^2\right )}{(4+x)^2 \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )} \, dx-\frac {1}{2} \int \frac {x^4 \left (12+251 x+440 x^2+175 x^3+20 x^4\right )}{(4+x)^2 \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2} \, dx-\int \frac {x^3 (16+3 x)}{(4+x)^2} \, dx\\ &=-\frac {x^4}{4+x}-\frac {1}{2} \int \frac {x^4 \left (12+251 x+440 x^2+175 x^3+20 x^4\right )}{(4+x)^2 \left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx+\frac {1}{2} \int \left (-\frac {432}{4 e^{e^4+\frac {3}{2 x}}-x+e^{e^4+\frac {3}{2 x}} x-20 x^2-5 x^3}-\frac {44 x}{-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3}-\frac {5 x^2}{-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3}+\frac {6 x^3}{-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3}+\frac {4096}{(4+x)^2 \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )}-\frac {2752}{(4+x) \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )}\right ) \, dx\\ &=-\frac {x^4}{4+x}-\frac {1}{2} \int \left (-\frac {2240}{\left (4 e^{e^4+\frac {3}{2 x}}-x+e^{e^4+\frac {3}{2 x}} x-20 x^2-5 x^3\right )^2}+\frac {432 x}{\left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2}-\frac {76 x^2}{\left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2}+\frac {11 x^3}{\left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2}+\frac {15 x^5}{\left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2}+\frac {20 x^6}{\left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2}-\frac {8192}{(4+x)^2 \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2}+\frac {11008}{(4+x) \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2}\right ) \, dx-\frac {5}{2} \int \frac {x^2}{-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3} \, dx+3 \int \frac {x^3}{-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3} \, dx-22 \int \frac {x}{-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3} \, dx-216 \int \frac {1}{4 e^{e^4+\frac {3}{2 x}}-x+e^{e^4+\frac {3}{2 x}} x-20 x^2-5 x^3} \, dx-1376 \int \frac {1}{(4+x) \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )} \, dx+2048 \int \frac {1}{(4+x)^2 \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )} \, dx\\ &=-\frac {x^4}{4+x}-\frac {5}{2} \int \frac {x^2}{-e^{e^4+\frac {3}{2 x}} (4+x)+x \left (1+20 x+5 x^2\right )} \, dx+3 \int \frac {x^3}{-e^{e^4+\frac {3}{2 x}} (4+x)+x \left (1+20 x+5 x^2\right )} \, dx-\frac {11}{2} \int \frac {x^3}{\left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2} \, dx-\frac {15}{2} \int \frac {x^5}{\left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2} \, dx-10 \int \frac {x^6}{\left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2} \, dx-22 \int \frac {x}{-e^{e^4+\frac {3}{2 x}} (4+x)+x \left (1+20 x+5 x^2\right )} \, dx+38 \int \frac {x^2}{\left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2} \, dx-216 \int \frac {x}{\left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2} \, dx-216 \int \frac {1}{e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )} \, dx+1120 \int \frac {1}{\left (4 e^{e^4+\frac {3}{2 x}}-x+e^{e^4+\frac {3}{2 x}} x-20 x^2-5 x^3\right )^2} \, dx-1376 \int \frac {1}{(4+x) \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )} \, dx+2048 \int \frac {1}{(4+x)^2 \left (-e^{e^4+\frac {3}{2 x}} (4+x)+x \left (1+20 x+5 x^2\right )\right )} \, dx+4096 \int \frac {1}{(4+x)^2 \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2} \, dx-5504 \int \frac {1}{(4+x) \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2} \, dx\\ &=-\frac {x^4}{4+x}-\frac {5}{2} \int \frac {x^2}{-e^{e^4+\frac {3}{2 x}} (4+x)+x \left (1+20 x+5 x^2\right )} \, dx+3 \int \frac {x^3}{-e^{e^4+\frac {3}{2 x}} (4+x)+x \left (1+20 x+5 x^2\right )} \, dx-\frac {11}{2} \int \frac {x^3}{\left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx-\frac {15}{2} \int \frac {x^5}{\left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx-10 \int \frac {x^6}{\left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx-22 \int \frac {x}{-e^{e^4+\frac {3}{2 x}} (4+x)+x \left (1+20 x+5 x^2\right )} \, dx+38 \int \frac {x^2}{\left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx-216 \int \frac {x}{\left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx-216 \int \frac {1}{e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )} \, dx+1120 \int \frac {1}{\left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx-1376 \int \frac {1}{(4+x) \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )} \, dx+2048 \int \frac {1}{(4+x)^2 \left (-e^{e^4+\frac {3}{2 x}} (4+x)+x \left (1+20 x+5 x^2\right )\right )} \, dx+4096 \int \frac {1}{(4+x)^2 \left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx-5504 \int \frac {1}{(4+x) \left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.16, size = 63, normalized size = 1.70 \begin {gather*} -16 x+4 x^2-x^3-\frac {256}{4+x}+\frac {x^5}{(4+x) \left (-e^{e^4+\frac {3}{2 x}} (4+x)+x \left (1+20 x+5 x^2\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 76, normalized size = 2.05 \begin {gather*} -\frac {5 \, x^{6} + 320 \, x^{3} + 1280 \, x^{2} - {\left (x^{4} + 64 \, x + 256\right )} e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )} + 64 \, x}{5 \, x^{3} + 20 \, x^{2} - {\left (x + 4\right )} e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )} + x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 115, normalized size = 3.11 \begin {gather*} -\frac {5 \, x^{6} - x^{4} e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )} + 320 \, x^{3} + 1280 \, x^{2} - 64 \, x e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )} + 64 \, x - 256 \, e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )}}{5 \, x^{3} + 20 \, x^{2} - x e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )} + x - 4 \, e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.53, size = 70, normalized size = 1.89
method | result | size |
norman | \(\frac {{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}} x^{4}-5 x^{6}}{5 x^{3}+20 x^{2}-x \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}}+x -4 \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}}}\) | \(70\) |
risch | \(-x^{3}+4 x^{2}-16 x -\frac {256}{4+x}+\frac {x^{5}}{\left (4+x \right ) \left (5 x^{3}+20 x^{2}-x \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}}+x -4 \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}}\right )}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 81, normalized size = 2.19 \begin {gather*} -\frac {5 \, x^{6} + 320 \, x^{3} + 1280 \, x^{2} - {\left (x^{4} e^{\left (e^{4}\right )} + 64 \, x e^{\left (e^{4}\right )} + 256 \, e^{\left (e^{4}\right )}\right )} e^{\left (\frac {3}{2 \, x}\right )} + 64 \, x}{5 \, x^{3} + 20 \, x^{2} - {\left (x e^{\left (e^{4}\right )} + 4 \, e^{\left (e^{4}\right )}\right )} e^{\left (\frac {3}{2 \, x}\right )} + x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {{\mathrm {e}}^{\frac {2\,\left (x\,{\mathrm {e}}^4+\frac {3}{2}\right )}{x}}\,\left (6\,x^4+32\,x^3\right )-{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^4+\frac {3}{2}}{x}}\,\left (60\,x^6+320\,x^5+6\,x^4-3\,x^3\right )+50\,x^6+800\,x^7+150\,x^8}{{\mathrm {e}}^{\frac {2\,\left (x\,{\mathrm {e}}^4+\frac {3}{2}\right )}{x}}\,\left (2\,x^2+16\,x+32\right )-{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^4+\frac {3}{2}}{x}}\,\left (20\,x^4+160\,x^3+324\,x^2+16\,x\right )+2\,x^2+80\,x^3+820\,x^4+400\,x^5+50\,x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.31, size = 58, normalized size = 1.57 \begin {gather*} - \frac {x^{5}}{- 5 x^{4} - 40 x^{3} - 81 x^{2} - 4 x + \left (x^{2} + 8 x + 16\right ) e^{\frac {x e^{4} + \frac {3}{2}}{x}}} - x^{3} + 4 x^{2} - 16 x - \frac {256}{x + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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