Optimal. Leaf size=30 \[ \frac {5}{-5 e^{(-4+x) \left (e^{2 x}-\frac {(1+x)^2}{x^2}\right )}+x} \]
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Rubi [F] time = 31.78, 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 {-5 x^3+\exp \left (\frac {4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )}{x^2}\right ) \left (-200-175 x-25 x^3+e^{2 x} \left (-175 x^3+50 x^4\right )\right )}{25 \exp \left (\frac {2 \left (4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )\right )}{x^2}\right ) x^3-10 \exp \left (\frac {4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )}{x^2}\right ) x^4+x^5} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 x} \left (-5 x^3+\exp \left (\frac {4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )}{x^2}\right ) \left (-200-175 x-25 x^3+e^{2 x} \left (-175 x^3+50 x^4\right )\right )\right )}{x^3 \left (5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}-e^x x\right )^2} \, dx\\ &=\int \left (-\frac {e^{-2-e^{2 x} (-4+x)-\frac {4}{x^2}-\frac {7}{x}+x} \left (8+7 x+x^3\right )}{x^3}-\frac {5 e^{2 x} \left (8+7 x+175 e^{4+2 e^{2 x} (-4+x)+\frac {8}{x^2}+\frac {14}{x}} x+x^2-50 e^{4+2 e^{2 x} (-4+x)+\frac {8}{x^2}+\frac {14}{x}} x^2+x^3\right )}{x^2 \left (-5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}+e^x x\right )^2}-\frac {e^{-2-e^{2 x} (-4+x)-\frac {4}{x^2}-\frac {7}{x}+2 x} \left (8+7 x-175 e^{4+2 e^{2 x} (-4+x)+\frac {8}{x^2}+\frac {14}{x}} x+50 e^{4+2 e^{2 x} (-4+x)+\frac {8}{x^2}+\frac {14}{x}} x^2+x^3\right )}{x^2 \left (5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}-e^x x\right )}\right ) \, dx\\ &=-\left (5 \int \frac {e^{2 x} \left (8+7 x+175 e^{4+2 e^{2 x} (-4+x)+\frac {8}{x^2}+\frac {14}{x}} x+x^2-50 e^{4+2 e^{2 x} (-4+x)+\frac {8}{x^2}+\frac {14}{x}} x^2+x^3\right )}{x^2 \left (-5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}+e^x x\right )^2} \, dx\right )-\int \frac {e^{-2-e^{2 x} (-4+x)-\frac {4}{x^2}-\frac {7}{x}+x} \left (8+7 x+x^3\right )}{x^3} \, dx-\int \frac {e^{-2-e^{2 x} (-4+x)-\frac {4}{x^2}-\frac {7}{x}+2 x} \left (8+7 x-175 e^{4+2 e^{2 x} (-4+x)+\frac {8}{x^2}+\frac {14}{x}} x+50 e^{4+2 e^{2 x} (-4+x)+\frac {8}{x^2}+\frac {14}{x}} x^2+x^3\right )}{x^2 \left (5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}-e^x x\right )} \, dx\\ &=-\left (5 \int \left (\frac {e^{2 x}}{\left (5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}-e^x x\right )^2}-\frac {50 e^{4+2 e^{2 x} (-4+x)+\frac {8}{x^2}+\frac {14}{x}+2 x}}{\left (5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}-e^x x\right )^2}+\frac {175 e^{4+2 e^{2 x} (-4+x)+\frac {8}{x^2}+\frac {14}{x}+2 x}}{x \left (5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}-e^x x\right )^2}+\frac {8 e^{2 x}}{x^2 \left (-5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}+e^x x\right )^2}+\frac {7 e^{2 x}}{x \left (-5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}+e^x x\right )^2}+\frac {e^{2 x} x}{\left (-5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}+e^x x\right )^2}\right ) \, dx\right )-\int \left (e^{-2-e^{2 x} (-4+x)-\frac {4}{x^2}-\frac {7}{x}+x}+\frac {8 e^{-2-e^{2 x} (-4+x)-\frac {4}{x^2}-\frac {7}{x}+x}}{x^3}+\frac {7 e^{-2-e^{2 x} (-4+x)-\frac {4}{x^2}-\frac {7}{x}+x}}{x^2}\right ) \, dx-\int \left (\frac {50 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}+2 x}}{5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}-e^x x}-\frac {175 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}+2 x}}{x \left (5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}-e^x x\right )}-\frac {8 e^{-2-e^{2 x} (-4+x)-\frac {4}{x^2}-\frac {7}{x}+2 x}}{x^2 \left (-5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}+e^x x\right )}-\frac {7 e^{-2-e^{2 x} (-4+x)-\frac {4}{x^2}-\frac {7}{x}+2 x}}{x \left (-5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}+e^x x\right )}-\frac {e^{-2-e^{2 x} (-4+x)-\frac {4}{x^2}-\frac {7}{x}+2 x} x}{-5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}+e^x x}\right ) \, dx\\ &=-\left (5 \int \frac {e^{2 x}}{\left (5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}-e^x x\right )^2} \, dx\right )-5 \int \frac {e^{2 x} x}{\left (-5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}+e^x x\right )^2} \, dx-7 \int \frac {e^{-2-e^{2 x} (-4+x)-\frac {4}{x^2}-\frac {7}{x}+x}}{x^2} \, dx+7 \int \frac {e^{-2-e^{2 x} (-4+x)-\frac {4}{x^2}-\frac {7}{x}+2 x}}{x \left (-5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}+e^x x\right )} \, dx-8 \int \frac {e^{-2-e^{2 x} (-4+x)-\frac {4}{x^2}-\frac {7}{x}+x}}{x^3} \, dx+8 \int \frac {e^{-2-e^{2 x} (-4+x)-\frac {4}{x^2}-\frac {7}{x}+2 x}}{x^2 \left (-5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}+e^x x\right )} \, dx-35 \int \frac {e^{2 x}}{x \left (-5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}+e^x x\right )^2} \, dx-40 \int \frac {e^{2 x}}{x^2 \left (-5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}+e^x x\right )^2} \, dx-50 \int \frac {e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}+2 x}}{5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}-e^x x} \, dx+175 \int \frac {e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}+2 x}}{x \left (5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}-e^x x\right )} \, dx+250 \int \frac {e^{4+2 e^{2 x} (-4+x)+\frac {8}{x^2}+\frac {14}{x}+2 x}}{\left (5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}-e^x x\right )^2} \, dx-875 \int \frac {e^{4+2 e^{2 x} (-4+x)+\frac {8}{x^2}+\frac {14}{x}+2 x}}{x \left (5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}-e^x x\right )^2} \, dx-\int e^{-2-e^{2 x} (-4+x)-\frac {4}{x^2}-\frac {7}{x}+x} \, dx+\int \frac {e^{-2-e^{2 x} (-4+x)-\frac {4}{x^2}-\frac {7}{x}+2 x} x}{-5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}+e^x x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.32, size = 39, normalized size = 1.30 \begin {gather*} -\frac {5 e^x}{5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}-e^x x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 42, normalized size = 1.40 \begin {gather*} \frac {5}{x - 5 \, e^{\left (-\frac {x^{3} - 2 \, x^{2} - {\left (x^{3} - 4 \, x^{2}\right )} e^{\left (2 \, x\right )} - 7 \, x - 4}{x^{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.52, size = 1342, normalized size = 44.73 result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 35, normalized size = 1.17
method | result | size |
risch | \(\frac {5}{x -5 \,{\mathrm e}^{\frac {\left (x -4\right ) \left ({\mathrm e}^{2 x} x^{2}-x^{2}-2 x -1\right )}{x^{2}}}}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 46, normalized size = 1.53 \begin {gather*} \frac {5 \, e^{\left (x + 4 \, e^{\left (2 \, x\right )}\right )}}{x e^{\left (x + 4 \, e^{\left (2 \, x\right )}\right )} - 5 \, e^{\left (x e^{\left (2 \, x\right )} + \frac {7}{x} + \frac {4}{x^{2}} + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.21, size = 40, normalized size = 1.33 \begin {gather*} \frac {5}{x-5\,{\mathrm {e}}^{-4\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^2\,{\mathrm {e}}^{x\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{\frac {4}{x^2}}\,{\mathrm {e}}^{7/x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 37, normalized size = 1.23 \begin {gather*} - \frac {1}{- \frac {x}{5} + e^{\frac {- x^{3} + 2 x^{2} + 7 x + \left (x^{3} - 4 x^{2}\right ) e^{2 x} + 4}{x^{2}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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