Optimal. Leaf size=33 \[ \frac {x \left (2+5 x+\frac {x+e^x x}{x}\right )}{-e^{(-1+x)^2 x}+x} \]
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Rubi [F] time = 22.87, 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^2+e^x x^2+e^{x-2 x^2+x^3} \left (-3-7 x-7 x^2-11 x^3+15 x^4+e^x \left (-1-4 x^2+3 x^3\right )\right )}{e^{2 x-4 x^2+2 x^3}-2 e^{x-2 x^2+x^3} x+x^2} \, 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^{4 x^2} \left (5 x^2+e^x x^2+e^{x-2 x^2+x^3} \left (-3-7 x-7 x^2-11 x^3+15 x^4+e^x \left (-1-4 x^2+3 x^3\right )\right )\right )}{\left (e^{x+x^3}-e^{2 x^2} x\right )^2} \, dx\\ &=\int \left (\frac {e^{4 x^2} x \left (3+e^x+5 x\right ) \left (-1+x-4 x^2+3 x^3\right )}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}+\frac {e^{4 x^2} \left (-3-e^x-7 x-7 x^2-4 e^x x^2-11 x^3+3 e^x x^3+15 x^4\right )}{e^{x (1+x)^2}-e^{4 x^2} x}\right ) \, dx\\ &=\int \frac {e^{4 x^2} x \left (3+e^x+5 x\right ) \left (-1+x-4 x^2+3 x^3\right )}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx+\int \frac {e^{4 x^2} \left (-3-e^x-7 x-7 x^2-4 e^x x^2-11 x^3+3 e^x x^3+15 x^4\right )}{e^{x (1+x)^2}-e^{4 x^2} x} \, dx\\ &=\int \left (-\frac {e^{4 x^2} x \left (3+e^x+5 x\right )}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}+\frac {e^{4 x^2} x^2 \left (3+e^x+5 x\right )}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}-\frac {4 e^{4 x^2} x^3 \left (3+e^x+5 x\right )}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}+\frac {3 e^{4 x^2} x^4 \left (3+e^x+5 x\right )}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}\right ) \, dx+\int \left (-\frac {3 e^{4 x^2}}{e^{x (1+x)^2}-e^{4 x^2} x}-\frac {e^{x+4 x^2}}{e^{x (1+x)^2}-e^{4 x^2} x}+\frac {7 e^{4 x^2} x}{-e^{x (1+x)^2}+e^{4 x^2} x}+\frac {7 e^{4 x^2} x^2}{-e^{x (1+x)^2}+e^{4 x^2} x}+\frac {4 e^{x+4 x^2} x^2}{-e^{x (1+x)^2}+e^{4 x^2} x}+\frac {11 e^{4 x^2} x^3}{-e^{x (1+x)^2}+e^{4 x^2} x}-\frac {3 e^{x+4 x^2} x^3}{-e^{x (1+x)^2}+e^{4 x^2} x}-\frac {15 e^{4 x^2} x^4}{-e^{x (1+x)^2}+e^{4 x^2} x}\right ) \, dx\\ &=3 \int \frac {e^{4 x^2} x^4 \left (3+e^x+5 x\right )}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx-3 \int \frac {e^{4 x^2}}{e^{x (1+x)^2}-e^{4 x^2} x} \, dx-3 \int \frac {e^{x+4 x^2} x^3}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx-4 \int \frac {e^{4 x^2} x^3 \left (3+e^x+5 x\right )}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx+4 \int \frac {e^{x+4 x^2} x^2}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx+7 \int \frac {e^{4 x^2} x}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx+7 \int \frac {e^{4 x^2} x^2}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx+11 \int \frac {e^{4 x^2} x^3}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx-15 \int \frac {e^{4 x^2} x^4}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx-\int \frac {e^{4 x^2} x \left (3+e^x+5 x\right )}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx+\int \frac {e^{4 x^2} x^2 \left (3+e^x+5 x\right )}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx-\int \frac {e^{x+4 x^2}}{e^{x (1+x)^2}-e^{4 x^2} x} \, dx\\ &=-\left (3 \int \frac {e^{4 x^2}}{e^{x (1+x)^2}-e^{4 x^2} x} \, dx\right )-3 \int \frac {e^{x+4 x^2} x^3}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx+3 \int \left (\frac {e^{x+4 x^2} x^4}{\left (e^{x+x^3}-e^{2 x^2} x\right )^2}+\frac {3 e^{4 x^2} x^4}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}+\frac {5 e^{4 x^2} x^5}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}\right ) \, dx+4 \int \frac {e^{x+4 x^2} x^2}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx-4 \int \left (\frac {e^{x+4 x^2} x^3}{\left (e^{x+x^3}-e^{2 x^2} x\right )^2}+\frac {3 e^{4 x^2} x^3}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}+\frac {5 e^{4 x^2} x^4}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}\right ) \, dx+7 \int \frac {e^{4 x^2} x}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx+7 \int \frac {e^{4 x^2} x^2}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx+11 \int \frac {e^{4 x^2} x^3}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx-15 \int \frac {e^{4 x^2} x^4}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx-\int \frac {e^{x+4 x^2}}{e^{x (1+x)^2}-e^{4 x^2} x} \, dx-\int \left (\frac {e^{x+4 x^2} x}{\left (e^{x+x^3}-e^{2 x^2} x\right )^2}+\frac {3 e^{4 x^2} x}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}+\frac {5 e^{4 x^2} x^2}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}\right ) \, dx+\int \left (\frac {e^{x+4 x^2} x^2}{\left (e^{x+x^3}-e^{2 x^2} x\right )^2}+\frac {3 e^{4 x^2} x^2}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}+\frac {5 e^{4 x^2} x^3}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}\right ) \, dx\\ &=3 \int \frac {e^{x+4 x^2} x^4}{\left (e^{x+x^3}-e^{2 x^2} x\right )^2} \, dx-3 \int \frac {e^{4 x^2} x}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx+3 \int \frac {e^{4 x^2} x^2}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx-3 \int \frac {e^{4 x^2}}{e^{x (1+x)^2}-e^{4 x^2} x} \, dx-3 \int \frac {e^{x+4 x^2} x^3}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx-4 \int \frac {e^{x+4 x^2} x^3}{\left (e^{x+x^3}-e^{2 x^2} x\right )^2} \, dx+4 \int \frac {e^{x+4 x^2} x^2}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx-5 \int \frac {e^{4 x^2} x^2}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx+5 \int \frac {e^{4 x^2} x^3}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx+7 \int \frac {e^{4 x^2} x}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx+7 \int \frac {e^{4 x^2} x^2}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx+9 \int \frac {e^{4 x^2} x^4}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx+11 \int \frac {e^{4 x^2} x^3}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx-12 \int \frac {e^{4 x^2} x^3}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx+15 \int \frac {e^{4 x^2} x^5}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx-15 \int \frac {e^{4 x^2} x^4}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx-20 \int \frac {e^{4 x^2} x^4}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx-\int \frac {e^{x+4 x^2} x}{\left (e^{x+x^3}-e^{2 x^2} x\right )^2} \, dx+\int \frac {e^{x+4 x^2} x^2}{\left (e^{x+x^3}-e^{2 x^2} x\right )^2} \, dx-\int \frac {e^{x+4 x^2}}{e^{x (1+x)^2}-e^{4 x^2} x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.13, size = 38, normalized size = 1.15 \begin {gather*} -\frac {e^{2 x^2} x \left (3+e^x+5 x\right )}{e^{x+x^3}-e^{2 x^2} x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 31, normalized size = 0.94 \begin {gather*} \frac {5 \, x^{2} + x e^{x} + 3 \, x}{x - e^{\left (x^{3} - 2 \, x^{2} + x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 31, normalized size = 0.94 \begin {gather*} \frac {5 \, x^{2} + x e^{x} + 3 \, x}{x - e^{\left (x^{3} - 2 \, x^{2} + x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 24, normalized size = 0.73
method | result | size |
risch | \(\frac {\left (5 x +{\mathrm e}^{x}+3\right ) x}{x -{\mathrm e}^{x \left (x -1\right )^{2}}}\) | \(24\) |
norman | \(\frac {3 \,{\mathrm e}^{x^{3}-2 x^{2}+x}+{\mathrm e}^{x} x +5 x^{2}}{x -{\mathrm e}^{x^{3}-2 x^{2}+x}}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 39, normalized size = 1.18 \begin {gather*} \frac {{\left (5 \, x^{2} + x e^{x} + 3 \, x\right )} e^{\left (2 \, x^{2}\right )}}{x e^{\left (2 \, x^{2}\right )} - e^{\left (x^{3} + x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.59, size = 31, normalized size = 0.94 \begin {gather*} \frac {3\,x+x\,{\mathrm {e}}^x+5\,x^2}{x-{\mathrm {e}}^{x^3-2\,x^2+x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 27, normalized size = 0.82 \begin {gather*} \frac {- 5 x^{2} - x e^{x} - 3 x}{- x + e^{x^{3} - 2 x^{2} + x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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