Optimal. Leaf size=30 \[ -3+\frac {3}{x \left (-e^{-4-x+5 x^2} x+x (3+x)\right )} \]
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Rubi [F] time = 4.32, 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 {e^{8+2 x-10 x^2} (-18-9 x)+e^{4+x-5 x^2} \left (6-3 x+30 x^2\right )}{x^3+e^{4+x-5 x^2} \left (-6 x^3-2 x^4\right )+e^{8+2 x-10 x^2} \left (9 x^3+6 x^4+x^5\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 {3 e^{4+x} \left (-3 e^{4+x} (2+x)-e^{5 x^2} \left (-2+x-10 x^2\right )\right )}{x^3 \left (e^{5 x^2}-e^{4+x} (3+x)\right )^2} \, dx\\ &=3 \int \frac {e^{4+x} \left (-3 e^{4+x} (2+x)-e^{5 x^2} \left (-2+x-10 x^2\right )\right )}{x^3 \left (e^{5 x^2}-e^{4+x} (3+x)\right )^2} \, dx\\ &=3 \int \left (-\frac {e^{4+x} \left (2-x+10 x^2\right )}{x^3 \left (-e^{5 x^2}+3 e^{4+x}+e^{4+x} x\right )}+\frac {e^{8+2 x} \left (-4+29 x+10 x^2\right )}{x^2 \left (-e^{5 x^2}+3 e^{4+x}+e^{4+x} x\right )^2}\right ) \, dx\\ &=-\left (3 \int \frac {e^{4+x} \left (2-x+10 x^2\right )}{x^3 \left (-e^{5 x^2}+3 e^{4+x}+e^{4+x} x\right )} \, dx\right )+3 \int \frac {e^{8+2 x} \left (-4+29 x+10 x^2\right )}{x^2 \left (-e^{5 x^2}+3 e^{4+x}+e^{4+x} x\right )^2} \, dx\\ &=3 \int \left (\frac {10 e^{8+2 x}}{\left (e^{5 x^2}-3 e^{4+x}-e^{4+x} x\right )^2}-\frac {4 e^{8+2 x}}{x^2 \left (-e^{5 x^2}+3 e^{4+x}+e^{4+x} x\right )^2}+\frac {29 e^{8+2 x}}{x \left (-e^{5 x^2}+3 e^{4+x}+e^{4+x} x\right )^2}\right ) \, dx-3 \int \left (\frac {2 e^{4+x}}{x^3 \left (-e^{5 x^2}+3 e^{4+x}+e^{4+x} x\right )}-\frac {e^{4+x}}{x^2 \left (-e^{5 x^2}+3 e^{4+x}+e^{4+x} x\right )}+\frac {10 e^{4+x}}{x \left (-e^{5 x^2}+3 e^{4+x}+e^{4+x} x\right )}\right ) \, dx\\ &=3 \int \frac {e^{4+x}}{x^2 \left (-e^{5 x^2}+3 e^{4+x}+e^{4+x} x\right )} \, dx-6 \int \frac {e^{4+x}}{x^3 \left (-e^{5 x^2}+3 e^{4+x}+e^{4+x} x\right )} \, dx-12 \int \frac {e^{8+2 x}}{x^2 \left (-e^{5 x^2}+3 e^{4+x}+e^{4+x} x\right )^2} \, dx+30 \int \frac {e^{8+2 x}}{\left (e^{5 x^2}-3 e^{4+x}-e^{4+x} x\right )^2} \, dx-30 \int \frac {e^{4+x}}{x \left (-e^{5 x^2}+3 e^{4+x}+e^{4+x} x\right )} \, dx+87 \int \frac {e^{8+2 x}}{x \left (-e^{5 x^2}+3 e^{4+x}+e^{4+x} x\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.87, size = 31, normalized size = 1.03 \begin {gather*} \frac {3 e^{4+x}}{x^2 \left (-e^{5 x^2}+e^{4+x} (3+x)\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 37, normalized size = 1.23 \begin {gather*} -\frac {3 \, e^{\left (-5 \, x^{2} + x + 4\right )}}{x^{2} - {\left (x^{3} + 3 \, x^{2}\right )} e^{\left (-5 \, x^{2} + x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 46, normalized size = 1.53 \begin {gather*} \frac {3 \, e^{\left (-5 \, x^{2} + x + 4\right )}}{x^{3} e^{\left (-5 \, x^{2} + x + 4\right )} + 3 \, x^{2} e^{\left (-5 \, x^{2} + x + 4\right )} - x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 41, normalized size = 1.37
method | result | size |
norman | \(\frac {3 \,{\mathrm e}^{-5 x^{2}+x +4}}{x^{2} \left ({\mathrm e}^{-5 x^{2}+x +4} x +3 \,{\mathrm e}^{-5 x^{2}+x +4}-1\right )}\) | \(41\) |
risch | \(\frac {3}{x^{2} \left (3+x \right )}+\frac {3}{x^{2} \left (3+x \right ) \left ({\mathrm e}^{-\left (4+5 x \right ) \left (x -1\right )} x +3 \,{\mathrm e}^{-\left (4+5 x \right ) \left (x -1\right )}-1\right )}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 37, normalized size = 1.23 \begin {gather*} -\frac {3 \, e^{\left (x + 4\right )}}{x^{2} e^{\left (5 \, x^{2}\right )} - {\left (x^{3} e^{4} + 3 \, x^{2} e^{4}\right )} e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.55, size = 49, normalized size = 1.63 \begin {gather*} \frac {3\,{\mathrm {e}}^4\,{\mathrm {e}}^{-5\,x^2}\,{\mathrm {e}}^x}{3\,x^2\,{\mathrm {e}}^4\,{\mathrm {e}}^{-5\,x^2}\,{\mathrm {e}}^x-x^2+x^3\,{\mathrm {e}}^4\,{\mathrm {e}}^{-5\,x^2}\,{\mathrm {e}}^x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.23, size = 42, normalized size = 1.40 \begin {gather*} \frac {3}{- x^{3} - 3 x^{2} + \left (x^{4} + 6 x^{3} + 9 x^{2}\right ) e^{- 5 x^{2} + x + 4}} + \frac {3}{x^{3} + 3 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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