Optimal. Leaf size=25 \[ \frac {5}{9-x-x^2+e^{-6-2 x} x^2} \]
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Rubi [F] time = 9.51, 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^{12+4 x} (5+10 x)+e^{6+2 x} \left (-10 x+10 x^2\right )}{x^4+e^{6+2 x} \left (18 x^2-2 x^3-2 x^4\right )+e^{12+4 x} \left (81-18 x-17 x^2+2 x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 e^{6+2 x} \left (2 (-1+x) x+e^{6+2 x} (1+2 x)\right )}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx\\ &=5 \int \frac {e^{6+2 x} \left (2 (-1+x) x+e^{6+2 x} (1+2 x)\right )}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx\\ &=5 \int \left (\frac {e^{6+2 x} (1+2 x)}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )}+\frac {e^{6+2 x} x \left (18-19 x+2 x^2+2 x^3\right )}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2}\right ) \, dx\\ &=5 \int \frac {e^{6+2 x} (1+2 x)}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )} \, dx+5 \int \frac {e^{6+2 x} x \left (18-19 x+2 x^2+2 x^3\right )}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2} \, dx\\ &=5 \int \frac {e^{6+2 x} (1+2 x)}{\left (9-x-x^2\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx+5 \int \left (-\frac {e^{6+2 x}}{\left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2}+\frac {2 e^{6+2 x} x^2}{\left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2}+\frac {e^{6+2 x} (-9+19 x)}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2}\right ) \, dx\\ &=-\left (5 \int \frac {e^{6+2 x}}{\left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2} \, dx\right )+5 \int \frac {e^{6+2 x} (-9+19 x)}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2} \, dx+5 \int \left (\frac {e^{6+2 x}}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )}+\frac {2 e^{6+2 x} x}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )}\right ) \, dx+10 \int \frac {e^{6+2 x} x^2}{\left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2} \, dx\\ &=5 \int \frac {e^{6+2 x}}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )} \, dx-5 \int \frac {e^{6+2 x}}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx+5 \int \left (-\frac {9 e^{6+2 x}}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2}+\frac {19 e^{6+2 x} x}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2}\right ) \, dx+10 \int \frac {e^{6+2 x} x}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )} \, dx+10 \int \frac {e^{6+2 x} x^2}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx\\ &=-\left (5 \int \frac {e^{6+2 x}}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx\right )+5 \int \frac {e^{6+2 x}}{\left (9-x-x^2\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx+10 \int \frac {e^{6+2 x} x^2}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx+10 \int \frac {e^{6+2 x} x}{\left (9-x-x^2\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx-45 \int \frac {e^{6+2 x}}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2} \, dx+95 \int \frac {e^{6+2 x} x}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2} \, dx\\ &=-\left (5 \int \frac {e^{6+2 x}}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx\right )+5 \int \left (\frac {2 e^{6+2 x}}{\sqrt {37} \left (-1+\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )}+\frac {2 e^{6+2 x}}{\sqrt {37} \left (1+\sqrt {37}+2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )}\right ) \, dx+10 \int \frac {e^{6+2 x} x^2}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx+10 \int \left (\frac {\left (1+\frac {1}{\sqrt {37}}\right ) e^{6+2 x}}{\left (-1-\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )}+\frac {\left (1-\frac {1}{\sqrt {37}}\right ) e^{6+2 x}}{\left (-1+\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )}\right ) \, dx-45 \int \frac {e^{6+2 x}}{\left (-9+x+x^2\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx+95 \int \frac {e^{6+2 x} x}{\left (-9+x+x^2\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx\\ &=-\left (5 \int \frac {e^{6+2 x}}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx\right )+10 \int \frac {e^{6+2 x} x^2}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx-45 \int \left (-\frac {2 e^{6+2 x}}{\sqrt {37} \left (-1+\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2}-\frac {2 e^{6+2 x}}{\sqrt {37} \left (1+\sqrt {37}+2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2}\right ) \, dx+95 \int \left (\frac {\left (1-\frac {1}{\sqrt {37}}\right ) e^{6+2 x}}{\left (1-\sqrt {37}+2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2}+\frac {\left (1+\frac {1}{\sqrt {37}}\right ) e^{6+2 x}}{\left (1+\sqrt {37}+2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2}\right ) \, dx+\frac {10 \int \frac {e^{6+2 x}}{\left (-1+\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx}{\sqrt {37}}+\frac {10 \int \frac {e^{6+2 x}}{\left (1+\sqrt {37}+2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx}{\sqrt {37}}+\frac {1}{37} \left (10 \left (37-\sqrt {37}\right )\right ) \int \frac {e^{6+2 x}}{\left (-1+\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx+\frac {1}{37} \left (10 \left (37+\sqrt {37}\right )\right ) \int \frac {e^{6+2 x}}{\left (-1-\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx\\ &=-\left (5 \int \frac {e^{6+2 x}}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx\right )+10 \int \frac {e^{6+2 x} x^2}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx+\frac {10 \int \frac {e^{6+2 x}}{\left (-1+\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx}{\sqrt {37}}+\frac {10 \int \frac {e^{6+2 x}}{\left (1+\sqrt {37}+2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx}{\sqrt {37}}+\frac {90 \int \frac {e^{6+2 x}}{\left (-1+\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx}{\sqrt {37}}+\frac {90 \int \frac {e^{6+2 x}}{\left (1+\sqrt {37}+2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx}{\sqrt {37}}+\frac {1}{37} \left (10 \left (37-\sqrt {37}\right )\right ) \int \frac {e^{6+2 x}}{\left (-1+\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx+\frac {1}{37} \left (95 \left (37-\sqrt {37}\right )\right ) \int \frac {e^{6+2 x}}{\left (1-\sqrt {37}+2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx+\frac {1}{37} \left (10 \left (37+\sqrt {37}\right )\right ) \int \frac {e^{6+2 x}}{\left (-1-\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx+\frac {1}{37} \left (95 \left (37+\sqrt {37}\right )\right ) \int \frac {e^{6+2 x}}{\left (1+\sqrt {37}+2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 1.62, size = 31, normalized size = 1.24 \begin {gather*} -\frac {5 e^{6+2 x}}{-x^2+e^{6+2 x} \left (-9+x+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 28, normalized size = 1.12 \begin {gather*} \frac {5 \, e^{\left (2 \, x + 6\right )}}{x^{2} - {\left (x^{2} + x - 9\right )} e^{\left (2 \, x + 6\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 42, normalized size = 1.68 \begin {gather*} -\frac {5 \, e^{\left (2 \, x + 6\right )}}{x^{2} e^{\left (2 \, x + 6\right )} - x^{2} + x e^{\left (2 \, x + 6\right )} - 9 \, e^{\left (2 \, x + 6\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 51, normalized size = 2.04
method | result | size |
norman | \(-\frac {5 \,{\mathrm e}^{6} {\mathrm e}^{2 x}}{x^{2} {\mathrm e}^{6} {\mathrm e}^{2 x}+{\mathrm e}^{6} {\mathrm e}^{2 x} x -9 \,{\mathrm e}^{6} {\mathrm e}^{2 x}-x^{2}}\) | \(51\) |
risch | \(-\frac {5}{x^{2}+x -9}-\frac {5 x^{2}}{\left (x^{2}+x -9\right ) \left (x^{2} {\mathrm e}^{2 x +6}+x \,{\mathrm e}^{2 x +6}-9 \,{\mathrm e}^{2 x +6}-x^{2}\right )}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 35, normalized size = 1.40 \begin {gather*} \frac {5 \, e^{\left (2 \, x + 6\right )}}{x^{2} - {\left (x^{2} e^{6} + x e^{6} - 9 \, e^{6}\right )} e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 57, normalized size = 2.28 \begin {gather*} \frac {5\,x\,\left ({\mathrm {e}}^{2\,x+6}-x+x\,{\mathrm {e}}^{2\,x+6}\right )}{9\,\left (9\,{\mathrm {e}}^{2\,x+6}-x\,{\mathrm {e}}^{2\,x+6}-x^2\,{\mathrm {e}}^{2\,x+6}+x^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.34, size = 66, normalized size = 2.64 \begin {gather*} - \frac {5 x^{2}}{- x^{4} - x^{3} + 9 x^{2} + \left (x^{4} e^{6} + 2 x^{3} e^{6} - 17 x^{2} e^{6} - 18 x e^{6} + 81 e^{6}\right ) e^{2 x}} - \frac {5}{x^{2} + x - 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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