Optimal. Leaf size=23 \[ \frac {4}{-26-\frac {e^{3+2 x}}{x^2}+\frac {5 x}{4}} \]
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Rubi [F]
time = 0.91, 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 {-80 x+\frac {e^{3+2 x} (-128+128 x)}{x^2}}{\frac {16 e^{6+4 x}}{x^3}+10816 x-1040 x^2+25 x^3+\frac {e^{3+2 x} \left (832 x-40 x^2\right )}{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 {128 e^{3+2 x} (-1+x) x-80 x^4}{\left (4 e^{3+2 x}+(104-5 x) x^2\right )^2} \, dx\\ &=\int \left (\frac {16 x^3 \left (208-223 x+10 x^2\right )}{\left (-4 e^{3+2 x}-104 x^2+5 x^3\right )^2}-\frac {32 (-1+x) x}{-4 e^{3+2 x}-104 x^2+5 x^3}\right ) \, dx\\ &=16 \int \frac {x^3 \left (208-223 x+10 x^2\right )}{\left (-4 e^{3+2 x}-104 x^2+5 x^3\right )^2} \, dx-32 \int \frac {(-1+x) x}{-4 e^{3+2 x}-104 x^2+5 x^3} \, dx\\ &=16 \int \left (\frac {208 x^3}{\left (-4 e^{3+2 x}-104 x^2+5 x^3\right )^2}-\frac {223 x^4}{\left (-4 e^{3+2 x}-104 x^2+5 x^3\right )^2}+\frac {10 x^5}{\left (-4 e^{3+2 x}-104 x^2+5 x^3\right )^2}\right ) \, dx-32 \int \left (-\frac {x}{-4 e^{3+2 x}-104 x^2+5 x^3}+\frac {x^2}{-4 e^{3+2 x}-104 x^2+5 x^3}\right ) \, dx\\ &=32 \int \frac {x}{-4 e^{3+2 x}-104 x^2+5 x^3} \, dx-32 \int \frac {x^2}{-4 e^{3+2 x}-104 x^2+5 x^3} \, dx+160 \int \frac {x^5}{\left (-4 e^{3+2 x}-104 x^2+5 x^3\right )^2} \, dx+3328 \int \frac {x^3}{\left (-4 e^{3+2 x}-104 x^2+5 x^3\right )^2} \, dx-3568 \int \frac {x^4}{\left (-4 e^{3+2 x}-104 x^2+5 x^3\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.96, size = 26, normalized size = 1.13 \begin {gather*} \frac {16 x^2}{-4 e^{3+2 x}+x^2 (-104+5 x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 21, normalized size = 0.91
method | result | size |
risch | \(\frac {16}{5 x -\frac {4 \,{\mathrm e}^{2 x +3}}{x^{2}}-104}\) | \(21\) |
norman | \(\frac {16}{5 x -4 \,{\mathrm e}^{-2 \ln \left (x \right )+2 x +3}-104}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 26, normalized size = 1.13 \begin {gather*} \frac {16 \, x^{2}}{5 \, x^{3} - 104 \, x^{2} - 4 \, e^{\left (2 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 21, normalized size = 0.91 \begin {gather*} \frac {16}{5 \, x - 4 \, e^{\left (2 \, x - 2 \, \log \left (x\right ) + 3\right )} - 104} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 24, normalized size = 1.04 \begin {gather*} - \frac {4 x^{2}}{- \frac {5 x^{3}}{4} + 26 x^{2} + e^{2 x + 3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 26, normalized size = 1.13 \begin {gather*} \frac {16 \, x^{2}}{5 \, x^{3} - 104 \, x^{2} - 4 \, e^{\left (2 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.81, size = 26, normalized size = 1.13 \begin {gather*} -\frac {16\,x^2}{4\,{\mathrm {e}}^{2\,x+3}+104\,x^2-5\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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