Optimal. Leaf size=23 \[ \frac {2 e^{-21+4 x} x}{25 (-x+(-1+x) x)^2} \]
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Rubi [B] time = 0.52, antiderivative size = 51, normalized size of antiderivative = 2.22, number of steps used = 13, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {6688, 12, 6742, 2178, 2177} \begin {gather*} \frac {e^{4 x-21}}{50 x}+\frac {e^{4 x-21}}{50 (2-x)}+\frac {e^{4 x-21}}{25 (2-x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2177
Rule 2178
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{-21+4 x} \left (-2+11 x-4 x^2\right )}{25 (2-x)^3 x^2} \, dx\\ &=\frac {2}{25} \int \frac {e^{-21+4 x} \left (-2+11 x-4 x^2\right )}{(2-x)^3 x^2} \, dx\\ &=\frac {2}{25} \int \left (\frac {e^{-21+4 x}}{2-x}-\frac {e^{-21+4 x}}{(-2+x)^3}+\frac {9 e^{-21+4 x}}{4 (-2+x)^2}-\frac {e^{-21+4 x}}{4 x^2}+\frac {e^{-21+4 x}}{x}\right ) \, dx\\ &=-\left (\frac {1}{50} \int \frac {e^{-21+4 x}}{x^2} \, dx\right )+\frac {2}{25} \int \frac {e^{-21+4 x}}{2-x} \, dx-\frac {2}{25} \int \frac {e^{-21+4 x}}{(-2+x)^3} \, dx+\frac {2}{25} \int \frac {e^{-21+4 x}}{x} \, dx+\frac {9}{50} \int \frac {e^{-21+4 x}}{(-2+x)^2} \, dx\\ &=\frac {e^{-21+4 x}}{25 (2-x)^2}+\frac {9 e^{-21+4 x}}{50 (2-x)}+\frac {e^{-21+4 x}}{50 x}-\frac {2 \text {Ei}(-4 (2-x))}{25 e^{13}}+\frac {2 \text {Ei}(4 x)}{25 e^{21}}-\frac {2}{25} \int \frac {e^{-21+4 x}}{x} \, dx-\frac {4}{25} \int \frac {e^{-21+4 x}}{(-2+x)^2} \, dx+\frac {18}{25} \int \frac {e^{-21+4 x}}{-2+x} \, dx\\ &=\frac {e^{-21+4 x}}{25 (2-x)^2}+\frac {e^{-21+4 x}}{50 (2-x)}+\frac {e^{-21+4 x}}{50 x}+\frac {16 \text {Ei}(-4 (2-x))}{25 e^{13}}-\frac {16}{25} \int \frac {e^{-21+4 x}}{-2+x} \, dx\\ &=\frac {e^{-21+4 x}}{25 (2-x)^2}+\frac {e^{-21+4 x}}{50 (2-x)}+\frac {e^{-21+4 x}}{50 x}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.11, size = 19, normalized size = 0.83 \begin {gather*} \frac {2 e^{-21+4 x}}{25 (-2+x)^2 x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 22, normalized size = 0.96 \begin {gather*} \frac {2 \, e^{\left (4 \, x - 21\right )}}{25 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 27, normalized size = 1.17 \begin {gather*} \frac {2 \, e^{\left (4 \, x\right )}}{25 \, {\left (x^{3} e^{21} - 4 \, x^{2} e^{21} + 4 \, x e^{21}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 17, normalized size = 0.74
method | result | size |
risch | \(\frac {2 \,{\mathrm e}^{4 x -21}}{25 x \left (x -2\right )^{2}}\) | \(17\) |
norman | \(\frac {2 \,{\mathrm e}^{4 x -21}}{25 x \left (x -2\right )^{2}}\) | \(19\) |
gosper | \(\frac {2 \,{\mathrm e}^{4 x -21}}{25 x \left (x^{2}-4 x +4\right )}\) | \(24\) |
derivativedivides | \(\frac {109 \,{\mathrm e}^{4 x -21} \left (16 \left (-x +\frac {21}{4}\right )^{2}+184 x -505\right )}{50 \left (64 \left (-x +\frac {21}{4}\right )^{3}-752 \left (-x +\frac {21}{4}\right )^{2}-2860 x +11466\right )}+\frac {31 \,{\mathrm e}^{4 x -21} \left (48 \left (-x +\frac {21}{4}\right )^{2}-472 x +21\right )}{100 \left (64 \left (-x +\frac {21}{4}\right )^{3}-752 \left (-x +\frac {21}{4}\right )^{2}-2860 x +11466\right )}-\frac {{\mathrm e}^{4 x -21} \left (4976 \left (-x +\frac {21}{4}\right )^{2}+25480 x -108927\right )}{100 \left (64 \left (-x +\frac {21}{4}\right )^{3}-752 \left (-x +\frac {21}{4}\right )^{2}-2860 x +11466\right )}\) | \(143\) |
default | \(\frac {109 \,{\mathrm e}^{4 x -21} \left (16 \left (-x +\frac {21}{4}\right )^{2}+184 x -505\right )}{50 \left (64 \left (-x +\frac {21}{4}\right )^{3}-752 \left (-x +\frac {21}{4}\right )^{2}-2860 x +11466\right )}+\frac {31 \,{\mathrm e}^{4 x -21} \left (48 \left (-x +\frac {21}{4}\right )^{2}-472 x +21\right )}{100 \left (64 \left (-x +\frac {21}{4}\right )^{3}-752 \left (-x +\frac {21}{4}\right )^{2}-2860 x +11466\right )}-\frac {{\mathrm e}^{4 x -21} \left (4976 \left (-x +\frac {21}{4}\right )^{2}+25480 x -108927\right )}{100 \left (64 \left (-x +\frac {21}{4}\right )^{3}-752 \left (-x +\frac {21}{4}\right )^{2}-2860 x +11466\right )}\) | \(143\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.80, size = 27, normalized size = 1.17 \begin {gather*} \frac {2 \, e^{\left (4 \, x\right )}}{25 \, {\left (x^{3} e^{21} - 4 \, x^{2} e^{21} + 4 \, x e^{21}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 24, normalized size = 1.04 \begin {gather*} \frac {2\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{-21}}{25\,\left (x^3-4\,x^2+4\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 20, normalized size = 0.87 \begin {gather*} \frac {2 e^{4 x - 21}}{25 x^{3} - 100 x^{2} + 100 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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