Optimal. Leaf size=22 \[ x \left (-1-e^{\frac {1}{-4+\frac {5}{x}}}+2 x+\log (5)\right ) \]
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Rubi [F]
time = 0.29, 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 {-25+140 x-176 x^2+64 x^3+e^{-\frac {x}{-5+4 x}} \left (-25+35 x-16 x^2\right )+\left (25-40 x+16 x^2\right ) \log (5)}{25-40 x+16 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 {-25+140 x-176 x^2+64 x^3+e^{-\frac {x}{-5+4 x}} \left (-25+35 x-16 x^2\right )+\left (25-40 x+16 x^2\right ) \log (5)}{(-5+4 x)^2} \, dx\\ &=\int \left (-1+4 x+\frac {e^{\frac {x}{5-4 x}} \left (-25+35 x-16 x^2\right )}{(5-4 x)^2}+\log (5)\right ) \, dx\\ &=2 x^2-x (1-\log (5))+\int \frac {e^{\frac {x}{5-4 x}} \left (-25+35 x-16 x^2\right )}{(5-4 x)^2} \, dx\\ &=2 x^2-x (1-\log (5))+\int \left (-e^{\frac {x}{5-4 x}}-\frac {25 e^{\frac {x}{5-4 x}}}{4 (-5+4 x)^2}-\frac {5 e^{\frac {x}{5-4 x}}}{4 (-5+4 x)}\right ) \, dx\\ &=2 x^2-x (1-\log (5))-\frac {5}{4} \int \frac {e^{\frac {x}{5-4 x}}}{-5+4 x} \, dx-\frac {25}{4} \int \frac {e^{\frac {x}{5-4 x}}}{(-5+4 x)^2} \, dx-\int e^{\frac {x}{5-4 x}} \, dx\\ &=2 x^2-x (1-\log (5))-\frac {5}{4} \int \frac {e^{-\frac {1}{4}+\frac {5}{4 (5-4 x)}}}{-5+4 x} \, dx-\frac {25}{4} \int \frac {e^{-\frac {1}{4}+\frac {5}{4 (5-4 x)}}}{(-5+4 x)^2} \, dx-\int e^{\frac {x}{5-4 x}} \, dx\\ &=-\frac {5}{4} e^{-\frac {1}{4}+\frac {5}{4 (5-4 x)}}+2 x^2+\frac {5 \text {Ei}\left (\frac {5}{4 (5-4 x)}\right )}{16 \sqrt [4]{e}}-x (1-\log (5))-\int e^{\frac {x}{5-4 x}} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.07, size = 22, normalized size = 1.00 \begin {gather*} x \left (-1-e^{\frac {x}{5-4 x}}+2 x+\log (5)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs.
\(2(21)=42\).
time = 0.68, size = 57, normalized size = 2.59
method | result | size |
risch | \(x \ln \left (5\right )+2 x^{2}-x \,{\mathrm e}^{-\frac {x}{4 x -5}}-x\) | \(28\) |
derivativedivides | \(\frac {\ln \left (5\right ) \left (4 x -5\right )}{4}+\frac {\left (4 x -5\right )^{2}}{8}+4 x -5-\frac {{\mathrm e}^{-\frac {1}{4}-\frac {5}{4 \left (4 x -5\right )}} \left (4 x -5\right )}{4}-\frac {5 \,{\mathrm e}^{-\frac {1}{4}-\frac {5}{4 \left (4 x -5\right )}}}{4}\) | \(57\) |
default | \(\frac {\ln \left (5\right ) \left (4 x -5\right )}{4}+\frac {\left (4 x -5\right )^{2}}{8}+4 x -5-\frac {{\mathrm e}^{-\frac {1}{4}-\frac {5}{4 \left (4 x -5\right )}} \left (4 x -5\right )}{4}-\frac {5 \,{\mathrm e}^{-\frac {1}{4}-\frac {5}{4 \left (4 x -5\right )}}}{4}\) | \(57\) |
norman | \(\frac {\left (-14+4 \ln \left (5\right )\right ) x^{2}+8 x^{3}+5 x \,{\mathrm e}^{-\frac {x}{4 x -5}}-4 x^{2} {\mathrm e}^{-\frac {x}{4 x -5}}+\frac {25}{4}-\frac {25 \ln \left (5\right )}{4}}{4 x -5}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 27, normalized size = 1.23 \begin {gather*} 2 \, x^{2} - x e^{\left (-\frac {x}{4 \, x - 5}\right )} + x \log \left (5\right ) - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 20, normalized size = 0.91 \begin {gather*} 2 x^{2} + x \left (-1 + \log {\left (5 \right )}\right ) - x e^{- \frac {x}{4 x - 5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs.
\(2 (21) = 42\).
time = 0.44, size = 100, normalized size = 4.55 \begin {gather*} -\frac {5 \, {\left (\frac {8 \, x e^{\left (-\frac {x}{4 \, x - 5}\right )}}{4 \, x - 5} - \frac {32 \, x^{2} e^{\left (-\frac {x}{4 \, x - 5}\right )}}{{\left (4 \, x - 5\right )}^{2}} + \frac {8 \, x \log \left (5\right )}{4 \, x - 5} + \frac {32 \, x}{4 \, x - 5} - 2 \, \log \left (5\right ) - 3\right )}}{8 \, {\left (\frac {8 \, x}{4 \, x - 5} - \frac {16 \, x^{2}}{{\left (4 \, x - 5\right )}^{2}} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 26, normalized size = 1.18 \begin {gather*} 2\,x^2-x\,\left ({\mathrm {e}}^{-\frac {x}{4\,x-5}}-\ln \left (5\right )+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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