Optimal. Leaf size=28 \[ 5-4 x^2+25 e^{4-\frac {6}{x}} \left (4+\frac {5}{x}\right ) (5+x) \]
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Rubi [A] time = 0.23, antiderivative size = 43, normalized size of antiderivative = 1.54, number of steps used = 10, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {14, 6742, 2206, 2210, 2212, 2209} \begin {gather*} -4 x^2+100 e^{4-\frac {6}{x}} x+625 e^{4-\frac {6}{x}}+\frac {625 e^{4-\frac {6}{x}}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2206
Rule 2209
Rule 2210
Rule 2212
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-8 x+\frac {25 e^{4-\frac {6}{x}} \left (150+125 x+24 x^2+4 x^3\right )}{x^3}\right ) \, dx\\ &=-4 x^2+25 \int \frac {e^{4-\frac {6}{x}} \left (150+125 x+24 x^2+4 x^3\right )}{x^3} \, dx\\ &=-4 x^2+25 \int \left (4 e^{4-\frac {6}{x}}+\frac {150 e^{4-\frac {6}{x}}}{x^3}+\frac {125 e^{4-\frac {6}{x}}}{x^2}+\frac {24 e^{4-\frac {6}{x}}}{x}\right ) \, dx\\ &=-4 x^2+100 \int e^{4-\frac {6}{x}} \, dx+600 \int \frac {e^{4-\frac {6}{x}}}{x} \, dx+3125 \int \frac {e^{4-\frac {6}{x}}}{x^2} \, dx+3750 \int \frac {e^{4-\frac {6}{x}}}{x^3} \, dx\\ &=\frac {3125}{6} e^{4-\frac {6}{x}}+\frac {625 e^{4-\frac {6}{x}}}{x}+100 e^{4-\frac {6}{x}} x-4 x^2-600 e^4 \text {Ei}\left (-\frac {6}{x}\right )-600 \int \frac {e^{4-\frac {6}{x}}}{x} \, dx+625 \int \frac {e^{4-\frac {6}{x}}}{x^2} \, dx\\ &=625 e^{4-\frac {6}{x}}+\frac {625 e^{4-\frac {6}{x}}}{x}+100 e^{4-\frac {6}{x}} x-4 x^2\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 35, normalized size = 1.25 \begin {gather*} -4 x^2+25 e^{-6/x} \left (25 e^4+\frac {25 e^4}{x}+4 e^4 x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 34, normalized size = 1.21 \begin {gather*} -\frac {4 \, x^{3} - 25 \, {\left (4 \, x^{2} + 25 \, x + 25\right )} e^{\left (\frac {2 \, {\left (2 \, x - 3\right )}}{x}\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 109, normalized size = 3.89 \begin {gather*} -\frac {\frac {625 \, {\left (2 \, x - 3\right )}^{3} e^{\left (\frac {2 \, {\left (2 \, x - 3\right )}}{x}\right )}}{x^{3}} - \frac {5625 \, {\left (2 \, x - 3\right )}^{2} e^{\left (\frac {2 \, {\left (2 \, x - 3\right )}}{x}\right )}}{x^{2}} + \frac {15900 \, {\left (2 \, x - 3\right )} e^{\left (\frac {2 \, {\left (2 \, x - 3\right )}}{x}\right )}}{x} - 14300 \, e^{\left (\frac {2 \, {\left (2 \, x - 3\right )}}{x}\right )} + 108}{3 \, {\left (\frac {{\left (2 \, x - 3\right )}^{2}}{x^{2}} - \frac {4 \, {\left (2 \, x - 3\right )}}{x} + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 33, normalized size = 1.18
method | result | size |
risch | \(-4 x^{2}+\frac {25 \left (4 x^{2}+25 x +25\right ) {\mathrm e}^{\frac {4 x -6}{x}}}{x}\) | \(33\) |
derivativedivides | \(\frac {3125 \,{\mathrm e}^{-\frac {6}{x}+4}}{3}-4 x^{2}-\frac {625 \,{\mathrm e}^{-\frac {6}{x}+4} \left (-\frac {3}{x}+2\right )}{3}+100 \,{\mathrm e}^{-\frac {6}{x}+4} x\) | \(51\) |
default | \(\frac {3125 \,{\mathrm e}^{-\frac {6}{x}+4}}{3}-4 x^{2}-\frac {625 \,{\mathrm e}^{-\frac {6}{x}+4} \left (-\frac {3}{x}+2\right )}{3}+100 \,{\mathrm e}^{-\frac {6}{x}+4} x\) | \(51\) |
norman | \(\frac {-4 x^{4}+625 \,{\mathrm e}^{\frac {4 x -6}{x}} x +625 \,{\mathrm e}^{\frac {4 x -6}{x}} x^{2}+100 \,{\mathrm e}^{\frac {4 x -6}{x}} x^{3}}{x^{2}}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.42, size = 48, normalized size = 1.71 \begin {gather*} -4 \, x^{2} - 600 \, {\rm Ei}\left (-\frac {6}{x}\right ) e^{4} + \frac {625}{6} \, e^{4} \Gamma \left (2, \frac {6}{x}\right ) + 600 \, e^{4} \Gamma \left (-1, \frac {6}{x}\right ) + \frac {3125}{6} \, e^{\left (-\frac {6}{x} + 4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.20, size = 40, normalized size = 1.43 \begin {gather*} 625\,{\mathrm {e}}^{4-\frac {6}{x}}+\frac {625\,{\mathrm {e}}^{4-\frac {6}{x}}}{x}+100\,x\,{\mathrm {e}}^{4-\frac {6}{x}}-4\,x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 24, normalized size = 0.86 \begin {gather*} - 4 x^{2} + \frac {\left (100 x^{2} + 625 x + 625\right ) e^{\frac {2 \left (2 x - 3\right )}{x}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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