Optimal. Leaf size=23 \[ \left (4+e^{\frac {2 e^2}{3 x}}\right ) \left (1+x+x^2\right )^2 \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.61, antiderivative size = 297, normalized size of antiderivative = 12.91, number of steps
used = 18, number of rules used = 8, integrand size = 86, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {12, 14, 6874,
2237, 2241, 2240, 2245, 2250} \begin {gather*} \frac {64}{81} e^8 \text {Gamma}\left (-4,-\frac {2 e^2}{3 x}\right )-\frac {8}{243} e^6 \left (9-e^2\right ) \text {ExpIntegralEi}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{3} e^2 \left (1-e^2\right ) \text {ExpIntegralEi}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{27} e^4 \left (9-2 e^2\right ) \text {ExpIntegralEi}\left (\frac {2 e^2}{3 x}\right )+\frac {4}{3} e^2 \text {ExpIntegralEi}\left (\frac {2 e^2}{3 x}\right )+4 x^4+\frac {2}{9} \left (9-e^2\right ) e^{\frac {2 e^2}{3 x}} x^3+8 x^3+\frac {2}{27} \left (9-e^2\right ) e^{\frac {2 e^2}{3 x}+2} x^2+\frac {1}{3} \left (9-2 e^2\right ) e^{\frac {2 e^2}{3 x}} x^2+12 x^2+\frac {4}{81} \left (9-e^2\right ) e^{\frac {2 e^2}{3 x}+4} x+2 \left (1-e^2\right ) e^{\frac {2 e^2}{3 x}} x+\frac {2}{9} \left (9-2 e^2\right ) e^{\frac {2 e^2}{3 x}+2} x+8 x+e^{\frac {2 e^2}{3 x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2237
Rule 2240
Rule 2241
Rule 2245
Rule 2250
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {24 x^2+72 x^3+72 x^4+48 x^5+e^{\frac {2 e^2}{3 x}} \left (6 x^2+18 x^3+18 x^4+12 x^5+e^2 \left (-2-4 x-6 x^2-4 x^3-2 x^4\right )\right )}{x^2} \, dx\\ &=\frac {1}{3} \int \left (24 \left (1+3 x+3 x^2+2 x^3\right )+\frac {2 e^{\frac {2 e^2}{3 x}} \left (1+x+x^2\right ) \left (-e^2-e^2 x+\left (3-e^2\right ) x^2+6 x^3\right )}{x^2}\right ) \, dx\\ &=\frac {2}{3} \int \frac {e^{\frac {2 e^2}{3 x}} \left (1+x+x^2\right ) \left (-e^2-e^2 x+\left (3-e^2\right ) x^2+6 x^3\right )}{x^2} \, dx+8 \int \left (1+3 x+3 x^2+2 x^3\right ) \, dx\\ &=8 x+12 x^2+8 x^3+4 x^4+\frac {2}{3} \int \left (-3 e^{\frac {2 e^2}{3 x}} \left (-1+e^2\right )-\frac {e^{2+\frac {2 e^2}{3 x}}}{x^2}-\frac {2 e^{2+\frac {2 e^2}{3 x}}}{x}-e^{\frac {2 e^2}{3 x}} \left (-9+2 e^2\right ) x-e^{\frac {2 e^2}{3 x}} \left (-9+e^2\right ) x^2+6 e^{\frac {2 e^2}{3 x}} x^3\right ) \, dx\\ &=8 x+12 x^2+8 x^3+4 x^4-\frac {2}{3} \int \frac {e^{2+\frac {2 e^2}{3 x}}}{x^2} \, dx-\frac {4}{3} \int \frac {e^{2+\frac {2 e^2}{3 x}}}{x} \, dx+4 \int e^{\frac {2 e^2}{3 x}} x^3 \, dx+\frac {1}{3} \left (2 \left (9-2 e^2\right )\right ) \int e^{\frac {2 e^2}{3 x}} x \, dx+\left (2 \left (1-e^2\right )\right ) \int e^{\frac {2 e^2}{3 x}} \, dx+\frac {1}{3} \left (2 \left (9-e^2\right )\right ) \int e^{\frac {2 e^2}{3 x}} x^2 \, dx\\ &=e^{\frac {2 e^2}{3 x}}+8 x+2 e^{\frac {2 e^2}{3 x}} \left (1-e^2\right ) x+12 x^2+\frac {1}{3} e^{\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x^2+8 x^3+\frac {2}{9} e^{\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^3+4 x^4+\frac {4}{3} e^2 \text {Ei}\left (\frac {2 e^2}{3 x}\right )+\frac {64}{81} e^8 \Gamma \left (-4,-\frac {2 e^2}{3 x}\right )+\frac {1}{9} \left (2 e^2 \left (9-2 e^2\right )\right ) \int e^{\frac {2 e^2}{3 x}} \, dx+\frac {1}{3} \left (4 e^2 \left (1-e^2\right )\right ) \int \frac {e^{\frac {2 e^2}{3 x}}}{x} \, dx+\frac {1}{27} \left (4 e^2 \left (9-e^2\right )\right ) \int e^{\frac {2 e^2}{3 x}} x \, dx\\ &=e^{\frac {2 e^2}{3 x}}+8 x+\frac {2}{9} e^{2+\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x+2 e^{\frac {2 e^2}{3 x}} \left (1-e^2\right ) x+12 x^2+\frac {1}{3} e^{\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x^2+\frac {2}{27} e^{2+\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^2+8 x^3+\frac {2}{9} e^{\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^3+4 x^4+\frac {4}{3} e^2 \text {Ei}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{3} e^2 \left (1-e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )+\frac {64}{81} e^8 \Gamma \left (-4,-\frac {2 e^2}{3 x}\right )+\frac {1}{27} \left (4 e^4 \left (9-2 e^2\right )\right ) \int \frac {e^{\frac {2 e^2}{3 x}}}{x} \, dx+\frac {1}{81} \left (4 e^4 \left (9-e^2\right )\right ) \int e^{\frac {2 e^2}{3 x}} \, dx\\ &=e^{\frac {2 e^2}{3 x}}+8 x+\frac {2}{9} e^{2+\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x+2 e^{\frac {2 e^2}{3 x}} \left (1-e^2\right ) x+\frac {4}{81} e^{4+\frac {2 e^2}{3 x}} \left (9-e^2\right ) x+12 x^2+\frac {1}{3} e^{\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x^2+\frac {2}{27} e^{2+\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^2+8 x^3+\frac {2}{9} e^{\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^3+4 x^4+\frac {4}{3} e^2 \text {Ei}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{27} e^4 \left (9-2 e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{3} e^2 \left (1-e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )+\frac {64}{81} e^8 \Gamma \left (-4,-\frac {2 e^2}{3 x}\right )+\frac {1}{243} \left (8 e^6 \left (9-e^2\right )\right ) \int \frac {e^{\frac {2 e^2}{3 x}}}{x} \, dx\\ &=e^{\frac {2 e^2}{3 x}}+8 x+\frac {2}{9} e^{2+\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x+2 e^{\frac {2 e^2}{3 x}} \left (1-e^2\right ) x+\frac {4}{81} e^{4+\frac {2 e^2}{3 x}} \left (9-e^2\right ) x+12 x^2+\frac {1}{3} e^{\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x^2+\frac {2}{27} e^{2+\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^2+8 x^3+\frac {2}{9} e^{\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^3+4 x^4+\frac {4}{3} e^2 \text {Ei}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{27} e^4 \left (9-2 e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{3} e^2 \left (1-e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )-\frac {8}{243} e^6 \left (9-e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )+\frac {64}{81} e^8 \Gamma \left (-4,-\frac {2 e^2}{3 x}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 40, normalized size = 1.74 \begin {gather*} 8 x+12 x^2+8 x^3+4 x^4+e^{\frac {2 e^2}{3 x}} \left (1+x+x^2\right )^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 9.55, size = 438, normalized size = 19.04 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.35, size = 130, normalized size = 5.65 \begin {gather*} 4 \, x^{4} + 8 \, x^{3} + 12 \, x^{2} + \frac {4}{3} \, {\rm Ei}\left (\frac {2 \, e^{2}}{3 \, x}\right ) e^{2} + \frac {4}{3} \, e^{4} \Gamma \left (-1, -\frac {2 \, e^{2}}{3 \, x}\right ) - \frac {4}{3} \, e^{2} \Gamma \left (-1, -\frac {2 \, e^{2}}{3 \, x}\right ) - \frac {16}{27} \, e^{6} \Gamma \left (-2, -\frac {2 \, e^{2}}{3 \, x}\right ) + \frac {8}{3} \, e^{4} \Gamma \left (-2, -\frac {2 \, e^{2}}{3 \, x}\right ) + \frac {16}{81} \, e^{8} \Gamma \left (-3, -\frac {2 \, e^{2}}{3 \, x}\right ) - \frac {16}{9} \, e^{6} \Gamma \left (-3, -\frac {2 \, e^{2}}{3 \, x}\right ) + \frac {64}{81} \, e^{8} \Gamma \left (-4, -\frac {2 \, e^{2}}{3 \, x}\right ) + 8 \, x + e^{\left (\frac {2 \, e^{2}}{3 \, x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs.
\(2 (19) = 38\).
time = 0.38, size = 46, normalized size = 2.00 \begin {gather*} 4 \, x^{4} + 8 \, x^{3} + 12 \, x^{2} + {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )} e^{\left (\frac {2 \, e^{2}}{3 \, x}\right )} + 8 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs.
\(2 (19) = 38\).
time = 0.14, size = 46, normalized size = 2.00 \begin {gather*} 4 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + \left (x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1\right ) e^{\frac {2 e^{2}}{3 x}} \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. 101 vs.
\(2 (19) = 38\).
time = 0.44, size = 101, normalized size = 4.39 \begin {gather*} x^{4} {\left (\frac {8 \, e^{10}}{x} + \frac {2 \, e^{\left (\frac {2 \, e^{2}}{3 \, x} + 10\right )}}{x} + \frac {12 \, e^{10}}{x^{2}} + \frac {3 \, e^{\left (\frac {2 \, e^{2}}{3 \, x} + 10\right )}}{x^{2}} + \frac {8 \, e^{10}}{x^{3}} + \frac {2 \, e^{\left (\frac {2 \, e^{2}}{3 \, x} + 10\right )}}{x^{3}} + \frac {e^{\left (\frac {2 \, e^{2}}{3 \, x} + 10\right )}}{x^{4}} + 4 \, e^{10} + e^{\left (\frac {2 \, e^{2}}{3 \, x} + 10\right )}\right )} e^{\left (-10\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.29, size = 69, normalized size = 3.00 \begin {gather*} {\mathrm {e}}^{\frac {2\,{\mathrm {e}}^2}{3\,x}}+x\,\left (2\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^2}{3\,x}}+8\right )+x^4\,\left ({\mathrm {e}}^{\frac {2\,{\mathrm {e}}^2}{3\,x}}+4\right )+x^3\,\left (2\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^2}{3\,x}}+8\right )+x^2\,\left (3\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^2}{3\,x}}+12\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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