Optimal. Leaf size=31 \[ 3+x+\left (-2+x+e^{2 x} x+\frac {\left (-e^x+x\right )^2}{9 x^2}\right )^2 \]
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Rubi [B] time = 0.68, antiderivative size = 127, normalized size of antiderivative = 4.10, number of steps used = 52, number of rules used = 7, integrand size = 126, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {12, 14, 2199, 2194, 2177, 2178, 2176} \begin {gather*} \frac {e^{4 x}}{81 x^4}-\frac {4 e^{3 x}}{81 x^3}+2 e^{2 x} x^2+e^{4 x} x^2-\frac {10 e^{2 x}}{27 x^2}+\frac {1}{324} (25-18 x)^2-\frac {4 e^x}{9}-\frac {4 e^{3 x}}{9}-\frac {34}{9} e^{2 x} x+\frac {68 e^x}{81 x}+\frac {2 e^{2 x}}{9 x}+\frac {2 e^{4 x}}{9 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2176
Rule 2177
Rule 2178
Rule 2194
Rule 2199
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{81} \int \frac {-225 x^5+162 x^6+e^{3 x} \left (12 x-12 x^2-108 x^5\right )+e^x \left (-68 x^3+68 x^4-36 x^5\right )+e^{2 x} \left (60 x^2-78 x^3+36 x^4-306 x^5-288 x^6+324 x^7\right )+e^{4 x} \left (-4+4 x-18 x^3+72 x^4+162 x^6+324 x^7\right )}{x^5} \, dx\\ &=\frac {1}{81} \int \left (9 (-25+18 x)-\frac {4 e^x \left (17-17 x+9 x^2\right )}{x^2}-\frac {12 e^{3 x} \left (-1+x+9 x^4\right )}{x^4}+\frac {2 e^{4 x} \left (1+9 x^3\right ) \left (-2+2 x+9 x^3+18 x^4\right )}{x^5}+\frac {6 e^{2 x} \left (10-13 x+6 x^2-51 x^3-48 x^4+54 x^5\right )}{x^3}\right ) \, dx\\ &=\frac {1}{324} (25-18 x)^2+\frac {2}{81} \int \frac {e^{4 x} \left (1+9 x^3\right ) \left (-2+2 x+9 x^3+18 x^4\right )}{x^5} \, dx-\frac {4}{81} \int \frac {e^x \left (17-17 x+9 x^2\right )}{x^2} \, dx+\frac {2}{27} \int \frac {e^{2 x} \left (10-13 x+6 x^2-51 x^3-48 x^4+54 x^5\right )}{x^3} \, dx-\frac {4}{27} \int \frac {e^{3 x} \left (-1+x+9 x^4\right )}{x^4} \, dx\\ &=\frac {1}{324} (25-18 x)^2+\frac {2}{81} \int \left (-\frac {2 e^{4 x}}{x^5}+\frac {2 e^{4 x}}{x^4}-\frac {9 e^{4 x}}{x^2}+\frac {36 e^{4 x}}{x}+81 e^{4 x} x+162 e^{4 x} x^2\right ) \, dx-\frac {4}{81} \int \left (9 e^x+\frac {17 e^x}{x^2}-\frac {17 e^x}{x}\right ) \, dx+\frac {2}{27} \int \left (-51 e^{2 x}+\frac {10 e^{2 x}}{x^3}-\frac {13 e^{2 x}}{x^2}+\frac {6 e^{2 x}}{x}-48 e^{2 x} x+54 e^{2 x} x^2\right ) \, dx-\frac {4}{27} \int \left (9 e^{3 x}-\frac {e^{3 x}}{x^4}+\frac {e^{3 x}}{x^3}\right ) \, dx\\ &=\frac {1}{324} (25-18 x)^2-\frac {4}{81} \int \frac {e^{4 x}}{x^5} \, dx+\frac {4}{81} \int \frac {e^{4 x}}{x^4} \, dx+\frac {4}{27} \int \frac {e^{3 x}}{x^4} \, dx-\frac {4}{27} \int \frac {e^{3 x}}{x^3} \, dx-\frac {2}{9} \int \frac {e^{4 x}}{x^2} \, dx-\frac {4 \int e^x \, dx}{9}+\frac {4}{9} \int \frac {e^{2 x}}{x} \, dx+\frac {20}{27} \int \frac {e^{2 x}}{x^3} \, dx-\frac {68}{81} \int \frac {e^x}{x^2} \, dx+\frac {68}{81} \int \frac {e^x}{x} \, dx+\frac {8}{9} \int \frac {e^{4 x}}{x} \, dx-\frac {26}{27} \int \frac {e^{2 x}}{x^2} \, dx-\frac {4}{3} \int e^{3 x} \, dx+2 \int e^{4 x} x \, dx-\frac {32}{9} \int e^{2 x} x \, dx-\frac {34}{9} \int e^{2 x} \, dx+4 \int e^{2 x} x^2 \, dx+4 \int e^{4 x} x^2 \, dx\\ &=-\frac {4 e^x}{9}-\frac {17 e^{2 x}}{9}-\frac {4 e^{3 x}}{9}+\frac {1}{324} (25-18 x)^2+\frac {e^{4 x}}{81 x^4}-\frac {4 e^{3 x}}{81 x^3}-\frac {4 e^{4 x}}{243 x^3}-\frac {10 e^{2 x}}{27 x^2}+\frac {2 e^{3 x}}{27 x^2}+\frac {68 e^x}{81 x}+\frac {26 e^{2 x}}{27 x}+\frac {2 e^{4 x}}{9 x}-\frac {16}{9} e^{2 x} x+\frac {1}{2} e^{4 x} x+2 e^{2 x} x^2+e^{4 x} x^2+\frac {68 \text {Ei}(x)}{81}+\frac {4 \text {Ei}(2 x)}{9}+\frac {8 \text {Ei}(4 x)}{9}-\frac {4}{81} \int \frac {e^{4 x}}{x^4} \, dx+\frac {16}{243} \int \frac {e^{4 x}}{x^3} \, dx+\frac {4}{27} \int \frac {e^{3 x}}{x^3} \, dx-\frac {2}{9} \int \frac {e^{3 x}}{x^2} \, dx-\frac {1}{2} \int e^{4 x} \, dx+\frac {20}{27} \int \frac {e^{2 x}}{x^2} \, dx-\frac {68}{81} \int \frac {e^x}{x} \, dx-\frac {8}{9} \int \frac {e^{4 x}}{x} \, dx+\frac {16}{9} \int e^{2 x} \, dx-\frac {52}{27} \int \frac {e^{2 x}}{x} \, dx-2 \int e^{4 x} x \, dx-4 \int e^{2 x} x \, dx\\ &=-\frac {4 e^x}{9}-e^{2 x}-\frac {4 e^{3 x}}{9}-\frac {e^{4 x}}{8}+\frac {1}{324} (25-18 x)^2+\frac {e^{4 x}}{81 x^4}-\frac {4 e^{3 x}}{81 x^3}-\frac {10 e^{2 x}}{27 x^2}-\frac {8 e^{4 x}}{243 x^2}+\frac {68 e^x}{81 x}+\frac {2 e^{2 x}}{9 x}+\frac {2 e^{3 x}}{9 x}+\frac {2 e^{4 x}}{9 x}-\frac {34}{9} e^{2 x} x+2 e^{2 x} x^2+e^{4 x} x^2-\frac {40 \text {Ei}(2 x)}{27}-\frac {16}{243} \int \frac {e^{4 x}}{x^3} \, dx+\frac {32}{243} \int \frac {e^{4 x}}{x^2} \, dx+\frac {2}{9} \int \frac {e^{3 x}}{x^2} \, dx+\frac {1}{2} \int e^{4 x} \, dx-\frac {2}{3} \int \frac {e^{3 x}}{x} \, dx+\frac {40}{27} \int \frac {e^{2 x}}{x} \, dx+2 \int e^{2 x} \, dx\\ &=-\frac {4 e^x}{9}-\frac {4 e^{3 x}}{9}+\frac {1}{324} (25-18 x)^2+\frac {e^{4 x}}{81 x^4}-\frac {4 e^{3 x}}{81 x^3}-\frac {10 e^{2 x}}{27 x^2}+\frac {68 e^x}{81 x}+\frac {2 e^{2 x}}{9 x}+\frac {22 e^{4 x}}{243 x}-\frac {34}{9} e^{2 x} x+2 e^{2 x} x^2+e^{4 x} x^2-\frac {2 \text {Ei}(3 x)}{3}-\frac {32}{243} \int \frac {e^{4 x}}{x^2} \, dx+\frac {128}{243} \int \frac {e^{4 x}}{x} \, dx+\frac {2}{3} \int \frac {e^{3 x}}{x} \, dx\\ &=-\frac {4 e^x}{9}-\frac {4 e^{3 x}}{9}+\frac {1}{324} (25-18 x)^2+\frac {e^{4 x}}{81 x^4}-\frac {4 e^{3 x}}{81 x^3}-\frac {10 e^{2 x}}{27 x^2}+\frac {68 e^x}{81 x}+\frac {2 e^{2 x}}{9 x}+\frac {2 e^{4 x}}{9 x}-\frac {34}{9} e^{2 x} x+2 e^{2 x} x^2+e^{4 x} x^2+\frac {128 \text {Ei}(4 x)}{243}-\frac {128}{243} \int \frac {e^{4 x}}{x} \, dx\\ &=-\frac {4 e^x}{9}-\frac {4 e^{3 x}}{9}+\frac {1}{324} (25-18 x)^2+\frac {e^{4 x}}{81 x^4}-\frac {4 e^{3 x}}{81 x^3}-\frac {10 e^{2 x}}{27 x^2}+\frac {68 e^x}{81 x}+\frac {2 e^{2 x}}{9 x}+\frac {2 e^{4 x}}{9 x}-\frac {34}{9} e^{2 x} x+2 e^{2 x} x^2+e^{4 x} x^2\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.06, size = 82, normalized size = 2.65 \begin {gather*} \frac {1}{81} \left (e^{3 x} \left (-36-\frac {4}{x^3}\right )+e^x \left (-36+\frac {68}{x}\right )-225 x+81 x^2+e^{4 x} \left (\frac {1}{x^4}+\frac {18}{x}+81 x^2\right )+e^{2 x} \left (-\frac {30}{x^2}+\frac {18}{x}-306 x+162 x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 88, normalized size = 2.84 \begin {gather*} \frac {81 \, x^{6} - 225 \, x^{5} + {\left (81 \, x^{6} + 18 \, x^{3} + 1\right )} e^{\left (4 \, x\right )} - 4 \, {\left (9 \, x^{4} + x\right )} e^{\left (3 \, x\right )} + 6 \, {\left (27 \, x^{6} - 51 \, x^{5} + 3 \, x^{3} - 5 \, x^{2}\right )} e^{\left (2 \, x\right )} - 4 \, {\left (9 \, x^{4} - 17 \, x^{3}\right )} e^{x}}{81 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 104, normalized size = 3.35 \begin {gather*} \frac {81 \, x^{6} e^{\left (4 \, x\right )} + 162 \, x^{6} e^{\left (2 \, x\right )} + 81 \, x^{6} - 306 \, x^{5} e^{\left (2 \, x\right )} - 225 \, x^{5} - 36 \, x^{4} e^{\left (3 \, x\right )} - 36 \, x^{4} e^{x} + 18 \, x^{3} e^{\left (4 \, x\right )} + 18 \, x^{3} e^{\left (2 \, x\right )} + 68 \, x^{3} e^{x} - 30 \, x^{2} e^{\left (2 \, x\right )} - 4 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}}{81 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 81, normalized size = 2.61
method | result | size |
risch | \(x^{2}-\frac {25 x}{9}+\frac {\left (81 x^{6}+18 x^{3}+1\right ) {\mathrm e}^{4 x}}{81 x^{4}}-\frac {4 \left (9 x^{3}+1\right ) {\mathrm e}^{3 x}}{81 x^{3}}+\frac {2 \left (27 x^{4}-51 x^{3}+3 x -5\right ) {\mathrm e}^{2 x}}{27 x^{2}}-\frac {4 \left (9 x -17\right ) {\mathrm e}^{x}}{81 x}\) | \(81\) |
default | \(x^{2}-\frac {25 x}{9}-\frac {4 \,{\mathrm e}^{3 x}}{9}+\frac {{\mathrm e}^{4 x}}{81 x^{4}}-\frac {4 \,{\mathrm e}^{3 x}}{81 x^{3}}-\frac {10 \,{\mathrm e}^{2 x}}{27 x^{2}}+\frac {2 \,{\mathrm e}^{2 x}}{9 x}-\frac {34 x \,{\mathrm e}^{2 x}}{9}+x^{2} {\mathrm e}^{4 x}+\frac {68 \,{\mathrm e}^{x}}{81 x}+2 \,{\mathrm e}^{2 x} x^{2}+\frac {2 \,{\mathrm e}^{4 x}}{9 x}-\frac {4 \,{\mathrm e}^{x}}{9}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.45, size = 148, normalized size = 4.77 \begin {gather*} x^{2} + \frac {1}{8} \, {\left (8 \, x^{2} - 4 \, x + 1\right )} e^{\left (4 \, x\right )} + \frac {1}{8} \, {\left (4 \, x - 1\right )} e^{\left (4 \, x\right )} + {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} - \frac {8}{9} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} - \frac {25}{9} \, x + \frac {8}{9} \, {\rm Ei}\left (4 \, x\right ) + \frac {4}{9} \, {\rm Ei}\left (2 \, x\right ) + \frac {68}{81} \, {\rm Ei}\relax (x) - \frac {4}{9} \, e^{\left (3 \, x\right )} - \frac {17}{9} \, e^{\left (2 \, x\right )} - \frac {4}{9} \, e^{x} - \frac {68}{81} \, \Gamma \left (-1, -x\right ) - \frac {52}{27} \, \Gamma \left (-1, -2 \, x\right ) - \frac {8}{9} \, \Gamma \left (-1, -4 \, x\right ) - \frac {80}{27} \, \Gamma \left (-2, -2 \, x\right ) + \frac {4}{3} \, \Gamma \left (-2, -3 \, x\right ) + 4 \, \Gamma \left (-3, -3 \, x\right ) + \frac {256}{81} \, \Gamma \left (-3, -4 \, x\right ) + \frac {1024}{81} \, \Gamma \left (-4, -4 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.81, size = 86, normalized size = 2.77 \begin {gather*} x^2\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )-\frac {4\,{\mathrm {e}}^x}{9}-\frac {4\,{\mathrm {e}}^{3\,x}}{9}+\frac {\frac {{\mathrm {e}}^{4\,x}}{81}+x^3\,\left (\frac {2\,{\mathrm {e}}^{2\,x}}{9}+\frac {2\,{\mathrm {e}}^{4\,x}}{9}+\frac {68\,{\mathrm {e}}^x}{81}\right )-\frac {4\,x\,{\mathrm {e}}^{3\,x}}{81}-\frac {10\,x^2\,{\mathrm {e}}^{2\,x}}{27}}{x^4}-x\,\left (\frac {34\,{\mathrm {e}}^{2\,x}}{9}+\frac {25}{9}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.24, size = 88, normalized size = 2.84 \begin {gather*} x^{2} - \frac {25 x}{9} + \frac {\left (- 6377292 x^{10} - 708588 x^{7}\right ) e^{3 x} + \left (- 6377292 x^{10} + 12045996 x^{9}\right ) e^{x} + \left (14348907 x^{12} + 3188646 x^{9} + 177147 x^{6}\right ) e^{4 x} + \left (28697814 x^{12} - 54206982 x^{11} + 3188646 x^{9} - 5314410 x^{8}\right ) e^{2 x}}{14348907 x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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