Optimal. Leaf size=28 \[ \frac {e^{2 x} x^2 \left (-1+e^{-2 x} x\right )^2}{25 (3+x)^2} \]
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Rubi [B] time = 0.99, antiderivative size = 109, normalized size of antiderivative = 3.89, number of steps used = 28, number of rules used = 9, integrand size = 83, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {6688, 12, 6742, 74, 2199, 2194, 2176, 2177, 2178} \begin {gather*} -\frac {2 x^3}{25 (x+3)^2}+\frac {1}{25} e^{-2 x} x^2-\frac {6}{25} e^{-2 x} x+\frac {27 e^{-2 x}}{25}+\frac {e^{2 x}}{25}-\frac {108 e^{-2 x}}{25 (x+3)}-\frac {6 e^{2 x}}{25 (x+3)}+\frac {81 e^{-2 x}}{25 (x+3)^2}+\frac {9 e^{2 x}}{25 (x+3)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 74
Rule 2176
Rule 2177
Rule 2178
Rule 2194
Rule 2199
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{-2 x} x \left (-e^{2 x} x (9+x)-x^2 \left (-6+2 x+x^2\right )+e^{4 x} \left (3+3 x+x^2\right )\right )}{25 (3+x)^3} \, dx\\ &=\frac {2}{25} \int \frac {e^{-2 x} x \left (-e^{2 x} x (9+x)-x^2 \left (-6+2 x+x^2\right )+e^{4 x} \left (3+3 x+x^2\right )\right )}{(3+x)^3} \, dx\\ &=\frac {2}{25} \int \left (-\frac {x^2 (9+x)}{(3+x)^3}-\frac {e^{-2 x} x^3 \left (-6+2 x+x^2\right )}{(3+x)^3}+\frac {e^{2 x} x \left (3+3 x+x^2\right )}{(3+x)^3}\right ) \, dx\\ &=-\left (\frac {2}{25} \int \frac {x^2 (9+x)}{(3+x)^3} \, dx\right )-\frac {2}{25} \int \frac {e^{-2 x} x^3 \left (-6+2 x+x^2\right )}{(3+x)^3} \, dx+\frac {2}{25} \int \frac {e^{2 x} x \left (3+3 x+x^2\right )}{(3+x)^3} \, dx\\ &=-\frac {2 x^3}{25 (3+x)^2}-\frac {2}{25} \int \left (30 e^{-2 x}-7 e^{-2 x} x+e^{-2 x} x^2+\frac {81 e^{-2 x}}{(3+x)^3}+\frac {27 e^{-2 x}}{(3+x)^2}-\frac {108 e^{-2 x}}{3+x}\right ) \, dx+\frac {2}{25} \int \left (e^{2 x}-\frac {9 e^{2 x}}{(3+x)^3}+\frac {12 e^{2 x}}{(3+x)^2}-\frac {6 e^{2 x}}{3+x}\right ) \, dx\\ &=-\frac {2 x^3}{25 (3+x)^2}+\frac {2}{25} \int e^{2 x} \, dx-\frac {2}{25} \int e^{-2 x} x^2 \, dx-\frac {12}{25} \int \frac {e^{2 x}}{3+x} \, dx+\frac {14}{25} \int e^{-2 x} x \, dx-\frac {18}{25} \int \frac {e^{2 x}}{(3+x)^3} \, dx+\frac {24}{25} \int \frac {e^{2 x}}{(3+x)^2} \, dx-\frac {54}{25} \int \frac {e^{-2 x}}{(3+x)^2} \, dx-\frac {12}{5} \int e^{-2 x} \, dx-\frac {162}{25} \int \frac {e^{-2 x}}{(3+x)^3} \, dx+\frac {216}{25} \int \frac {e^{-2 x}}{3+x} \, dx\\ &=\frac {6 e^{-2 x}}{5}+\frac {e^{2 x}}{25}-\frac {7}{25} e^{-2 x} x+\frac {1}{25} e^{-2 x} x^2+\frac {81 e^{-2 x}}{25 (3+x)^2}+\frac {9 e^{2 x}}{25 (3+x)^2}-\frac {2 x^3}{25 (3+x)^2}+\frac {54 e^{-2 x}}{25 (3+x)}-\frac {24 e^{2 x}}{25 (3+x)}+\frac {216}{25} e^6 \text {Ei}(-2 (3+x))-\frac {12 \text {Ei}(2 (3+x))}{25 e^6}-\frac {2}{25} \int e^{-2 x} x \, dx+\frac {7}{25} \int e^{-2 x} \, dx-\frac {18}{25} \int \frac {e^{2 x}}{(3+x)^2} \, dx+\frac {48}{25} \int \frac {e^{2 x}}{3+x} \, dx+\frac {108}{25} \int \frac {e^{-2 x}}{3+x} \, dx+\frac {162}{25} \int \frac {e^{-2 x}}{(3+x)^2} \, dx\\ &=\frac {53 e^{-2 x}}{50}+\frac {e^{2 x}}{25}-\frac {6}{25} e^{-2 x} x+\frac {1}{25} e^{-2 x} x^2+\frac {81 e^{-2 x}}{25 (3+x)^2}+\frac {9 e^{2 x}}{25 (3+x)^2}-\frac {2 x^3}{25 (3+x)^2}-\frac {108 e^{-2 x}}{25 (3+x)}-\frac {6 e^{2 x}}{25 (3+x)}+\frac {324}{25} e^6 \text {Ei}(-2 (3+x))+\frac {36 \text {Ei}(2 (3+x))}{25 e^6}-\frac {1}{25} \int e^{-2 x} \, dx-\frac {36}{25} \int \frac {e^{2 x}}{3+x} \, dx-\frac {324}{25} \int \frac {e^{-2 x}}{3+x} \, dx\\ &=\frac {27 e^{-2 x}}{25}+\frac {e^{2 x}}{25}-\frac {6}{25} e^{-2 x} x+\frac {1}{25} e^{-2 x} x^2+\frac {81 e^{-2 x}}{25 (3+x)^2}+\frac {9 e^{2 x}}{25 (3+x)^2}-\frac {2 x^3}{25 (3+x)^2}-\frac {108 e^{-2 x}}{25 (3+x)}-\frac {6 e^{2 x}}{25 (3+x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.23, size = 50, normalized size = 1.79 \begin {gather*} -\frac {e^{-2 x} \left (-e^{4 x} x^2-x^4+2 e^{2 x} \left (54+36 x+6 x^2+x^3\right )\right )}{25 (3+x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 47, normalized size = 1.68 \begin {gather*} \frac {{\left (x^{4} + x^{2} e^{\left (4 \, x\right )} - 2 \, {\left (x^{3} + 6 \, x^{2} + 36 \, x + 54\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )}}{25 \, {\left (x^{2} + 6 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 43, normalized size = 1.54 \begin {gather*} \frac {x^{4} e^{\left (-2 \, x\right )} - 2 \, x^{3} + x^{2} e^{\left (2 \, x\right )} - 12 \, x^{2} - 72 \, x - 108}{25 \, {\left (x^{2} + 6 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 49, normalized size = 1.75
method | result | size |
risch | \(-\frac {2 x}{25}+\frac {-\frac {54 x}{25}-\frac {108}{25}}{x^{2}+6 x +9}+\frac {x^{2} {\mathrm e}^{2 x}}{25 \left (3+x \right )^{2}}+\frac {x^{4} {\mathrm e}^{-2 x}}{25 \left (3+x \right )^{2}}\) | \(49\) |
default | \(-\frac {54}{25 \left (3+x \right )}+\frac {54}{25 \left (3+x \right )^{2}}-\frac {2 x}{25}-\frac {6 \,{\mathrm e}^{-2 x}}{25}-\frac {162 \,{\mathrm e}^{-2 x} \left (4 x +11\right )}{25 \left (x^{2}+6 x +9\right )}+\frac {\left (2 x -17\right ) {\mathrm e}^{-2 x}}{25}-\frac {54 \,{\mathrm e}^{-2 x} \left (14 x +39\right )}{25 \left (x^{2}+6 x +9\right )}+\frac {\left (x^{2}-8 x +50\right ) {\mathrm e}^{-2 x}}{25}+\frac {81 \,{\mathrm e}^{-2 x} \left (16 x +45\right )}{25 \left (x^{2}+6 x +9\right )}-\frac {6 \,{\mathrm e}^{2 x}}{25 \left (3+x \right )}+\frac {9 \,{\mathrm e}^{2 x}}{25 \left (3+x \right )^{2}}+\frac {{\mathrm e}^{2 x}}{25}\) | \(141\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 43, normalized size = 1.54 \begin {gather*} \frac {x^{4} e^{\left (-2 \, x\right )} - 2 \, x^{3} + x^{2} e^{\left (2 \, x\right )} - 12 \, x^{2} - 72 \, x - 108}{25 \, {\left (x^{2} + 6 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 24, normalized size = 0.86 \begin {gather*} \frac {x^2\,{\mathrm {e}}^{-2\,x}\,{\left (x-{\mathrm {e}}^{2\,x}\right )}^2}{25\,{\left (x+3\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.20, size = 80, normalized size = 2.86 \begin {gather*} - \frac {2 x}{25} - \frac {54 x + 108}{25 x^{2} + 150 x + 225} + \frac {\left (25 x^{4} + 150 x^{3} + 225 x^{2}\right ) e^{2 x} + \left (25 x^{6} + 150 x^{5} + 225 x^{4}\right ) e^{- 2 x}}{625 x^{4} + 7500 x^{3} + 33750 x^{2} + 67500 x + 50625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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