Optimal. Leaf size=35 \[ \left (e^x+x-\frac {(-1+x)^2-x-\frac {5+x}{5 (1-x)}}{x}\right )^2 \]
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Rubi [A] time = 0.20, antiderivative size = 43, normalized size of antiderivative = 1.23, number of steps used = 10, number of rules used = 6, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6688, 2194, 37, 2199, 2177, 2178} \begin {gather*} \frac {9 (7-5 x)^2}{25 (1-x)^2}+6 e^x+e^{2 x}+\frac {12 e^x}{5 (1-x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 2177
Rule 2178
Rule 2194
Rule 2199
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 e^{2 x}+\frac {36 (-7+5 x)}{25 (-1+x)^3}+\frac {6 e^x \left (9-12 x+5 x^2\right )}{5 (-1+x)^2}\right ) \, dx\\ &=\frac {6}{5} \int \frac {e^x \left (9-12 x+5 x^2\right )}{(-1+x)^2} \, dx+\frac {36}{25} \int \frac {-7+5 x}{(-1+x)^3} \, dx+2 \int e^{2 x} \, dx\\ &=e^{2 x}+\frac {9 (7-5 x)^2}{25 (1-x)^2}+\frac {6}{5} \int \left (5 e^x+\frac {2 e^x}{(-1+x)^2}-\frac {2 e^x}{-1+x}\right ) \, dx\\ &=e^{2 x}+\frac {9 (7-5 x)^2}{25 (1-x)^2}+\frac {12}{5} \int \frac {e^x}{(-1+x)^2} \, dx-\frac {12}{5} \int \frac {e^x}{-1+x} \, dx+6 \int e^x \, dx\\ &=6 e^x+e^{2 x}+\frac {9 (7-5 x)^2}{25 (1-x)^2}+\frac {12 e^x}{5 (1-x)}-\frac {12}{5} e \text {Ei}(-1+x)+\frac {12}{5} \int \frac {e^x}{-1+x} \, dx\\ &=6 e^x+e^{2 x}+\frac {9 (7-5 x)^2}{25 (1-x)^2}+\frac {12 e^x}{5 (1-x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.08, size = 43, normalized size = 1.23 \begin {gather*} 6 e^x+e^{2 x}+\frac {12 e^x}{5 (1-x)}+\frac {36}{25 (-1+x)^2}-\frac {36}{5 (-1+x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 45, normalized size = 1.29 \begin {gather*} \frac {25 \, {\left (x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + 30 \, {\left (5 \, x^{2} - 12 \, x + 7\right )} e^{x} - 180 \, x + 216}{25 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 55, normalized size = 1.57 \begin {gather*} \frac {25 \, x^{2} e^{\left (2 \, x\right )} + 150 \, x^{2} e^{x} - 50 \, x e^{\left (2 \, x\right )} - 360 \, x e^{x} - 180 \, x + 25 \, e^{\left (2 \, x\right )} + 210 \, e^{x} + 216}{25 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 33, normalized size = 0.94
method | result | size |
default | \(\frac {36}{25 \left (x -1\right )^{2}}-\frac {36}{5 \left (x -1\right )}-\frac {12 \,{\mathrm e}^{x}}{5 \left (x -1\right )}+6 \,{\mathrm e}^{x}+{\mathrm e}^{2 x}\) | \(33\) |
risch | \(\frac {-\frac {36 x}{5}+\frac {216}{25}}{x^{2}-2 x +1}+{\mathrm e}^{2 x}+\frac {6 \left (5 x -7\right ) {\mathrm e}^{x}}{5 \left (x -1\right )}\) | \(36\) |
norman | \(\frac {{\mathrm e}^{2 x}-\frac {36 x}{5}+{\mathrm e}^{2 x} x^{2}-2 x \,{\mathrm e}^{2 x}-\frac {72 \,{\mathrm e}^{x} x}{5}+6 \,{\mathrm e}^{x} x^{2}+\frac {42 \,{\mathrm e}^{x}}{5}+\frac {216}{25}}{\left (x -1\right )^{2}}\) | \(47\) |
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*} \frac {5 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )} e^{\left (2 \, x\right )} + 6 \, {\left (5 \, x^{3} - 17 \, x^{2} + 19 \, x\right )} e^{x}}{5 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {18 \, {\left (2 \, x - 1\right )}}{5 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {54 \, e E_{3}\left (-x + 1\right )}{5 \, {\left (x - 1\right )}^{2}} + \frac {126}{25 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {2}{25} \, \int \frac {15 \, {\left (2 \, x + 19\right )} e^{x}}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 30, normalized size = 0.86 \begin {gather*} {\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^x+\frac {\frac {12\,{\mathrm {e}}^x}{5}-x\,\left (\frac {12\,{\mathrm {e}}^x}{5}+\frac {36}{5}\right )+\frac {216}{25}}{{\left (x-1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 37, normalized size = 1.06 \begin {gather*} \frac {216 - 180 x}{25 x^{2} - 50 x + 25} + \frac {\left (5 x - 5\right ) e^{2 x} + \left (30 x - 42\right ) e^{x}}{5 x - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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