Optimal. Leaf size=23 \[ e^{-6+x+4 \left (x+\frac {16 x^2}{25}\right )} (5-x) x \]
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Rubi [A] time = 0.36, antiderivative size = 37, normalized size of antiderivative = 1.61, number of steps used = 28, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {12, 6742, 2235, 2234, 2204, 2244, 2240, 2241} \begin {gather*} 5 e^{\frac {64 x^2}{25}+5 x-6} x-e^{\frac {64 x^2}{25}+5 x-6} x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2204
Rule 2234
Rule 2235
Rule 2240
Rule 2241
Rule 2244
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{25} \int e^{\frac {1}{25} \left (-150+125 x+64 x^2\right )} \left (125+575 x+515 x^2-128 x^3\right ) \, dx\\ &=\frac {1}{25} \int \left (125 e^{\frac {1}{25} \left (-150+125 x+64 x^2\right )}+575 e^{\frac {1}{25} \left (-150+125 x+64 x^2\right )} x+515 e^{\frac {1}{25} \left (-150+125 x+64 x^2\right )} x^2-128 e^{\frac {1}{25} \left (-150+125 x+64 x^2\right )} x^3\right ) \, dx\\ &=5 \int e^{\frac {1}{25} \left (-150+125 x+64 x^2\right )} \, dx-\frac {128}{25} \int e^{\frac {1}{25} \left (-150+125 x+64 x^2\right )} x^3 \, dx+\frac {103}{5} \int e^{\frac {1}{25} \left (-150+125 x+64 x^2\right )} x^2 \, dx+23 \int e^{\frac {1}{25} \left (-150+125 x+64 x^2\right )} x \, dx\\ &=5 \int e^{-6+5 x+\frac {64 x^2}{25}} \, dx-\frac {128}{25} \int e^{-6+5 x+\frac {64 x^2}{25}} x^3 \, dx+\frac {103}{5} \int e^{-6+5 x+\frac {64 x^2}{25}} x^2 \, dx+23 \int e^{-6+5 x+\frac {64 x^2}{25}} x \, dx\\ &=\frac {575}{128} e^{-6+5 x+\frac {64 x^2}{25}}+\frac {515}{128} e^{-6+5 x+\frac {64 x^2}{25}} x-e^{-6+5 x+\frac {64 x^2}{25}} x^2+2 \int e^{-6+5 x+\frac {64 x^2}{25}} x \, dx-\frac {515}{128} \int e^{-6+5 x+\frac {64 x^2}{25}} \, dx+5 \int e^{-6+5 x+\frac {64 x^2}{25}} x^2 \, dx-\frac {2575}{128} \int e^{-6+5 x+\frac {64 x^2}{25}} x \, dx-\frac {2875}{128} \int e^{-6+5 x+\frac {64 x^2}{25}} \, dx+\frac {5 \int e^{\frac {25}{256} \left (5+\frac {128 x}{25}\right )^2} \, dx}{e^{2161/256}}\\ &=\frac {15625 e^{-6+5 x+\frac {64 x^2}{25}}}{16384}+5 e^{-6+5 x+\frac {64 x^2}{25}} x-e^{-6+5 x+\frac {64 x^2}{25}} x^2+\frac {25 \sqrt {\pi } \text {erfi}\left (\frac {1}{80} (125+128 x)\right )}{16 e^{2161/256}}-\frac {125}{128} \int e^{-6+5 x+\frac {64 x^2}{25}} \, dx-\frac {125}{64} \int e^{-6+5 x+\frac {64 x^2}{25}} \, dx-\frac {625}{128} \int e^{-6+5 x+\frac {64 x^2}{25}} x \, dx+\frac {321875 \int e^{-6+5 x+\frac {64 x^2}{25}} \, dx}{16384}-\frac {515 \int e^{\frac {25}{256} \left (5+\frac {128 x}{25}\right )^2} \, dx}{128 e^{2161/256}}-\frac {2875 \int e^{\frac {25}{256} \left (5+\frac {128 x}{25}\right )^2} \, dx}{128 e^{2161/256}}\\ &=5 e^{-6+5 x+\frac {64 x^2}{25}} x-e^{-6+5 x+\frac {64 x^2}{25}} x^2-\frac {6875 \sqrt {\pi } \text {erfi}\left (\frac {1}{80} (125+128 x)\right )}{1024 e^{2161/256}}+\frac {78125 \int e^{-6+5 x+\frac {64 x^2}{25}} \, dx}{16384}-\frac {125 \int e^{\frac {25}{256} \left (5+\frac {128 x}{25}\right )^2} \, dx}{128 e^{2161/256}}-\frac {125 \int e^{\frac {25}{256} \left (5+\frac {128 x}{25}\right )^2} \, dx}{64 e^{2161/256}}+\frac {321875 \int e^{\frac {25}{256} \left (5+\frac {128 x}{25}\right )^2} \, dx}{16384 e^{2161/256}}\\ &=5 e^{-6+5 x+\frac {64 x^2}{25}} x-e^{-6+5 x+\frac {64 x^2}{25}} x^2-\frac {390625 \sqrt {\pi } \text {erfi}\left (\frac {1}{80} (125+128 x)\right )}{262144 e^{2161/256}}+\frac {78125 \int e^{\frac {25}{256} \left (5+\frac {128 x}{25}\right )^2} \, dx}{16384 e^{2161/256}}\\ &=5 e^{-6+5 x+\frac {64 x^2}{25}} x-e^{-6+5 x+\frac {64 x^2}{25}} x^2\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 20, normalized size = 0.87 \begin {gather*} -e^{-6+5 x+\frac {64 x^2}{25}} (-5+x) x \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 20, normalized size = 0.87 \begin {gather*} -{\left (x^{2} - 5 \, x\right )} e^{\left (\frac {64}{25} \, x^{2} + 5 \, x - 6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 25, normalized size = 1.09 \begin {gather*} -\frac {1}{16384} \, {\left ({\left (128 \, x + 125\right )}^{2} - 113920 \, x - 15625\right )} e^{\left (\frac {64}{25} \, x^{2} + 5 \, x - 6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 18, normalized size = 0.78
method | result | size |
gosper | \(-{\mathrm e}^{\frac {64}{25} x^{2}+5 x -6} \left (x -5\right ) x\) | \(18\) |
risch | \(\frac {\left (-25 x^{2}+125 x \right ) {\mathrm e}^{\frac {64}{25} x^{2}+5 x -6}}{25}\) | \(23\) |
default | \(5 x \,{\mathrm e}^{\frac {64}{25} x^{2}+5 x -6}-x^{2} {\mathrm e}^{\frac {64}{25} x^{2}+5 x -6}\) | \(32\) |
norman | \(5 x \,{\mathrm e}^{\frac {64}{25} x^{2}+5 x -6}-x^{2} {\mathrm e}^{\frac {64}{25} x^{2}+5 x -6}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.53, size = 251, normalized size = 10.91 \begin {gather*} -\frac {25}{16} i \, \sqrt {\pi } \operatorname {erf}\left (\frac {8}{5} i \, x + \frac {25}{16} i\right ) e^{\left (-\frac {2161}{256}\right )} - \frac {25}{262144} \, {\left (\frac {19200 \, {\left (128 \, x + 125\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {1}{6400} \, {\left (128 \, x + 125\right )}^{2}\right )}{\left (-{\left (128 \, x + 125\right )}^{2}\right )^{\frac {3}{2}}} - \frac {15625 \, \sqrt {\pi } {\left (128 \, x + 125\right )} {\left (\operatorname {erf}\left (\frac {1}{80} \, \sqrt {-{\left (128 \, x + 125\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (128 \, x + 125\right )}^{2}}} + 30000 \, e^{\left (\frac {1}{6400} \, {\left (128 \, x + 125\right )}^{2}\right )} - 4096 \, \Gamma \left (2, -\frac {1}{6400} \, {\left (128 \, x + 125\right )}^{2}\right )\right )} e^{\left (-\frac {2161}{256}\right )} - \frac {2575}{262144} \, {\left (\frac {256 \, {\left (128 \, x + 125\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {1}{6400} \, {\left (128 \, x + 125\right )}^{2}\right )}{\left (-{\left (128 \, x + 125\right )}^{2}\right )^{\frac {3}{2}}} - \frac {625 \, \sqrt {\pi } {\left (128 \, x + 125\right )} {\left (\operatorname {erf}\left (\frac {1}{80} \, \sqrt {-{\left (128 \, x + 125\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (128 \, x + 125\right )}^{2}}} + 800 \, e^{\left (\frac {1}{6400} \, {\left (128 \, x + 125\right )}^{2}\right )}\right )} e^{\left (-\frac {2161}{256}\right )} - \frac {575}{2048} \, {\left (\frac {25 \, \sqrt {\pi } {\left (128 \, x + 125\right )} {\left (\operatorname {erf}\left (\frac {1}{80} \, \sqrt {-{\left (128 \, x + 125\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (128 \, x + 125\right )}^{2}}} - 16 \, e^{\left (\frac {1}{6400} \, {\left (128 \, x + 125\right )}^{2}\right )}\right )} e^{\left (-\frac {2161}{256}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 17, normalized size = 0.74 \begin {gather*} -x\,{\mathrm {e}}^{\frac {64\,x^2}{25}+5\,x-6}\,\left (x-5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 19, normalized size = 0.83 \begin {gather*} \left (- x^{2} + 5 x\right ) e^{\frac {64 x^{2}}{25} + 5 x - 6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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