Integrand size = 35, antiderivative size = 23 \[ \int \frac {1}{25} e^{\frac {1}{25} \left (-150+125 x+64 x^2\right )} \left (125+575 x+515 x^2-128 x^3\right ) \, dx=e^{-6+x+4 \left (x+\frac {16 x^2}{25}\right )} (5-x) x \]
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Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.61, number of steps used = 28, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {12, 6874, 2267, 2266, 2235, 2276, 2272, 2273} \[ \int \frac {1}{25} e^{\frac {1}{25} \left (-150+125 x+64 x^2\right )} \left (125+575 x+515 x^2-128 x^3\right ) \, dx=5 e^{\frac {64 x^2}{25}+5 x-6} x-e^{\frac {64 x^2}{25}+5 x-6} x^2 \]
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Rule 12
Rule 2235
Rule 2266
Rule 2267
Rule 2272
Rule 2273
Rule 2276
Rule 6874
Rubi steps \begin{align*} \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{align*}
Time = 0.49 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {1}{25} e^{\frac {1}{25} \left (-150+125 x+64 x^2\right )} \left (125+575 x+515 x^2-128 x^3\right ) \, dx=-e^{-6+5 x+\frac {64 x^2}{25}} (-5+x) x \]
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Time = 0.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78
method | result | size |
gosper | \(-{\mathrm e}^{\frac {64}{25} x^{2}+5 x -6} \left (-5+x \right ) x\) | \(18\) |
risch | \(\frac {\left (-25 x^{2}+125 x \right ) {\mathrm e}^{\frac {64}{25} x^{2}+5 x -6}}{25}\) | \(23\) |
norman | \(5 x \,{\mathrm e}^{\frac {64}{25} x^{2}+5 x -6}-x^{2} {\mathrm e}^{\frac {64}{25} x^{2}+5 x -6}\) | \(32\) |
parallelrisch | \(5 x \,{\mathrm e}^{\frac {64}{25} x^{2}+5 x -6}-x^{2} {\mathrm e}^{\frac {64}{25} x^{2}+5 x -6}\) | \(32\) |
default | \(-\frac {25 i {\mathrm e}^{-6} \sqrt {\pi }\, {\mathrm e}^{-\frac {625}{256}} \operatorname {erf}\left (\frac {8}{5} i x +\frac {25}{16} i\right )}{16}+23 \,{\mathrm e}^{-6} \left (\frac {25 \,{\mathrm e}^{\frac {64}{25} x^{2}+5 x}}{128}+\frac {625 i \sqrt {\pi }\, {\mathrm e}^{-\frac {625}{256}} \operatorname {erf}\left (\frac {8}{5} i x +\frac {25}{16} i\right )}{2048}\right )+\frac {103 \,{\mathrm e}^{-6} \left (\frac {25 x \,{\mathrm e}^{\frac {64}{25} x^{2}+5 x}}{128}-\frac {3125 \,{\mathrm e}^{\frac {64}{25} x^{2}+5 x}}{16384}-\frac {62125 i \sqrt {\pi }\, {\mathrm e}^{-\frac {625}{256}} \operatorname {erf}\left (\frac {8}{5} i x +\frac {25}{16} i\right )}{262144}\right )}{5}-\frac {128 \,{\mathrm e}^{-6} \left (\frac {25 x^{2} {\mathrm e}^{\frac {64}{25} x^{2}+5 x}}{128}-\frac {3125 x \,{\mathrm e}^{\frac {64}{25} x^{2}+5 x}}{16384}+\frac {230625 \,{\mathrm e}^{\frac {64}{25} x^{2}+5 x}}{2097152}+\frac {3765625 i \sqrt {\pi }\, {\mathrm e}^{-\frac {625}{256}} \operatorname {erf}\left (\frac {8}{5} i x +\frac {25}{16} i\right )}{33554432}\right )}{25}\) | \(160\) |
parts | \(\frac {8 i \sqrt {\pi }\, {\mathrm e}^{-\frac {2161}{256}} \operatorname {erf}\left (\frac {8}{5} i x +\frac {25}{16} i\right ) x^{3}}{5}-\frac {103 i \sqrt {\pi }\, {\mathrm e}^{-\frac {2161}{256}} \operatorname {erf}\left (\frac {8}{5} i x +\frac {25}{16} i\right ) x^{2}}{16}-\frac {115 i \sqrt {\pi }\, {\mathrm e}^{-\frac {2161}{256}} \operatorname {erf}\left (\frac {8}{5} i x +\frac {25}{16} i\right ) x}{16}-\frac {25 i \sqrt {\pi }\, {\mathrm e}^{-\frac {2161}{256}} \operatorname {erf}\left (\frac {8}{5} i x +\frac {25}{16} i\right )}{16}-\frac {i {\mathrm e}^{-\frac {2161}{256}} \left (128 x^{3} \operatorname {erf}\left (\frac {i \left (128 x +125\right )}{80}\right ) \sqrt {\pi }-515 \sqrt {\pi }\, x^{2} \operatorname {erf}\left (\frac {i \left (128 x +125\right )}{80}\right )-80 i x^{2} {\mathrm e}^{\frac {\left (128 x +125\right )^{2}}{6400}}-575 \sqrt {\pi }\, x \,\operatorname {erf}\left (\frac {i \left (128 x +125\right )}{80}\right )+400 i x \,{\mathrm e}^{\frac {\left (128 x +125\right )^{2}}{6400}}-125 \,\operatorname {erf}\left (\frac {i \left (128 x +125\right )}{80}\right ) \sqrt {\pi }\right )}{80}\) | \(172\) |
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Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {1}{25} e^{\frac {1}{25} \left (-150+125 x+64 x^2\right )} \left (125+575 x+515 x^2-128 x^3\right ) \, dx=-{\left (x^{2} - 5 \, x\right )} e^{\left (\frac {64}{25} \, x^{2} + 5 \, x - 6\right )} \]
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Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{25} e^{\frac {1}{25} \left (-150+125 x+64 x^2\right )} \left (125+575 x+515 x^2-128 x^3\right ) \, dx=\left (- x^{2} + 5 x\right ) e^{\frac {64 x^{2}}{25} + 5 x - 6} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.33 (sec) , antiderivative size = 251, normalized size of antiderivative = 10.91 \[ \int \frac {1}{25} e^{\frac {1}{25} \left (-150+125 x+64 x^2\right )} \left (125+575 x+515 x^2-128 x^3\right ) \, dx=-\frac {25}{16} i \, \sqrt {\pi } \operatorname {erf}\left (\frac {8}{5} i \, x + \frac {25}{16} i\right ) e^{\left (-\frac {2161}{256}\right )} - \frac {25}{262144} \, {\left (\frac {19200 \, {\left (128 \, x + 125\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {1}{6400} \, {\left (128 \, x + 125\right )}^{2}\right )}{\left (-{\left (128 \, x + 125\right )}^{2}\right )^{\frac {3}{2}}} - \frac {15625 \, \sqrt {\pi } {\left (128 \, x + 125\right )} {\left (\operatorname {erf}\left (\frac {1}{80} \, \sqrt {-{\left (128 \, x + 125\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (128 \, x + 125\right )}^{2}}} + 30000 \, e^{\left (\frac {1}{6400} \, {\left (128 \, x + 125\right )}^{2}\right )} - 4096 \, \Gamma \left (2, -\frac {1}{6400} \, {\left (128 \, x + 125\right )}^{2}\right )\right )} e^{\left (-\frac {2161}{256}\right )} - \frac {2575}{262144} \, {\left (\frac {256 \, {\left (128 \, x + 125\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {1}{6400} \, {\left (128 \, x + 125\right )}^{2}\right )}{\left (-{\left (128 \, x + 125\right )}^{2}\right )^{\frac {3}{2}}} - \frac {625 \, \sqrt {\pi } {\left (128 \, x + 125\right )} {\left (\operatorname {erf}\left (\frac {1}{80} \, \sqrt {-{\left (128 \, x + 125\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (128 \, x + 125\right )}^{2}}} + 800 \, e^{\left (\frac {1}{6400} \, {\left (128 \, x + 125\right )}^{2}\right )}\right )} e^{\left (-\frac {2161}{256}\right )} - \frac {575}{2048} \, {\left (\frac {25 \, \sqrt {\pi } {\left (128 \, x + 125\right )} {\left (\operatorname {erf}\left (\frac {1}{80} \, \sqrt {-{\left (128 \, x + 125\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (128 \, x + 125\right )}^{2}}} - 16 \, e^{\left (\frac {1}{6400} \, {\left (128 \, x + 125\right )}^{2}\right )}\right )} e^{\left (-\frac {2161}{256}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{25} e^{\frac {1}{25} \left (-150+125 x+64 x^2\right )} \left (125+575 x+515 x^2-128 x^3\right ) \, dx=-\frac {1}{16384} \, {\left ({\left (128 \, x + 125\right )}^{2} - 113920 \, x - 15625\right )} e^{\left (\frac {64}{25} \, x^{2} + 5 \, x - 6\right )} \]
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Time = 0.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {1}{25} e^{\frac {1}{25} \left (-150+125 x+64 x^2\right )} \left (125+575 x+515 x^2-128 x^3\right ) \, dx=-x\,{\mathrm {e}}^{\frac {64\,x^2}{25}+5\,x-6}\,\left (x-5\right ) \]
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