Integrand size = 8, antiderivative size = 96 \[ \int x^5 \text {erf}(b x) \, dx=\frac {5 e^{-b^2 x^2} x}{8 b^5 \sqrt {\pi }}+\frac {5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}-\frac {5 \text {erf}(b x)}{16 b^6}+\frac {1}{6} x^6 \text {erf}(b x) \]
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Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6496, 2243, 2236} \[ \int x^5 \text {erf}(b x) \, dx=-\frac {5 \text {erf}(b x)}{16 b^6}+\frac {x^5 e^{-b^2 x^2}}{6 \sqrt {\pi } b}+\frac {5 x e^{-b^2 x^2}}{8 \sqrt {\pi } b^5}+\frac {5 x^3 e^{-b^2 x^2}}{12 \sqrt {\pi } b^3}+\frac {1}{6} x^6 \text {erf}(b x) \]
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Rule 2236
Rule 2243
Rule 6496
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^6 \text {erf}(b x)-\frac {b \int e^{-b^2 x^2} x^6 \, dx}{3 \sqrt {\pi }} \\ & = \frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erf}(b x)-\frac {5 \int e^{-b^2 x^2} x^4 \, dx}{6 b \sqrt {\pi }} \\ & = \frac {5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erf}(b x)-\frac {5 \int e^{-b^2 x^2} x^2 \, dx}{4 b^3 \sqrt {\pi }} \\ & = \frac {5 e^{-b^2 x^2} x}{8 b^5 \sqrt {\pi }}+\frac {5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erf}(b x)-\frac {5 \int e^{-b^2 x^2} \, dx}{8 b^5 \sqrt {\pi }} \\ & = \frac {5 e^{-b^2 x^2} x}{8 b^5 \sqrt {\pi }}+\frac {5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}-\frac {5 \text {erf}(b x)}{16 b^6}+\frac {1}{6} x^6 \text {erf}(b x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.75 \[ \int x^5 \text {erf}(b x) \, dx=\frac {e^{-b^2 x^2} \left (30 b x+20 b^3 x^3+8 b^5 x^5+e^{b^2 x^2} \sqrt {\pi } \left (-15+8 b^6 x^6\right ) \text {erf}(b x)\right )}{48 b^6 \sqrt {\pi }} \]
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Time = 0.42 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.62
method | result | size |
meijerg | \(\frac {\frac {x b \left (28 b^{4} x^{4}+70 b^{2} x^{2}+105\right ) {\mathrm e}^{-b^{2} x^{2}}}{84}-\frac {\left (-56 b^{6} x^{6}+105\right ) \operatorname {erf}\left (b x \right ) \sqrt {\pi }}{168}}{2 b^{6} \sqrt {\pi }}\) | \(60\) |
parallelrisch | \(\frac {8 \,\operatorname {erf}\left (b x \right ) x^{6} b^{6} \sqrt {\pi }+8 \,{\mathrm e}^{-b^{2} x^{2}} x^{5} b^{5}+20 x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}+30 \,{\mathrm e}^{-b^{2} x^{2}} b x -15 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{48 b^{6} \sqrt {\pi }}\) | \(81\) |
derivativedivides | \(\frac {\frac {\operatorname {erf}\left (b x \right ) b^{6} x^{6}}{6}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{5} b^{5}}{2}-\frac {5 x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}}{4}-\frac {15 \,{\mathrm e}^{-b^{2} x^{2}} b x}{8}+\frac {15 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{16}}{3 \sqrt {\pi }}}{b^{6}}\) | \(83\) |
default | \(\frac {\frac {\operatorname {erf}\left (b x \right ) b^{6} x^{6}}{6}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{5} b^{5}}{2}-\frac {5 x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}}{4}-\frac {15 \,{\mathrm e}^{-b^{2} x^{2}} b x}{8}+\frac {15 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{16}}{3 \sqrt {\pi }}}{b^{6}}\) | \(83\) |
parts | \(\frac {x^{6} \operatorname {erf}\left (b x \right )}{6}-\frac {b \left (-\frac {x^{5} {\mathrm e}^{-b^{2} x^{2}}}{2 b^{2}}+\frac {-\frac {5 x^{3} {\mathrm e}^{-b^{2} x^{2}}}{4 b^{2}}+\frac {5 \left (-\frac {3 x \,{\mathrm e}^{-b^{2} x^{2}}}{4 b^{2}}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (b x \right )}{8 b^{3}}\right )}{2 b^{2}}}{b^{2}}\right )}{3 \sqrt {\pi }}\) | \(91\) |
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Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.66 \[ \int x^5 \text {erf}(b x) \, dx=\frac {2 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} + 10 \, b^{3} x^{3} + 15 \, b x\right )} e^{\left (-b^{2} x^{2}\right )} - {\left (15 \, \pi - 8 \, \pi b^{6} x^{6}\right )} \operatorname {erf}\left (b x\right )}{48 \, \pi b^{6}} \]
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Time = 0.40 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.92 \[ \int x^5 \text {erf}(b x) \, dx=\begin {cases} \frac {x^{6} \operatorname {erf}{\left (b x \right )}}{6} + \frac {x^{5} e^{- b^{2} x^{2}}}{6 \sqrt {\pi } b} + \frac {5 x^{3} e^{- b^{2} x^{2}}}{12 \sqrt {\pi } b^{3}} + \frac {5 x e^{- b^{2} x^{2}}}{8 \sqrt {\pi } b^{5}} - \frac {5 \operatorname {erf}{\left (b x \right )}}{16 b^{6}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.66 \[ \int x^5 \text {erf}(b x) \, dx=\frac {1}{6} \, x^{6} \operatorname {erf}\left (b x\right ) + \frac {b {\left (\frac {2 \, {\left (4 \, b^{4} x^{5} + 10 \, b^{2} x^{3} + 15 \, x\right )} e^{\left (-b^{2} x^{2}\right )}}{b^{6}} - \frac {15 \, \sqrt {\pi } \operatorname {erf}\left (b x\right )}{b^{7}}\right )}}{48 \, \sqrt {\pi }} \]
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Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.67 \[ \int x^5 \text {erf}(b x) \, dx=\frac {1}{6} \, x^{6} \operatorname {erf}\left (b x\right ) + \frac {b {\left (\frac {2 \, {\left (4 \, b^{4} x^{5} + 10 \, b^{2} x^{3} + 15 \, x\right )} e^{\left (-b^{2} x^{2}\right )}}{b^{6}} + \frac {15 \, \sqrt {\pi } \operatorname {erf}\left (-b x\right )}{b^{7}}\right )}}{48 \, \sqrt {\pi }} \]
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Time = 5.19 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.12 \[ \int x^5 \text {erf}(b x) \, dx=\frac {x^6\,\mathrm {erf}\left (b\,x\right )}{6}-\frac {5\,b\,x^7}{16\,{\left (b^2\,x^2\right )}^{7/2}}+\frac {x^5\,{\mathrm {e}}^{-b^2\,x^2}}{6\,b\,\sqrt {\pi }}+\frac {5\,x^3\,{\mathrm {e}}^{-b^2\,x^2}}{12\,b^3\,\sqrt {\pi }}+\frac {5\,x\,{\mathrm {e}}^{-b^2\,x^2}}{8\,b^5\,\sqrt {\pi }}+\frac {5\,b\,x^7\,\mathrm {erfc}\left (\sqrt {b^2\,x^2}\right )}{16\,{\left (b^2\,x^2\right )}^{7/2}} \]
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