Optimal. Leaf size=96 \[ \frac{5 \text{Erf}(b x)}{16 b^6}-\frac{x^5 e^{-b^2 x^2}}{6 \sqrt{\pi } b}-\frac{5 x^3 e^{-b^2 x^2}}{12 \sqrt{\pi } b^3}-\frac{5 x e^{-b^2 x^2}}{8 \sqrt{\pi } b^5}+\frac{1}{6} x^6 \text{Erfc}(b x) \]
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Rubi [A] time = 0.0897058, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6362, 2212, 2205} \[ \frac{5 \text{Erf}(b x)}{16 b^6}-\frac{x^5 e^{-b^2 x^2}}{6 \sqrt{\pi } b}-\frac{5 x^3 e^{-b^2 x^2}}{12 \sqrt{\pi } b^3}-\frac{5 x e^{-b^2 x^2}}{8 \sqrt{\pi } b^5}+\frac{1}{6} x^6 \text{Erfc}(b x) \]
Antiderivative was successfully verified.
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Rule 6362
Rule 2212
Rule 2205
Rubi steps
\begin{align*} \int x^5 \text{erfc}(b x) \, dx &=\frac{1}{6} x^6 \text{erfc}(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{erfc}(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{erfc}(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{erfc}(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{erfc}(b x)\\ \end{align*}
Mathematica [A] time = 0.0581878, size = 62, normalized size = 0.65 \[ \frac{1}{48} \left (\frac{15 \text{Erf}(b x)}{b^6}-\frac{2 x e^{-b^2 x^2} \left (4 b^4 x^4+10 b^2 x^2+15\right )}{\sqrt{\pi } b^5}+8 x^6 \text{Erfc}(b x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 83, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{6}} \left ({\frac{{b}^{6}{x}^{6}{\it erfc} \left ( bx \right ) }{6}}+{\frac{1}{3\,\sqrt{\pi }} \left ( -{\frac{{b}^{5}{x}^{5}}{2\,{{\rm e}^{{b}^{2}{x}^{2}}}}}-{\frac{5\,{x}^{3}{b}^{3}}{4\,{{\rm e}^{{b}^{2}{x}^{2}}}}}-{\frac{15\,bx}{8\,{{\rm e}^{{b}^{2}{x}^{2}}}}}+{\frac{15\,\sqrt{\pi }{\it Erf} \left ( bx \right ) }{16}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00688, size = 85, normalized size = 0.89 \begin{align*} \frac{1}{6} \, x^{6} \operatorname{erfc}\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 }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.40923, size = 167, normalized size = 1.74 \begin{align*} \frac{8 \, \pi b^{6} x^{6} - 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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.05136, size = 92, normalized size = 0.96 \begin{align*} \begin{cases} \frac{x^{6} \operatorname{erfc}{\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{erfc}{\left (b x \right )}}{16 b^{6}} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37257, size = 93, normalized size = 0.97 \begin{align*} -\frac{1}{6} \, x^{6} \operatorname{erf}\left (b x\right ) + \frac{1}{6} \, x^{6} - \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 }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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