Optimal. Leaf size=96 \[ -\frac{5 \text{Erf}(b x)}{16 b^6}+\frac{x^5 e^{-b^2 x^2}}{6 \sqrt{\pi } b}+\frac{5 x^3 e^{-b^2 x^2}}{12 \sqrt{\pi } b^3}+\frac{5 x e^{-b^2 x^2}}{8 \sqrt{\pi } b^5}+\frac{1}{6} x^6 \text{Erf}(b x) \]
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Rubi [A] time = 0.0961821, 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 = {6361, 2212, 2205} \[ -\frac{5 \text{Erf}(b x)}{16 b^6}+\frac{x^5 e^{-b^2 x^2}}{6 \sqrt{\pi } b}+\frac{5 x^3 e^{-b^2 x^2}}{12 \sqrt{\pi } b^3}+\frac{5 x e^{-b^2 x^2}}{8 \sqrt{\pi } b^5}+\frac{1}{6} x^6 \text{Erf}(b x) \]
Antiderivative was successfully verified.
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Rule 6361
Rule 2212
Rule 2205
Rubi steps
\begin{align*} \int x^5 \text{erf}(b x) \, dx &=\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*}
Mathematica [A] time = 0.0254642, size = 72, normalized size = 0.75 \[ \frac{e^{-b^2 x^2} \left (\sqrt{\pi } e^{b^2 x^2} \left (8 b^6 x^6-15\right ) \text{Erf}(b x)+8 b^5 x^5+20 b^3 x^3+30 b x\right )}{48 \sqrt{\pi } b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 83, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{6}} \left ({\frac{{\it Erf} \left ( bx \right ){b}^{6}{x}^{6}}{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 = 1.0966, size = 85, normalized size = 0.89 \begin{align*} \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 }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.51827, size = 147, normalized size = 1.53 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.46549, size = 88, normalized size = 0.92 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17509, size = 86, normalized size = 0.9 \begin{align*} \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 }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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