Optimal. Leaf size=118 \[ \frac{x^4 e^{b^2 x^2+c} \text{Erf}(b x)}{2 b^2}-\frac{x^2 e^{b^2 x^2+c} \text{Erf}(b x)}{b^4}+\frac{e^{b^2 x^2+c} \text{Erf}(b x)}{b^6}+\frac{2 e^c x^3}{3 \sqrt{\pi } b^3}-\frac{2 e^c x}{\sqrt{\pi } b^5}-\frac{e^c x^5}{5 \sqrt{\pi } b} \]
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Rubi [A] time = 0.141572, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {6385, 6382, 8, 12, 30} \[ \frac{x^4 e^{b^2 x^2+c} \text{Erf}(b x)}{2 b^2}-\frac{x^2 e^{b^2 x^2+c} \text{Erf}(b x)}{b^4}+\frac{e^{b^2 x^2+c} \text{Erf}(b x)}{b^6}+\frac{2 e^c x^3}{3 \sqrt{\pi } b^3}-\frac{2 e^c x}{\sqrt{\pi } b^5}-\frac{e^c x^5}{5 \sqrt{\pi } b} \]
Antiderivative was successfully verified.
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Rule 6385
Rule 6382
Rule 8
Rule 12
Rule 30
Rubi steps
\begin{align*} \int e^{c+b^2 x^2} x^5 \text{erf}(b x) \, dx &=\frac{e^{c+b^2 x^2} x^4 \text{erf}(b x)}{2 b^2}-\frac{2 \int e^{c+b^2 x^2} x^3 \text{erf}(b x) \, dx}{b^2}-\frac{\int e^c x^4 \, dx}{b \sqrt{\pi }}\\ &=-\frac{e^{c+b^2 x^2} x^2 \text{erf}(b x)}{b^4}+\frac{e^{c+b^2 x^2} x^4 \text{erf}(b x)}{2 b^2}+\frac{2 \int e^{c+b^2 x^2} x \text{erf}(b x) \, dx}{b^4}+\frac{2 \int e^c x^2 \, dx}{b^3 \sqrt{\pi }}-\frac{e^c \int x^4 \, dx}{b \sqrt{\pi }}\\ &=-\frac{e^c x^5}{5 b \sqrt{\pi }}+\frac{e^{c+b^2 x^2} \text{erf}(b x)}{b^6}-\frac{e^{c+b^2 x^2} x^2 \text{erf}(b x)}{b^4}+\frac{e^{c+b^2 x^2} x^4 \text{erf}(b x)}{2 b^2}-\frac{2 \int e^c \, dx}{b^5 \sqrt{\pi }}+\frac{\left (2 e^c\right ) \int x^2 \, dx}{b^3 \sqrt{\pi }}\\ &=-\frac{2 e^c x}{b^5 \sqrt{\pi }}+\frac{2 e^c x^3}{3 b^3 \sqrt{\pi }}-\frac{e^c x^5}{5 b \sqrt{\pi }}+\frac{e^{c+b^2 x^2} \text{erf}(b x)}{b^6}-\frac{e^{c+b^2 x^2} x^2 \text{erf}(b x)}{b^4}+\frac{e^{c+b^2 x^2} x^4 \text{erf}(b x)}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0460526, size = 73, normalized size = 0.62 \[ \frac{e^c \left (15 \sqrt{\pi } e^{b^2 x^2} \left (b^4 x^4-2 b^2 x^2+2\right ) \text{Erf}(b x)-6 b^5 x^5+20 b^3 x^3-60 b x\right )}{30 \sqrt{\pi } b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.131, size = 88, normalized size = 0.8 \begin{align*}{\frac{1}{b} \left ({\frac{{\it Erf} \left ( bx \right ){{\rm e}^{c}}}{{b}^{5}} \left ({\frac{{{\rm e}^{{b}^{2}{x}^{2}}}{b}^{4}{x}^{4}}{2}}-{b}^{2}{x}^{2}{{\rm e}^{{b}^{2}{x}^{2}}}+{{\rm e}^{{b}^{2}{x}^{2}}} \right ) }-{\frac{{{\rm e}^{c}}}{\sqrt{\pi }{b}^{5}} \left ({\frac{{b}^{5}{x}^{5}}{5}}-{\frac{2\,{x}^{3}{b}^{3}}{3}}+2\,bx \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00364, size = 111, normalized size = 0.94 \begin{align*} -\frac{6 \, b^{5} x^{5} e^{c} - 20 \, b^{3} x^{3} e^{c} - 15 \,{\left (\sqrt{\pi } b^{4} x^{4} e^{c} - 2 \, \sqrt{\pi } b^{2} x^{2} e^{c} + 2 \, \sqrt{\pi } e^{c}\right )} \operatorname{erf}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} + 60 \, b x e^{c}}{30 \, \sqrt{\pi } b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.02796, size = 177, normalized size = 1.5 \begin{align*} \frac{15 \,{\left (2 \, \pi + \pi b^{4} x^{4} - 2 \, \pi b^{2} x^{2}\right )} \operatorname{erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} - 2 \, \sqrt{\pi }{\left (3 \, b^{5} x^{5} - 10 \, b^{3} x^{3} + 30 \, b x\right )} e^{c}}{30 \, \pi b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27515, size = 139, normalized size = 1.18 \begin{align*} -\frac{{\left (2 \, b^{2} x^{2} -{\left (b^{2} x^{2} + c\right )}^{2} + 2 \,{\left (b^{2} x^{2} + c\right )} c - c^{2} - 2\right )} \operatorname{erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{2 \, b^{6}} - \frac{3 \, \sqrt{\pi } b^{4} x^{5} e^{c} - 10 \, \sqrt{\pi } b^{2} x^{3} e^{c} + 30 \, \sqrt{\pi } x e^{c}}{15 \, \pi b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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