Optimal. Leaf size=79 \[ \frac{x^2 e^{b^2 x^2+c} \text{Erf}(b x)}{2 b^2}-\frac{e^{b^2 x^2+c} \text{Erf}(b x)}{2 b^4}+\frac{e^c x}{\sqrt{\pi } b^3}-\frac{e^c x^3}{3 \sqrt{\pi } b} \]
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Rubi [A] time = 0.0799707, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ \frac{x^2 e^{b^2 x^2+c} \text{Erf}(b x)}{2 b^2}-\frac{e^{b^2 x^2+c} \text{Erf}(b x)}{2 b^4}+\frac{e^c x}{\sqrt{\pi } b^3}-\frac{e^c x^3}{3 \sqrt{\pi } b} \]
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^3 \text{erf}(b x) \, dx &=\frac{e^{c+b^2 x^2} x^2 \text{erf}(b x)}{2 b^2}-\frac{\int e^{c+b^2 x^2} x \text{erf}(b x) \, dx}{b^2}-\frac{\int e^c x^2 \, dx}{b \sqrt{\pi }}\\ &=-\frac{e^{c+b^2 x^2} \text{erf}(b x)}{2 b^4}+\frac{e^{c+b^2 x^2} x^2 \text{erf}(b x)}{2 b^2}+\frac{\int e^c \, dx}{b^3 \sqrt{\pi }}-\frac{e^c \int x^2 \, dx}{b \sqrt{\pi }}\\ &=\frac{e^c x}{b^3 \sqrt{\pi }}-\frac{e^c x^3}{3 b \sqrt{\pi }}-\frac{e^{c+b^2 x^2} \text{erf}(b x)}{2 b^4}+\frac{e^{c+b^2 x^2} x^2 \text{erf}(b x)}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0348345, size = 57, normalized size = 0.72 \[ \frac{e^c \left (3 \sqrt{\pi } e^{b^2 x^2} \left (b^2 x^2-1\right ) \text{Erf}(b x)-2 b^3 x^3+6 b x\right )}{6 \sqrt{\pi } b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.218, size = 66, normalized size = 0.8 \begin{align*}{\frac{1}{b} \left ({\frac{{\it Erf} \left ( bx \right ){{\rm e}^{c}}}{{b}^{3}} \left ({\frac{{b}^{2}{x}^{2}{{\rm e}^{{b}^{2}{x}^{2}}}}{2}}-{\frac{{{\rm e}^{{b}^{2}{x}^{2}}}}{2}} \right ) }-{\frac{{{\rm e}^{c}}}{{b}^{3}\sqrt{\pi }} \left ({\frac{{x}^{3}{b}^{3}}{3}}-bx \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01245, size = 80, normalized size = 1.01 \begin{align*} -\frac{2 \, b^{3} x^{3} e^{c} - 3 \,{\left (\sqrt{\pi } b^{2} x^{2} e^{c} - \sqrt{\pi } e^{c}\right )} \operatorname{erf}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - 6 \, b x e^{c}}{6 \, \sqrt{\pi } b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.71925, size = 131, normalized size = 1.66 \begin{align*} -\frac{3 \,{\left (\pi - \pi b^{2} x^{2}\right )} \operatorname{erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} + 2 \, \sqrt{\pi }{\left (b^{3} x^{3} - 3 \, b x\right )} e^{c}}{6 \, \pi b^{4}} \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 = 1.27213, size = 78, normalized size = 0.99 \begin{align*} \frac{{\left (b^{2} x^{2} - 1\right )} \operatorname{erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{2 \, b^{4}} - \frac{\sqrt{\pi } b^{2} x^{3} e^{c} - 3 \, \sqrt{\pi } x e^{c}}{3 \, \pi b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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