Optimal. Leaf size=83 \[ -\frac{\text{Unintegrable}\left (\text{Erf}(b x) e^{c+d x^2},x\right )}{2 d}+\frac{b e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt{\pi } d \left (b^2-d\right )}+\frac{x \text{Erf}(b x) e^{c+d x^2}}{2 d} \]
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Rubi [A] time = 0.0948577, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int e^{c+d x^2} x^2 \text{Erf}(b x) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int e^{c+d x^2} x^2 \text{erf}(b x) \, dx &=\frac{e^{c+d x^2} x \text{erf}(b x)}{2 d}-\frac{\int e^{c+d x^2} \text{erf}(b x) \, dx}{2 d}-\frac{b \int e^{c-\left (b^2-d\right ) x^2} x \, dx}{d \sqrt{\pi }}\\ &=\frac{b e^{c-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right ) d \sqrt{\pi }}+\frac{e^{c+d x^2} x \text{erf}(b x)}{2 d}-\frac{\int e^{c+d x^2} \text{erf}(b x) \, dx}{2 d}\\ \end{align*}
Mathematica [A] time = 0.215184, size = 0, normalized size = 0. \[ \int e^{c+d x^2} x^2 \text{Erf}(b x) \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.231, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{d{x}^{2}+c}}{x}^{2}{\it Erf} \left ( bx \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} \operatorname{erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} e^{c} \int x^{2} e^{d x^{2}} \operatorname{erf}{\left (b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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