Optimal. Leaf size=59 \[ \frac{x^2 e^{-b^2 x^2}}{3 \sqrt{\pi } b}+\frac{e^{-b^2 x^2}}{3 \sqrt{\pi } b^3}+\frac{1}{3} x^3 \text{Erf}(b x) \]
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Rubi [A] time = 0.0489509, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6361, 2212, 2209} \[ \frac{x^2 e^{-b^2 x^2}}{3 \sqrt{\pi } b}+\frac{e^{-b^2 x^2}}{3 \sqrt{\pi } b^3}+\frac{1}{3} x^3 \text{Erf}(b x) \]
Antiderivative was successfully verified.
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Rule 6361
Rule 2212
Rule 2209
Rubi steps
\begin{align*} \int x^2 \text{erf}(b x) \, dx &=\frac{1}{3} x^3 \text{erf}(b x)-\frac{(2 b) \int e^{-b^2 x^2} x^3 \, dx}{3 \sqrt{\pi }}\\ &=\frac{e^{-b^2 x^2} x^2}{3 b \sqrt{\pi }}+\frac{1}{3} x^3 \text{erf}(b x)-\frac{2 \int e^{-b^2 x^2} x \, dx}{3 b \sqrt{\pi }}\\ &=\frac{e^{-b^2 x^2}}{3 b^3 \sqrt{\pi }}+\frac{e^{-b^2 x^2} x^2}{3 b \sqrt{\pi }}+\frac{1}{3} x^3 \text{erf}(b x)\\ \end{align*}
Mathematica [A] time = 0.024314, size = 41, normalized size = 0.69 \[ \frac{1}{3} \left (\frac{e^{-b^2 x^2} \left (b^2 x^2+1\right )}{\sqrt{\pi } b^3}+x^3 \text{Erf}(b x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 54, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{3}} \left ({\frac{{\it Erf} \left ( bx \right ){b}^{3}{x}^{3}}{3}}-{\frac{2}{3\,\sqrt{\pi }} \left ( -{\frac{{b}^{2}{x}^{2}}{2\,{{\rm e}^{{b}^{2}{x}^{2}}}}}-{\frac{1}{2\,{{\rm e}^{{b}^{2}{x}^{2}}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10549, size = 49, normalized size = 0.83 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{erf}\left (b x\right ) + \frac{{\left (b^{2} x^{2} + 1\right )} e^{\left (-b^{2} x^{2}\right )}}{3 \, \sqrt{\pi } b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.55151, size = 100, normalized size = 1.69 \begin{align*} \frac{\pi b^{3} x^{3} \operatorname{erf}\left (b x\right ) + \sqrt{\pi }{\left (b^{2} x^{2} + 1\right )} e^{\left (-b^{2} x^{2}\right )}}{3 \, \pi b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.804436, size = 51, normalized size = 0.86 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{erf}{\left (b x \right )}}{3} + \frac{x^{2} e^{- b^{2} x^{2}}}{3 \sqrt{\pi } b} + \frac{e^{- b^{2} x^{2}}}{3 \sqrt{\pi } b^{3}} & \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 = 1.27415, size = 49, normalized size = 0.83 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{erf}\left (b x\right ) + \frac{{\left (b^{2} x^{2} + 1\right )} e^{\left (-b^{2} x^{2}\right )}}{3 \, \sqrt{\pi } b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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