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 = 0.05, antiderivative size = 59, normalized size of antiderivative = 1.00, 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 2209
Rule 2212
Rule 6361
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.39, size = 43, normalized size = 0.73 \[ \frac {\pi b^{3} x^{3} \operatorname {erf}\left (b x\right ) + \sqrt {\pi } {\left (b^{2} x^{2} + 1\right )} e^{\left (-b^{2} x^{2}\right )}}{3 \, \pi b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 36, normalized size = 0.61 \[ \frac {1}{3} \, x^{3} \operatorname {erf}\left (b x\right ) + \frac {{\left (b^{2} x^{2} + 1\right )} e^{\left (-b^{2} x^{2}\right )}}{3 \, \sqrt {\pi } b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 0.92 \[ \frac {\frac {b^{3} x^{3} \erf \left (b x \right )}{3}-\frac {2 \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} b^{2} x^{2}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}\right )}{3 \sqrt {\pi }}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 36, normalized size = 0.61 \[ \frac {1}{3} \, x^{3} \operatorname {erf}\left (b x\right ) + \frac {{\left (b^{2} x^{2} + 1\right )} e^{\left (-b^{2} x^{2}\right )}}{3 \, \sqrt {\pi } b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 36, normalized size = 0.61 \[ \frac {x^3\,\mathrm {erf}\left (b\,x\right )}{3}+\frac {{\mathrm {e}}^{-b^2\,x^2}\,\left (b^2\,x^2+1\right )}{3\,b^3\,\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.65, size = 51, normalized size = 0.86 \[ \begin {cases} \frac {x^{3} \operatorname {erf}{\left (b x \right )}}{3} + \frac {x^{2} e^{- b^{2} x^{2}}}{3 \sqrt {\pi } b} + \frac {e^{- b^{2} x^{2}}}{3 \sqrt {\pi } b^{3}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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