Optimal. Leaf size=84 \[ \frac {x^4 e^{-b^2 x^2}}{5 \sqrt {\pi } b}+\frac {2 e^{-b^2 x^2}}{5 \sqrt {\pi } b^5}+\frac {2 x^2 e^{-b^2 x^2}}{5 \sqrt {\pi } b^3}+\frac {1}{5} x^5 \text {erf}(b x) \]
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Rubi [A] time = 0.07, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6361, 2212, 2209} \[ \frac {x^4 e^{-b^2 x^2}}{5 \sqrt {\pi } b}+\frac {2 x^2 e^{-b^2 x^2}}{5 \sqrt {\pi } b^3}+\frac {2 e^{-b^2 x^2}}{5 \sqrt {\pi } b^5}+\frac {1}{5} x^5 \text {Erf}(b x) \]
Antiderivative was successfully verified.
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Rule 2209
Rule 2212
Rule 6361
Rubi steps
\begin {align*} \int x^4 \text {erf}(b x) \, dx &=\frac {1}{5} x^5 \text {erf}(b x)-\frac {(2 b) \int e^{-b^2 x^2} x^5 \, dx}{5 \sqrt {\pi }}\\ &=\frac {e^{-b^2 x^2} x^4}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erf}(b x)-\frac {4 \int e^{-b^2 x^2} x^3 \, dx}{5 b \sqrt {\pi }}\\ &=\frac {2 e^{-b^2 x^2} x^2}{5 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^4}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erf}(b x)-\frac {4 \int e^{-b^2 x^2} x \, dx}{5 b^3 \sqrt {\pi }}\\ &=\frac {2 e^{-b^2 x^2}}{5 b^5 \sqrt {\pi }}+\frac {2 e^{-b^2 x^2} x^2}{5 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^4}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erf}(b x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 66, normalized size = 0.79 \[ e^{-b^2 x^2} \left (\frac {2}{5 \sqrt {\pi } b^5}+\frac {2 x^2}{5 \sqrt {\pi } b^3}+\frac {x^4}{5 \sqrt {\pi } b}\right )+\frac {1}{5} x^5 \text {erf}(b x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 51, normalized size = 0.61 \[ \frac {\pi b^{5} x^{5} \operatorname {erf}\left (b x\right ) + \sqrt {\pi } {\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} e^{\left (-b^{2} x^{2}\right )}}{5 \, \pi b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 44, normalized size = 0.52 \[ \frac {1}{5} \, x^{5} \operatorname {erf}\left (b x\right ) + \frac {{\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} e^{\left (-b^{2} x^{2}\right )}}{5 \, \sqrt {\pi } b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 72, normalized size = 0.86 \[ \frac {\frac {b^{5} x^{5} \erf \left (b x \right )}{5}-\frac {2 \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} b^{4} x^{4}}{2}-{\mathrm e}^{-b^{2} x^{2}} b^{2} x^{2}-{\mathrm e}^{-b^{2} x^{2}}\right )}{5 \sqrt {\pi }}}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 44, normalized size = 0.52 \[ \frac {1}{5} \, x^{5} \operatorname {erf}\left (b x\right ) + \frac {{\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} e^{\left (-b^{2} x^{2}\right )}}{5 \, \sqrt {\pi } b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 44, normalized size = 0.52 \[ \frac {x^5\,\mathrm {erf}\left (b\,x\right )}{5}+\frac {{\mathrm {e}}^{-b^2\,x^2}\,\left (b^4\,x^4+2\,b^2\,x^2+2\right )}{5\,b^5\,\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.25, size = 75, normalized size = 0.89 \[ \begin {cases} \frac {x^{5} \operatorname {erf}{\left (b x \right )}}{5} + \frac {x^{4} e^{- b^{2} x^{2}}}{5 \sqrt {\pi } b} + \frac {2 x^{2} e^{- b^{2} x^{2}}}{5 \sqrt {\pi } b^{3}} + \frac {2 e^{- b^{2} x^{2}}}{5 \sqrt {\pi } b^{5}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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