Optimal. Leaf size=96 \[ -\frac {5 \text {erf}(b x)}{16 b^6}+\frac {x^5 e^{-b^2 x^2}}{6 \sqrt {\pi } b}+\frac {5 x e^{-b^2 x^2}}{8 \sqrt {\pi } b^5}+\frac {5 x^3 e^{-b^2 x^2}}{12 \sqrt {\pi } b^3}+\frac {1}{6} x^6 \text {erf}(b x) \]
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Rubi [A] time = 0.10, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6361, 2212, 2205} \[ -\frac {5 \text {Erf}(b x)}{16 b^6}+\frac {x^5 e^{-b^2 x^2}}{6 \sqrt {\pi } b}+\frac {5 x^3 e^{-b^2 x^2}}{12 \sqrt {\pi } b^3}+\frac {5 x e^{-b^2 x^2}}{8 \sqrt {\pi } b^5}+\frac {1}{6} x^6 \text {Erf}(b x) \]
Antiderivative was successfully verified.
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Rule 2205
Rule 2212
Rule 6361
Rubi steps
\begin {align*} \int x^5 \text {erf}(b x) \, dx &=\frac {1}{6} x^6 \text {erf}(b x)-\frac {b \int e^{-b^2 x^2} x^6 \, dx}{3 \sqrt {\pi }}\\ &=\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erf}(b x)-\frac {5 \int e^{-b^2 x^2} x^4 \, dx}{6 b \sqrt {\pi }}\\ &=\frac {5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erf}(b x)-\frac {5 \int e^{-b^2 x^2} x^2 \, dx}{4 b^3 \sqrt {\pi }}\\ &=\frac {5 e^{-b^2 x^2} x}{8 b^5 \sqrt {\pi }}+\frac {5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erf}(b x)-\frac {5 \int e^{-b^2 x^2} \, dx}{8 b^5 \sqrt {\pi }}\\ &=\frac {5 e^{-b^2 x^2} x}{8 b^5 \sqrt {\pi }}+\frac {5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}-\frac {5 \text {erf}(b x)}{16 b^6}+\frac {1}{6} x^6 \text {erf}(b x)\\ \end {align*}
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Mathematica [A] time = 0.13, size = 72, normalized size = 0.75 \[ \frac {e^{-b^2 x^2} \left (8 b^5 x^5+20 b^3 x^3+\sqrt {\pi } e^{b^2 x^2} \left (8 b^6 x^6-15\right ) \text {erf}(b x)+30 b x\right )}{48 \sqrt {\pi } b^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.23, size = 63, normalized size = 0.66 \[ \frac {2 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} + 10 \, b^{3} x^{3} + 15 \, b x\right )} e^{\left (-b^{2} x^{2}\right )} - {\left (15 \, \pi - 8 \, \pi b^{6} x^{6}\right )} \operatorname {erf}\left (b x\right )}{48 \, \pi b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 64, normalized size = 0.67 \[ \frac {1}{6} \, x^{6} \operatorname {erf}\left (b x\right ) + \frac {b {\left (\frac {2 \, {\left (4 \, b^{4} x^{5} + 10 \, b^{2} x^{3} + 15 \, x\right )} e^{\left (-b^{2} x^{2}\right )}}{b^{6}} + \frac {15 \, \sqrt {\pi } \operatorname {erf}\left (-b x\right )}{b^{7}}\right )}}{48 \, \sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 83, normalized size = 0.86 \[ \frac {\frac {\erf \left (b x \right ) b^{6} x^{6}}{6}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} b^{5} x^{5}}{2}-\frac {5 b^{3} x^{3} {\mathrm e}^{-b^{2} x^{2}}}{4}-\frac {15 \,{\mathrm e}^{-b^{2} x^{2}} b x}{8}+\frac {15 \sqrt {\pi }\, \erf \left (b x \right )}{16}}{3 \sqrt {\pi }}}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 63, normalized size = 0.66 \[ \frac {1}{6} \, x^{6} \operatorname {erf}\left (b x\right ) + \frac {b {\left (\frac {2 \, {\left (4 \, b^{4} x^{5} + 10 \, b^{2} x^{3} + 15 \, x\right )} e^{\left (-b^{2} x^{2}\right )}}{b^{6}} - \frac {15 \, \sqrt {\pi } \operatorname {erf}\left (b x\right )}{b^{7}}\right )}}{48 \, \sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 108, normalized size = 1.12 \[ \frac {x^6\,\mathrm {erf}\left (b\,x\right )}{6}-\frac {5\,b\,x^7}{16\,{\left (b^2\,x^2\right )}^{7/2}}+\frac {x^5\,{\mathrm {e}}^{-b^2\,x^2}}{6\,b\,\sqrt {\pi }}+\frac {5\,x^3\,{\mathrm {e}}^{-b^2\,x^2}}{12\,b^3\,\sqrt {\pi }}+\frac {5\,x\,{\mathrm {e}}^{-b^2\,x^2}}{8\,b^5\,\sqrt {\pi }}+\frac {5\,b\,x^7\,\mathrm {erfc}\left (\sqrt {b^2\,x^2}\right )}{16\,{\left (b^2\,x^2\right )}^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.94, size = 88, normalized size = 0.92 \[ \begin {cases} \frac {x^{6} \operatorname {erf}{\left (b x \right )}}{6} + \frac {x^{5} e^{- b^{2} x^{2}}}{6 \sqrt {\pi } b} + \frac {5 x^{3} e^{- b^{2} x^{2}}}{12 \sqrt {\pi } b^{3}} + \frac {5 x e^{- b^{2} x^{2}}}{8 \sqrt {\pi } b^{5}} - \frac {5 \operatorname {erf}{\left (b x \right )}}{16 b^{6}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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