Optimal. Leaf size=118 \[ \frac {2 e^c x}{\sqrt {\pi } b^5}-\frac {2 e^c x^3}{3 \sqrt {\pi } b^3}+\frac {x^4 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{b^6}-\frac {x^2 e^{b^2 x^2+c} \text {erfc}(b x)}{b^4}+\frac {e^c x^5}{5 \sqrt {\pi } b} \]
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Rubi [A] time = 0.15, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6386, 6383, 8, 12, 30} \[ \frac {x^4 e^{b^2 x^2+c} \text {Erfc}(b x)}{2 b^2}-\frac {x^2 e^{b^2 x^2+c} \text {Erfc}(b x)}{b^4}+\frac {e^{b^2 x^2+c} \text {Erfc}(b x)}{b^6}-\frac {2 e^c x^3}{3 \sqrt {\pi } b^3}+\frac {2 e^c x}{\sqrt {\pi } b^5}+\frac {e^c x^5}{5 \sqrt {\pi } b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 30
Rule 6383
Rule 6386
Rubi steps
\begin {align*} \int e^{c+b^2 x^2} x^5 \text {erfc}(b x) \, dx &=\frac {e^{c+b^2 x^2} x^4 \text {erfc}(b x)}{2 b^2}-\frac {2 \int e^{c+b^2 x^2} x^3 \text {erfc}(b x) \, dx}{b^2}+\frac {\int e^c x^4 \, dx}{b \sqrt {\pi }}\\ &=-\frac {e^{c+b^2 x^2} x^2 \text {erfc}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erfc}(b x)}{2 b^2}+\frac {2 \int e^{c+b^2 x^2} x \text {erfc}(b x) \, dx}{b^4}-\frac {2 \int e^c x^2 \, dx}{b^3 \sqrt {\pi }}+\frac {e^c \int x^4 \, dx}{b \sqrt {\pi }}\\ &=\frac {e^c x^5}{5 b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{b^6}-\frac {e^{c+b^2 x^2} x^2 \text {erfc}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erfc}(b x)}{2 b^2}+\frac {2 \int e^c \, dx}{b^5 \sqrt {\pi }}-\frac {\left (2 e^c\right ) \int x^2 \, dx}{b^3 \sqrt {\pi }}\\ &=\frac {2 e^c x}{b^5 \sqrt {\pi }}-\frac {2 e^c x^3}{3 b^3 \sqrt {\pi }}+\frac {e^c x^5}{5 b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{b^6}-\frac {e^{c+b^2 x^2} x^2 \text {erfc}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erfc}(b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 73, normalized size = 0.62 \[ \frac {e^c \left (6 b^5 x^5-20 b^3 x^3+15 \sqrt {\pi } e^{b^2 x^2} \left (b^4 x^4-2 b^2 x^2+2\right ) \text {erfc}(b x)+60 b x\right )}{30 \sqrt {\pi } b^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 97, normalized size = 0.82 \[ \frac {2 \, \sqrt {\pi } {\left (3 \, b^{5} x^{5} - 10 \, b^{3} x^{3} + 30 \, b x\right )} e^{c} + 15 \, {\left (2 \, \pi + \pi b^{4} x^{4} - 2 \, \pi b^{2} x^{2} - {\left (2 \, \pi + \pi b^{4} x^{4} - 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (b^{2} x^{2} + c\right )}}{30 \, \pi b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{5} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 135, normalized size = 1.14 \[ \frac {\frac {{\mathrm e}^{c} \left (\frac {{\mathrm e}^{b^{2} x^{2}} b^{4} x^{4}}{2}-b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}+{\mathrm e}^{b^{2} x^{2}}\right )}{b^{5}}-\frac {\erf \left (b x \right ) {\mathrm e}^{c} \left (\frac {{\mathrm e}^{b^{2} x^{2}} b^{4} x^{4}}{2}-b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}+{\mathrm e}^{b^{2} x^{2}}\right )}{b^{5}}+\frac {{\mathrm e}^{c} \left (\frac {1}{5} b^{5} x^{5}-\frac {2}{3} b^{3} x^{3}+2 b x \right )}{\sqrt {\pi }\, b^{5}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{5} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 94, normalized size = 0.80 \[ \frac {{\mathrm {e}}^c\,\left (60\,b\,x-20\,b^3\,x^3+6\,b^5\,x^5+30\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )-30\,b^2\,x^2\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )+15\,b^4\,x^4\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )\right )}{30\,b^6\,\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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