Optimal. Leaf size=107 \[ \frac {2 x}{\sqrt {\pi } b^5}+\frac {2 x^3}{3 \sqrt {\pi } b^3}-\frac {x^4 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{b^6}-\frac {x^2 e^{-b^2 x^2} \text {erfi}(b x)}{b^4}+\frac {x^5}{5 \sqrt {\pi } b} \]
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Rubi [A] time = 0.12, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6387, 6384, 8, 30} \[ -\frac {x^4 e^{-b^2 x^2} \text {Erfi}(b x)}{2 b^2}-\frac {x^2 e^{-b^2 x^2} \text {Erfi}(b x)}{b^4}-\frac {e^{-b^2 x^2} \text {Erfi}(b x)}{b^6}+\frac {2 x^3}{3 \sqrt {\pi } b^3}+\frac {2 x}{\sqrt {\pi } b^5}+\frac {x^5}{5 \sqrt {\pi } b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 6384
Rule 6387
Rubi steps
\begin {align*} \int e^{-b^2 x^2} x^5 \text {erfi}(b x) \, dx &=-\frac {e^{-b^2 x^2} x^4 \text {erfi}(b x)}{2 b^2}+\frac {2 \int e^{-b^2 x^2} x^3 \text {erfi}(b x) \, dx}{b^2}+\frac {\int x^4 \, dx}{b \sqrt {\pi }}\\ &=\frac {x^5}{5 b \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^2 \text {erfi}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erfi}(b x)}{2 b^2}+\frac {2 \int e^{-b^2 x^2} x \text {erfi}(b x) \, dx}{b^4}+\frac {2 \int x^2 \, dx}{b^3 \sqrt {\pi }}\\ &=\frac {2 x^3}{3 b^3 \sqrt {\pi }}+\frac {x^5}{5 b \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{b^6}-\frac {e^{-b^2 x^2} x^2 \text {erfi}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erfi}(b x)}{2 b^2}+\frac {2 \int 1 \, dx}{b^5 \sqrt {\pi }}\\ &=\frac {2 x}{b^5 \sqrt {\pi }}+\frac {2 x^3}{3 b^3 \sqrt {\pi }}+\frac {x^5}{5 b \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{b^6}-\frac {e^{-b^2 x^2} x^2 \text {erfi}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erfi}(b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 68, normalized size = 0.64 \[ \frac {\frac {6 b^5 x^5+20 b^3 x^3+60 b x}{\sqrt {\pi }}-15 e^{-b^2 x^2} \left (b^4 x^4+2 b^2 x^2+2\right ) \text {erfi}(b x)}{30 b^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 79, normalized size = 0.74 \[ \frac {{\left (2 \, \sqrt {\pi } {\left (3 \, b^{5} x^{5} + 10 \, b^{3} x^{3} + 30 \, b x\right )} e^{\left (b^{2} x^{2}\right )} - 15 \, {\left (2 \, \pi + \pi b^{4} x^{4} + 2 \, \pi b^{2} x^{2}\right )} \operatorname {erfi}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{30 \, \pi b^{6}} \]
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 \[ \int x^{5} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 103, normalized size = 0.96 \[ \frac {\left (6 \,{\mathrm e}^{b^{2} x^{2}} b^{5} x^{5}-15 \erfi \left (b x \right ) x^{4} b^{4} \sqrt {\pi }+20 \,{\mathrm e}^{b^{2} x^{2}} b^{3} x^{3}-30 \sqrt {\pi }\, \erfi \left (b x \right ) b^{2} x^{2}+60 \,{\mathrm e}^{b^{2} x^{2}} b x -30 \sqrt {\pi }\, \erfi \left (b x \right )\right ) {\mathrm e}^{-b^{2} x^{2}}}{30 b^{6} \sqrt {\pi }} \]
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 \[ \int x^{5} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 82, normalized size = 0.77 \[ \frac {3\,b^4\,x^5+10\,b^2\,x^3+30\,x}{15\,b^5\,\sqrt {\pi }}-\mathrm {erfi}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{-b^2\,x^2}}{b^6}+\frac {x^4\,{\mathrm {e}}^{-b^2\,x^2}}{2\,b^2}+\frac {x^2\,{\mathrm {e}}^{-b^2\,x^2}}{b^4}\right ) \]
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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