Optimal. Leaf size=44 \[ -\frac {\sqrt {\pi } e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{2 \sqrt {c}} \]
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Rubi [A] time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2234, 2205} \[ -\frac {\sqrt {\pi } e^{a+\frac {b^2}{4 c}} \text {Erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{2 \sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 2205
Rule 2234
Rubi steps
\begin {align*} \int e^{a+b x-c x^2} \, dx &=e^{a+\frac {b^2}{4 c}} \int e^{-\frac {(b-2 c x)^2}{4 c}} \, dx\\ &=-\frac {e^{a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{2 \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 46, normalized size = 1.05 \[ \frac {\sqrt {\pi } e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {2 c x-b}{2 \sqrt {c}}\right )}{2 \sqrt {c}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 36, normalized size = 0.82 \[ \frac {\sqrt {\pi } \operatorname {erf}\left (\frac {2 \, c x - b}{2 \, \sqrt {c}}\right ) e^{\left (\frac {b^{2} + 4 \, a c}{4 \, c}\right )}}{2 \, \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 38, normalized size = 0.86 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x - \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} + 4 \, a c}{4 \, c}\right )}}{2 \, \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 34, normalized size = 0.77 \[ -\frac {\sqrt {\pi }\, \erf \left (-\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right ) {\mathrm e}^{a +\frac {b^{2}}{4 c}}}{2 \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 32, normalized size = 0.73 \[ \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {c} x - \frac {b}{2 \, \sqrt {c}}\right ) e^{\left (a + \frac {b^{2}}{4 \, c}\right )}}{2 \, \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 40, normalized size = 0.91 \[ -\frac {\sqrt {\pi }\,\mathrm {erf}\left (\frac {b\,1{}\mathrm {i}-c\,x\,2{}\mathrm {i}}{2\,\sqrt {-c}}\right )\,{\mathrm {e}}^{a+\frac {b^2}{4\,c}}\,1{}\mathrm {i}}{2\,\sqrt {-c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.70, size = 41, normalized size = 0.93 \[ \frac {\sqrt {\pi } \sqrt {- \frac {1}{c}} e^{a + \frac {b^{2}}{4 c}} \operatorname {erfi}{\left (\frac {b - 2 c x}{2 \sqrt {- c}} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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