Optimal. Leaf size=66 \[ -\frac {\sqrt {\pi } b e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 c^{3/2}}-\frac {e^{a+b x-c x^2}}{2 c} \]
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Rubi [A] time = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2240, 2234, 2205} \[ -\frac {\sqrt {\pi } b e^{a+\frac {b^2}{4 c}} \text {Erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 c^{3/2}}-\frac {e^{a+b x-c x^2}}{2 c} \]
Antiderivative was successfully verified.
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Rule 2205
Rule 2234
Rule 2240
Rubi steps
\begin {align*} \int e^{a+b x-c x^2} x \, dx &=-\frac {e^{a+b x-c x^2}}{2 c}+\frac {b \int e^{a+b x-c x^2} \, dx}{2 c}\\ &=-\frac {e^{a+b x-c x^2}}{2 c}+\frac {\left (b e^{a+\frac {b^2}{4 c}}\right ) \int e^{-\frac {(b-2 c x)^2}{4 c}} \, dx}{2 c}\\ &=-\frac {e^{a+b x-c x^2}}{2 c}-\frac {b e^{a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 68, normalized size = 1.03 \[ \frac {\sqrt {\pi } b e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {2 c x-b}{2 \sqrt {c}}\right )}{4 c^{3/2}}-\frac {e^{a+b x-c x^2}}{2 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 57, normalized size = 0.86 \[ \frac {\sqrt {\pi } b \sqrt {c} \operatorname {erf}\left (\frac {2 \, c x - b}{2 \, \sqrt {c}}\right ) e^{\left (\frac {b^{2} + 4 \, a c}{4 \, c}\right )} - 2 \, c e^{\left (-c x^{2} + b x + a\right )}}{4 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 58, normalized size = 0.88 \[ -\frac {\frac {\sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x - \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} + 4 \, a c}{4 \, c}\right )}}{\sqrt {c}} + 2 \, e^{\left (-c x^{2} + b x + a\right )}}{4 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 53, normalized size = 0.80 \[ -\frac {\sqrt {\pi }\, b \erf \left (-\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right ) {\mathrm e}^{a +\frac {b^{2}}{4 c}}}{4 c^{\frac {3}{2}}}-\frac {{\mathrm e}^{-c \,x^{2}+b x +a}}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.49, size = 98, normalized size = 1.48 \[ \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x - b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {{\left (2 \, c x - b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, c x - b\right )}^{2}}{c}} \left (-c\right )^{\frac {3}{2}}} - \frac {2 \, c e^{\left (-\frac {{\left (2 \, c x - b\right )}^{2}}{4 \, c}\right )}}{\left (-c\right )^{\frac {3}{2}}}\right )} e^{\left (a + \frac {b^{2}}{4 \, c}\right )}}{4 \, \sqrt {-c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.47, size = 58, normalized size = 0.88 \[ -\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,{\mathrm {e}}^{-c\,x^2}}{2\,c}-\frac {b\,\sqrt {\pi }\,{\mathrm {e}}^{\frac {b^2}{4\,c}}\,{\mathrm {e}}^a\,\mathrm {erfi}\left (\frac {b}{2\,\sqrt {-c}}+\sqrt {-c}\,x\right )}{4\,{\left (-c\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{a} \int x e^{b x} e^{- c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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