Optimal. Leaf size=134 \[ -\frac {\sqrt {\pi } b^2 e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{8 c^{5/2}}-\frac {\sqrt {\pi } e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 c^{3/2}}-\frac {b e^{a+b x-c x^2}}{4 c^2}-\frac {x e^{a+b x-c x^2}}{2 c} \]
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Rubi [A] time = 0.08, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2241, 2240, 2234, 2205} \[ -\frac {\sqrt {\pi } b^2 e^{a+\frac {b^2}{4 c}} \text {Erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{8 c^{5/2}}-\frac {\sqrt {\pi } e^{a+\frac {b^2}{4 c}} \text {Erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 c^{3/2}}-\frac {b e^{a+b x-c x^2}}{4 c^2}-\frac {x e^{a+b x-c x^2}}{2 c} \]
Antiderivative was successfully verified.
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Rule 2205
Rule 2234
Rule 2240
Rule 2241
Rubi steps
\begin {align*} \int e^{a+b x-c x^2} x^2 \, dx &=-\frac {e^{a+b x-c x^2} x}{2 c}+\frac {\int e^{a+b x-c x^2} \, dx}{2 c}+\frac {b \int e^{a+b x-c x^2} x \, dx}{2 c}\\ &=-\frac {b e^{a+b x-c x^2}}{4 c^2}-\frac {e^{a+b x-c x^2} x}{2 c}+\frac {b^2 \int e^{a+b x-c x^2} \, dx}{4 c^2}+\frac {e^{a+\frac {b^2}{4 c}} \int e^{-\frac {(b-2 c x)^2}{4 c}} \, dx}{2 c}\\ &=-\frac {b e^{a+b x-c x^2}}{4 c^2}-\frac {e^{a+b x-c x^2} x}{2 c}-\frac {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}}+\frac {\left (b^2 e^{a+\frac {b^2}{4 c}}\right ) \int e^{-\frac {(b-2 c x)^2}{4 c}} \, dx}{4 c^2}\\ &=-\frac {b e^{a+b x-c x^2}}{4 c^2}-\frac {e^{a+b x-c x^2} x}{2 c}-\frac {b^2 e^{a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{8 c^{5/2}}-\frac {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 = 0.12, size = 79, normalized size = 0.59 \[ \frac {e^a \left (\sqrt {\pi } \left (b^2+2 c\right ) e^{\frac {b^2}{4 c}} \text {erf}\left (\frac {2 c x-b}{2 \sqrt {c}}\right )-2 \sqrt {c} e^{x (b-c x)} (b+2 c x)\right )}{8 c^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 72, normalized size = 0.54 \[ \frac {\sqrt {\pi } {\left (b^{2} + 2 \, c\right )} \sqrt {c} \operatorname {erf}\left (\frac {2 \, c x - b}{2 \, \sqrt {c}}\right ) e^{\left (\frac {b^{2} + 4 \, a c}{4 \, c}\right )} - 2 \, {\left (2 \, c^{2} x + b c\right )} e^{\left (-c x^{2} + b x + a\right )}}{8 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 80, normalized size = 0.60 \[ -\frac {\frac {\sqrt {\pi } {\left (b^{2} + 2 \, c\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x - \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} + 4 \, a c}{4 \, c}\right )}}{\sqrt {c}} + 2 \, {\left (c {\left (2 \, x - \frac {b}{c}\right )} + 2 \, b\right )} e^{\left (-c x^{2} + b x + a\right )}}{8 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 111, normalized size = 0.83 \[ -\frac {x \,{\mathrm e}^{-c \,x^{2}+b x +a}}{2 c}-\frac {\sqrt {\pi }\, \erf \left (-\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right ) {\mathrm e}^{a +\frac {b^{2}}{4 c}}}{4 c^{\frac {3}{2}}}+\frac {\left (-\frac {\sqrt {\pi }\, b \erf \left (-\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right ) {\mathrm e}^{a +\frac {b^{2}}{4 c}}}{4 c^{\frac {3}{2}}}-\frac {{\mathrm e}^{-c \,x^{2}+b x +a}}{2 c}\right ) b}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.13, size = 151, normalized size = 1.13 \[ -\frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x - b\right )} b^{2} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {{\left (2 \, c x - b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, c x - b\right )}^{2}}{c}} \left (-c\right )^{\frac {5}{2}}} - \frac {4 \, b c e^{\left (-\frac {{\left (2 \, c x - b\right )}^{2}}{4 \, c}\right )}}{\left (-c\right )^{\frac {5}{2}}} - \frac {4 \, {\left (2 \, c x - b\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {{\left (2 \, c x - b\right )}^{2}}{4 \, c}\right )}{\left (\frac {{\left (2 \, c x - b\right )}^{2}}{c}\right )^{\frac {3}{2}} \left (-c\right )^{\frac {5}{2}}}\right )} e^{\left (a + \frac {b^{2}}{4 \, c}\right )}}{8 \, \sqrt {-c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.72, size = 80, normalized size = 0.60 \[ \frac {\sqrt {\pi }\,\mathrm {erfi}\left (\frac {\frac {b}{2}-c\,x}{\sqrt {-c}}\right )\,{\mathrm {e}}^{a+\frac {b^2}{4\,c}}\,\left (b^2+2\,c\right )}{8\,{\left (-c\right )}^{5/2}}-\frac {x\,{\mathrm {e}}^{-c\,x^2+b\,x+a}}{2\,c}-\frac {b\,{\mathrm {e}}^{-c\,x^2+b\,x+a}}{4\,c^2} \]
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 \[ e^{a} \int x^{2} e^{b x} e^{- c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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