Optimal. Leaf size=181 \[ -\frac{\sqrt{\pi } b^3 e^{a+\frac{b^2}{4 c}} \text{Erf}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{16 c^{7/2}}-\frac{3 \sqrt{\pi } b e^{a+\frac{b^2}{4 c}} \text{Erf}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{8 c^{5/2}}-\frac{b^2 e^{a+b x-c x^2}}{8 c^3}-\frac{b x e^{a+b x-c x^2}}{4 c^2}-\frac{e^{a+b x-c x^2}}{2 c^2}-\frac{x^2 e^{a+b x-c x^2}}{2 c} \]
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Rubi [A] time = 0.178745, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2241, 2240, 2234, 2205} \[ -\frac{\sqrt{\pi } b^3 e^{a+\frac{b^2}{4 c}} \text{Erf}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{16 c^{7/2}}-\frac{3 \sqrt{\pi } b e^{a+\frac{b^2}{4 c}} \text{Erf}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{8 c^{5/2}}-\frac{b^2 e^{a+b x-c x^2}}{8 c^3}-\frac{b x e^{a+b x-c x^2}}{4 c^2}-\frac{e^{a+b x-c x^2}}{2 c^2}-\frac{x^2 e^{a+b x-c x^2}}{2 c} \]
Antiderivative was successfully verified.
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Rule 2241
Rule 2240
Rule 2234
Rule 2205
Rubi steps
\begin{align*} \int e^{a+b x-c x^2} x^3 \, dx &=-\frac{e^{a+b x-c x^2} x^2}{2 c}+\frac{\int e^{a+b x-c x^2} x \, dx}{c}+\frac{b \int e^{a+b x-c x^2} x^2 \, dx}{2 c}\\ &=-\frac{e^{a+b x-c x^2}}{2 c^2}-\frac{b e^{a+b x-c x^2} x}{4 c^2}-\frac{e^{a+b x-c x^2} x^2}{2 c}+\frac{b \int e^{a+b x-c x^2} \, dx}{4 c^2}+\frac{b \int e^{a+b x-c x^2} \, dx}{2 c^2}+\frac{b^2 \int e^{a+b x-c x^2} x \, dx}{4 c^2}\\ &=-\frac{b^2 e^{a+b x-c x^2}}{8 c^3}-\frac{e^{a+b x-c x^2}}{2 c^2}-\frac{b e^{a+b x-c x^2} x}{4 c^2}-\frac{e^{a+b x-c x^2} x^2}{2 c}+\frac{b^3 \int e^{a+b x-c x^2} \, dx}{8 c^3}+\frac{\left (b e^{a+\frac{b^2}{4 c}}\right ) \int e^{-\frac{(b-2 c x)^2}{4 c}} \, dx}{4 c^2}+\frac{\left (b e^{a+\frac{b^2}{4 c}}\right ) \int e^{-\frac{(b-2 c x)^2}{4 c}} \, dx}{2 c^2}\\ &=-\frac{b^2 e^{a+b x-c x^2}}{8 c^3}-\frac{e^{a+b x-c x^2}}{2 c^2}-\frac{b e^{a+b x-c x^2} x}{4 c^2}-\frac{e^{a+b x-c x^2} x^2}{2 c}-\frac{3 b 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{\left (b^3 e^{a+\frac{b^2}{4 c}}\right ) \int e^{-\frac{(b-2 c x)^2}{4 c}} \, dx}{8 c^3}\\ &=-\frac{b^2 e^{a+b x-c x^2}}{8 c^3}-\frac{e^{a+b x-c x^2}}{2 c^2}-\frac{b e^{a+b x-c x^2} x}{4 c^2}-\frac{e^{a+b x-c x^2} x^2}{2 c}-\frac{b^3 e^{a+\frac{b^2}{4 c}} \sqrt{\pi } \text{erf}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{16 c^{7/2}}-\frac{3 b 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}}\\ \end{align*}
Mathematica [A] time = 0.217722, size = 91, normalized size = 0.5 \[ -\frac{e^a \left (\sqrt{\pi } b \left (b^2+6 c\right ) e^{\frac{b^2}{4 c}} \text{Erf}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )+2 \sqrt{c} e^{x (b-c x)} \left (b^2+2 b c x+4 c \left (c x^2+1\right )\right )\right )}{16 c^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 194, normalized size = 1.1 \begin{align*} -{\frac{{{\rm e}^{-c{x}^{2}+bx+a}}{x}^{2}}{2\,c}}+{\frac{b}{2\,c} \left ( -{\frac{{{\rm e}^{-c{x}^{2}+bx+a}}x}{2\,c}}+{\frac{b}{2\,c} \left ( -{\frac{{{\rm e}^{-c{x}^{2}+bx+a}}}{2\,c}}-{\frac{b\sqrt{\pi }}{4}{{\rm e}^{a+{\frac{{b}^{2}}{4\,c}}}}{\it Erf} \left ( -\sqrt{c}x+{\frac{b}{2}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}} \right ) }-{\frac{\sqrt{\pi }}{4}{{\rm e}^{a+{\frac{{b}^{2}}{4\,c}}}}{\it Erf} \left ( -\sqrt{c}x+{\frac{b}{2}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}} \right ) }+{\frac{1}{c} \left ( -{\frac{{{\rm e}^{-c{x}^{2}+bx+a}}}{2\,c}}-{\frac{b\sqrt{\pi }}{4}{{\rm e}^{a+{\frac{{b}^{2}}{4\,c}}}}{\it Erf} \left ( -\sqrt{c}x+{\frac{b}{2}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15163, size = 244, normalized size = 1.35 \begin{align*} \frac{{\left (\frac{\sqrt{\pi }{\left (2 \, c x - b\right )} b^{3}{\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{7}{2}}} - \frac{6 \, b^{2} c e^{\left (-\frac{{\left (2 \, c x - b\right )}^{2}}{4 \, c}\right )}}{\left (-c\right )^{\frac{7}{2}}} - \frac{12 \,{\left (2 \, c x - b\right )}^{3} b \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{7}{2}}} - \frac{8 \, c^{2} \Gamma \left (2, \frac{{\left (2 \, c x - b\right )}^{2}}{4 \, c}\right )}{\left (-c\right )^{\frac{7}{2}}}\right )} e^{\left (a + \frac{b^{2}}{4 \, c}\right )}}{16 \, \sqrt{-c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54274, size = 217, normalized size = 1.2 \begin{align*} \frac{\sqrt{\pi }{\left (b^{3} + 6 \, b 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 (4 \, c^{3} x^{2} + 2 \, b c^{2} x + b^{2} c + 4 \, c^{2}\right )} e^{\left (-c x^{2} + b x + a\right )}}{16 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} e^{a} \int x^{3} e^{b x} e^{- c x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26629, size = 140, normalized size = 0.77 \begin{align*} -\frac{\frac{\sqrt{\pi }{\left (b^{3} + 6 \, b 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^{2}{\left (2 \, x - \frac{b}{c}\right )}^{2} + 3 \, b c{\left (2 \, x - \frac{b}{c}\right )} + 3 \, b^{2} + 4 \, c\right )} e^{\left (-c x^{2} + b x + a\right )}}{16 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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