3.434 \(\int e^{a+b x-c x^2} x \, dx\)

Optimal. Leaf size=66 \[ -\frac{\sqrt{\pi } b e^{a+\frac{b^2}{4 c}} \text{Erf}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{4 c^{3/2}}-\frac{e^{a+b x-c x^2}}{2 c} \]

[Out]

-E^(a + b*x - c*x^2)/(2*c) - (b*E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])])/(4*c^(3/2))

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Rubi [A]  time = 0.0323039, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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.

[In]

Int[E^(a + b*x - c*x^2)*x,x]

[Out]

-E^(a + b*x - c*x^2)/(2*c) - (b*E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])])/(4*c^(3/2))

Rule 2240

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] - Dist[(b*e - 2*c*d)/(2*c), Int[F^(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e}, x] &&
NeQ[b*e - 2*c*d, 0]

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

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*}

Mathematica [A]  time = 0.0464995, 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.

[In]

Integrate[E^(a + b*x - c*x^2)*x,x]

[Out]

-E^(a + b*x - c*x^2)/(2*c) + (b*E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(-b + 2*c*x)/(2*Sqrt[c])])/(4*c^(3/2))

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Maple [A]  time = 0.003, size = 53, normalized size = 0.8 \begin{align*} -{\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}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-c*x^2+b*x+a)*x,x)

[Out]

-1/2*exp(-c*x^2+b*x+a)/c-1/4*b/c^(3/2)*Pi^(1/2)*exp(a+1/4*b^2/c)*erf(-c^(1/2)*x+1/2*b/c^(1/2))

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Maxima [A]  time = 1.12839, size = 132, normalized size = 2. \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-c*x^2+b*x+a)*x,x, algorithm="maxima")

[Out]

1/4*(sqrt(pi)*(2*c*x - b)*b*(erf(1/2*sqrt((2*c*x - b)^2/c)) - 1)/(sqrt((2*c*x - b)^2/c)*(-c)^(3/2)) - 2*c*e^(-
1/4*(2*c*x - b)^2/c)/(-c)^(3/2))*e^(a + 1/4*b^2/c)/sqrt(-c)

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Fricas [A]  time = 1.5284, size = 149, normalized size = 2.26 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-c*x^2+b*x+a)*x,x, algorithm="fricas")

[Out]

1/4*(sqrt(pi)*b*sqrt(c)*erf(1/2*(2*c*x - b)/sqrt(c))*e^(1/4*(b^2 + 4*a*c)/c) - 2*c*e^(-c*x^2 + b*x + a))/c^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{a} \int x e^{b x} e^{- c x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-c*x**2+b*x+a)*x,x)

[Out]

exp(a)*Integral(x*exp(b*x)*exp(-c*x**2), x)

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Giac [A]  time = 1.3829, size = 78, normalized size = 1.18 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-c*x^2+b*x+a)*x,x, algorithm="giac")

[Out]

-1/4*(sqrt(pi)*b*erf(-1/2*sqrt(c)*(2*x - b/c))*e^(1/4*(b^2 + 4*a*c)/c)/sqrt(c) + 2*e^(-c*x^2 + b*x + a))/c