3.87 \(\int e^{c+d x^2} x \text{Erf}(a+b x) \, dx\)

Optimal. Leaf size=86 \[ \frac{e^{c+d x^2} \text{Erf}(a+b x)}{2 d}-\frac{b e^{\frac{a^2 d}{b^2-d}+c} \text{Erf}\left (\frac{a b+x \left (b^2-d\right )}{\sqrt{b^2-d}}\right )}{2 d \sqrt{b^2-d}} \]

[Out]

(E^(c + d*x^2)*Erf[a + b*x])/(2*d) - (b*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(2*S
qrt[b^2 - d]*d)

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Rubi [A]  time = 0.0570083, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {6382, 2234, 2205} \[ \frac{e^{c+d x^2} \text{Erf}(a+b x)}{2 d}-\frac{b e^{\frac{a^2 d}{b^2-d}+c} \text{Erf}\left (\frac{a b+x \left (b^2-d\right )}{\sqrt{b^2-d}}\right )}{2 d \sqrt{b^2-d}} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x^2)*x*Erf[a + b*x],x]

[Out]

(E^(c + d*x^2)*Erf[a + b*x])/(2*d) - (b*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(2*S
qrt[b^2 - d]*d)

Rule 6382

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

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^{c+d x^2} x \text{erf}(a+b x) \, dx &=\frac{e^{c+d x^2} \text{erf}(a+b x)}{2 d}-\frac{b \int e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} \, dx}{d \sqrt{\pi }}\\ &=\frac{e^{c+d x^2} \text{erf}(a+b x)}{2 d}-\frac{\left (b e^{\frac{b^2 c+a^2 d-c d}{b^2-d}}\right ) \int \exp \left (\frac{\left (-2 a b+2 \left (-b^2+d\right ) x\right )^2}{4 \left (-b^2+d\right )}\right ) \, dx}{d \sqrt{\pi }}\\ &=\frac{e^{c+d x^2} \text{erf}(a+b x)}{2 d}-\frac{b e^{\frac{b^2 c+a^2 d-c d}{b^2-d}} \text{erf}\left (\frac{a b+\left (b^2-d\right ) x}{\sqrt{b^2-d}}\right )}{2 \sqrt{b^2-d} d}\\ \end{align*}

Mathematica [A]  time = 0.105384, size = 82, normalized size = 0.95 \[ \frac{e^c \left (e^{d x^2} \text{Erf}(a+b x)-\frac{b e^{\frac{a^2 d}{b^2-d}} \text{Erfi}\left (\frac{x \left (d-b^2\right )-a b}{\sqrt{d-b^2}}\right )}{\sqrt{d-b^2}}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(c + d*x^2)*x*Erf[a + b*x],x]

[Out]

(E^c*(E^(d*x^2)*Erf[a + b*x] - (b*E^((a^2*d)/(b^2 - d))*Erfi[(-(a*b) + (-b^2 + d)*x)/Sqrt[-b^2 + d]])/Sqrt[-b^
2 + d]))/(2*d)

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Maple [A]  time = 0.373, size = 134, normalized size = 1.6 \begin{align*}{\frac{1}{b} \left ({\frac{{\it Erf} \left ( bx+a \right ) b}{2\,d}{{\rm e}^{{\frac{d \left ( bx+a \right ) ^{2}}{{b}^{2}}}-2\,{\frac{ad \left ( bx+a \right ) }{{b}^{2}}}+{\frac{{a}^{2}d}{{b}^{2}}}+c}}}-{\frac{b}{2\,d}{{\rm e}^{{\frac{{a}^{2}d}{{b}^{2}}}+c-{\frac{{a}^{2}{d}^{2}}{{b}^{4}} \left ( -1+{\frac{d}{{b}^{2}}} \right ) ^{-1}}}}{\it Erf} \left ( \sqrt{1-{\frac{d}{{b}^{2}}}} \left ( bx+a \right ) +{\frac{ad}{{b}^{2}}{\frac{1}{\sqrt{1-{\frac{d}{{b}^{2}}}}}}} \right ){\frac{1}{\sqrt{1-{\frac{d}{{b}^{2}}}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0529, size = 113, normalized size = 1.31 \begin{align*} -\frac{b \operatorname{erf}\left (\frac{a b}{\sqrt{b^{2} - d}} + \sqrt{b^{2} - d} x\right ) e^{\left (\frac{a^{2} b^{2}}{b^{2} - d} - a^{2} + c\right )}}{2 \, \sqrt{b^{2} - d} d} + \frac{\operatorname{erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.73982, size = 205, normalized size = 2.38 \begin{align*} -\frac{\sqrt{b^{2} - d} b \operatorname{erf}\left (\frac{a b +{\left (b^{2} - d\right )} x}{\sqrt{b^{2} - d}}\right ) e^{\left (\frac{b^{2} c +{\left (a^{2} - c\right )} d}{b^{2} - d}\right )} -{\left (b^{2} - d\right )} \operatorname{erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{2 \,{\left (b^{2} d - d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.24279, size = 117, normalized size = 1.36 \begin{align*} \frac{b \operatorname{erf}\left (-\sqrt{b^{2} - d}{\left (\frac{a b}{b^{2} - d} + x\right )}\right ) e^{\left (\frac{b^{2} c + a^{2} d - c d}{b^{2} - d}\right )}}{2 \, \sqrt{b^{2} - d} d} + \frac{\operatorname{erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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