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

Optimal. Leaf size=78 \[ \frac{e^{c+d x^2} \text{Erfi}(a+b x)}{2 d}-\frac{b e^{\frac{a^2 d}{b^2+d}+c} \text{Erfi}\left (\frac{a b+x \left (b^2+d\right )}{\sqrt{b^2+d}}\right )}{2 d \sqrt{b^2+d}} \]

[Out]

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

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Rubi [A]  time = 0.0532274, antiderivative size = 78, 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 = {6384, 2234, 2204} \[ \frac{e^{c+d x^2} \text{Erfi}(a+b x)}{2 d}-\frac{b e^{\frac{a^2 d}{b^2+d}+c} \text{Erfi}\left (\frac{a b+x \left (b^2+d\right )}{\sqrt{b^2+d}}\right )}{2 d \sqrt{b^2+d}} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x^2)*x*Erfi[a + b*x],x]

[Out]

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

Rule 6384

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Simp[(E^(c + d*x^2)*Erfi[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 2204

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

Rubi steps

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

Mathematica [A]  time = 0.0707114, size = 73, normalized size = 0.94 \[ \frac{e^c \left (e^{d x^2} \text{Erfi}(a+b x)-\frac{b e^{\frac{a^2 d}{b^2+d}} \text{Erfi}\left (\frac{a b+x \left (b^2+d\right )}{\sqrt{b^2+d}}\right )}{\sqrt{b^2+d}}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^c*(E^(d*x^2)*Erfi[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 [F]  time = 0.261, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{d{x}^{2}+c}}x{\it erfi} \left ( bx+a \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*erfi(b*x + a)*e^(d*x^2 + c), x)

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Fricas [A]  time = 2.67521, size = 221, normalized size = 2.83 \begin{align*} \frac{\sqrt{-b^{2} - d} b \operatorname{erf}\left (\frac{{\left (a b +{\left (b^{2} + d\right )} x\right )} \sqrt{-b^{2} - d}}{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{erfi}\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*erfi(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)/(b^2 + d))*e^((b^2*c + (a^2 + c)*d)/(b^2 + d)) +
(b^2 + d)*erfi(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*erfi(b*x+a),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*erfi(b*x + a)*e^(d*x^2 + c), x)