3.294 \(\int e^{c+d x^2} x^3 \text{Erfi}(a+b x) \, dx\)

Optimal. Leaf size=304 \[ \frac{b e^{\frac{a^2 d}{b^2+d}+c} \text{Erfi}\left (\frac{a b+x \left (b^2+d\right )}{\sqrt{b^2+d}}\right )}{2 d^2 \sqrt{b^2+d}}-\frac{a^2 b^3 e^{\frac{a^2 d}{b^2+d}+c} \text{Erfi}\left (\frac{a b+x \left (b^2+d\right )}{\sqrt{b^2+d}}\right )}{2 d \left (b^2+d\right )^{5/2}}+\frac{b e^{\frac{a^2 d}{b^2+d}+c} \text{Erfi}\left (\frac{a b+x \left (b^2+d\right )}{\sqrt{b^2+d}}\right )}{4 d \left (b^2+d\right )^{3/2}}+\frac{a b^2 e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 \sqrt{\pi } d \left (b^2+d\right )^2}-\frac{b x e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 \sqrt{\pi } d \left (b^2+d\right )}-\frac{e^{c+d x^2} \text{Erfi}(a+b x)}{2 d^2}+\frac{x^2 e^{c+d x^2} \text{Erfi}(a+b x)}{2 d} \]

[Out]

(a*b^2*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2))/(2*d*(b^2 + d)^2*Sqrt[Pi]) - (b*E^(a^2 + c + 2*a*b*x + (b^2 + d)
*x^2)*x)/(2*d*(b^2 + d)*Sqrt[Pi]) - (E^(c + d*x^2)*Erfi[a + b*x])/(2*d^2) + (E^(c + d*x^2)*x^2*Erfi[a + b*x])/
(2*d) - (a^2*b^3*E^(c + (a^2*d)/(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]])/(2*d*(b^2 + d)^(5/2)) + (b
*E^(c + (a^2*d)/(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]])/(4*d*(b^2 + d)^(3/2)) + (b*E^(c + (a^2*d)/
(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]])/(2*d^2*Sqrt[b^2 + d])

________________________________________________________________________________________

Rubi [A]  time = 0.468279, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {6387, 6384, 2234, 2204, 2241, 2240} \[ \frac{b e^{\frac{a^2 d}{b^2+d}+c} \text{Erfi}\left (\frac{a b+x \left (b^2+d\right )}{\sqrt{b^2+d}}\right )}{2 d^2 \sqrt{b^2+d}}-\frac{a^2 b^3 e^{\frac{a^2 d}{b^2+d}+c} \text{Erfi}\left (\frac{a b+x \left (b^2+d\right )}{\sqrt{b^2+d}}\right )}{2 d \left (b^2+d\right )^{5/2}}+\frac{b e^{\frac{a^2 d}{b^2+d}+c} \text{Erfi}\left (\frac{a b+x \left (b^2+d\right )}{\sqrt{b^2+d}}\right )}{4 d \left (b^2+d\right )^{3/2}}+\frac{a b^2 e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 \sqrt{\pi } d \left (b^2+d\right )^2}-\frac{b x e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 \sqrt{\pi } d \left (b^2+d\right )}-\frac{e^{c+d x^2} \text{Erfi}(a+b x)}{2 d^2}+\frac{x^2 e^{c+d x^2} \text{Erfi}(a+b x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x^2)*x^3*Erfi[a + b*x],x]

[Out]

(a*b^2*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2))/(2*d*(b^2 + d)^2*Sqrt[Pi]) - (b*E^(a^2 + c + 2*a*b*x + (b^2 + d)
*x^2)*x)/(2*d*(b^2 + d)*Sqrt[Pi]) - (E^(c + d*x^2)*Erfi[a + b*x])/(2*d^2) + (E^(c + d*x^2)*x^2*Erfi[a + b*x])/
(2*d) - (a^2*b^3*E^(c + (a^2*d)/(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]])/(2*d*(b^2 + d)^(5/2)) + (b
*E^(c + (a^2*d)/(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]])/(4*d*(b^2 + d)^(3/2)) + (b*E^(c + (a^2*d)/
(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]])/(2*d^2*Sqrt[b^2 + d])

Rule 6387

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m - 1)*E^(c + d*x^2)*Er
fi[a + b*x])/(2*d), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erfi[a + b*x], x], x] - Dist[b/(d*S
qrt[Pi]), Int[x^(m - 1)*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2), x], x]) /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1]

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]

Rule 2241

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*F^(a + b*x + c*x^2))/(2*c*Log[F]), x] + (-Dist[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*F^(a + b*x + c*x^2)
, x], x] - Dist[((m - 1)*e^2)/(2*c*Log[F]), Int[(d + e*x)^(m - 2)*F^(a + b*x + c*x^2), x], x]) /; FreeQ[{F, a,
b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]

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]

Rubi steps

\begin{align*} \int e^{c+d x^2} x^3 \text{erfi}(a+b x) \, dx &=\frac{e^{c+d x^2} x^2 \text{erfi}(a+b x)}{2 d}-\frac{\int e^{c+d x^2} x \text{erfi}(a+b x) \, dx}{d}-\frac{b \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x^2 \, dx}{d \sqrt{\pi }}\\ &=-\frac{b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right ) \sqrt{\pi }}-\frac{e^{c+d x^2} \text{erfi}(a+b x)}{2 d^2}+\frac{e^{c+d x^2} 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^2 \sqrt{\pi }}+\frac{b \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} \, dx}{2 d \left (b^2+d\right ) \sqrt{\pi }}+\frac{\left (a b^2\right ) \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x \, dx}{d \left (b^2+d\right ) \sqrt{\pi }}\\ &=\frac{a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^2 \sqrt{\pi }}-\frac{b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right ) \sqrt{\pi }}-\frac{e^{c+d x^2} \text{erfi}(a+b x)}{2 d^2}+\frac{e^{c+d x^2} x^2 \text{erfi}(a+b x)}{2 d}-\frac{\left (a^2 b^3\right ) \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} \, dx}{d \left (b^2+d\right )^2 \sqrt{\pi }}+\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^2 \sqrt{\pi }}+\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}{2 d \left (b^2+d\right ) \sqrt{\pi }}\\ &=\frac{a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^2 \sqrt{\pi }}-\frac{b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right ) \sqrt{\pi }}-\frac{e^{c+d x^2} \text{erfi}(a+b x)}{2 d^2}+\frac{e^{c+d x^2} 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 )}{4 d \left (b^2+d\right )^{3/2}}+\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^2 \sqrt{b^2+d}}-\frac{\left (a^2 b^3 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 \left (b^2+d\right )^2 \sqrt{\pi }}\\ &=\frac{a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^2 \sqrt{\pi }}-\frac{b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right ) \sqrt{\pi }}-\frac{e^{c+d x^2} \text{erfi}(a+b x)}{2 d^2}+\frac{e^{c+d x^2} x^2 \text{erfi}(a+b x)}{2 d}-\frac{a^2 b^3 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 \left (b^2+d\right )^{5/2}}+\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 )}{4 d \left (b^2+d\right )^{3/2}}+\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^2 \sqrt{b^2+d}}\\ \end{align*}

Mathematica [A]  time = 2.11667, size = 206, normalized size = 0.68 \[ \frac{e^c \left (\frac{2 b e^{\frac{a^2 d}{b^2+d}} \text{Erfi}\left (\frac{a b+x \left (b^2+d\right )}{\sqrt{b^2+d}}\right )}{\sqrt{b^2+d}}-\frac{b d e^{\frac{a^2 d}{b^2+d}} \left (\sqrt{\pi } \sqrt{b^2+d} \left (\left (2 a^2-1\right ) b^2-d\right ) \text{Erfi}\left (\frac{a b+x \left (b^2+d\right )}{\sqrt{b^2+d}}\right )+2 \left (b^2+d\right ) e^{\frac{\left (a b+x \left (b^2+d\right )\right )^2}{b^2+d}} \left (x \left (b^2+d\right )-a b\right )\right )}{\sqrt{\pi } \left (b^2+d\right )^3}+2 e^{d x^2} \left (d x^2-1\right ) \text{Erfi}(a+b x)\right )}{4 d^2} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(c + d*x^2)*x^3*Erfi[a + b*x],x]

[Out]

(E^c*(2*E^(d*x^2)*(-1 + d*x^2)*Erfi[a + b*x] + (2*b*E^((a^2*d)/(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 +
d]])/Sqrt[b^2 + d] - (b*d*E^((a^2*d)/(b^2 + d))*(2*(b^2 + d)*E^((a*b + (b^2 + d)*x)^2/(b^2 + d))*(-(a*b) + (b^
2 + d)*x) + ((-1 + 2*a^2)*b^2 - d)*Sqrt[b^2 + d]*Sqrt[Pi]*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]]))/((b^2 + d)
^3*Sqrt[Pi])))/(4*d^2)

________________________________________________________________________________________

Maple [F]  time = 0.383, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{d{x}^{2}+c}}{x}^{3}{\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^3*erfi(b*x+a),x)

[Out]

int(exp(d*x^2+c)*x^3*erfi(b*x+a),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \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^3*erfi(b*x+a),x, algorithm="maxima")

[Out]

integrate(x^3*erfi(b*x + a)*e^(d*x^2 + c), x)

________________________________________________________________________________________

Fricas [A]  time = 2.71188, size = 567, normalized size = 1.87 \begin{align*} -\frac{\pi{\left (2 \, b^{5} -{\left (2 \, a^{2} - 5\right )} b^{3} d + 3 \, b d^{2}\right )} \sqrt{-b^{2} - d} \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 )} - 2 \,{\left (\pi{\left (b^{6} d + 3 \, b^{4} d^{2} + 3 \, b^{2} d^{3} + d^{4}\right )} x^{2} - \pi{\left (b^{6} + 3 \, b^{4} d + 3 \, b^{2} d^{2} + d^{3}\right )}\right )} \operatorname{erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} - 2 \, \sqrt{\pi }{\left (a b^{4} d + a b^{2} d^{2} -{\left (b^{5} d + 2 \, b^{3} d^{2} + b d^{3}\right )} x\right )} e^{\left (b^{2} x^{2} + 2 \, a b x + d x^{2} + a^{2} + c\right )}}{4 \, \pi{\left (b^{6} d^{2} + 3 \, b^{4} d^{3} + 3 \, b^{2} d^{4} + d^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x^2+c)*x^3*erfi(b*x+a),x, algorithm="fricas")

[Out]

-1/4*(pi*(2*b^5 - (2*a^2 - 5)*b^3*d + 3*b*d^2)*sqrt(-b^2 - d)*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)) - 2*(pi*(b^6*d + 3*b^4*d^2 + 3*b^2*d^3 + d^4)*x^2 - pi*(b^6 + 3*b^4*d +
3*b^2*d^2 + d^3))*erfi(b*x + a)*e^(d*x^2 + c) - 2*sqrt(pi)*(a*b^4*d + a*b^2*d^2 - (b^5*d + 2*b^3*d^2 + b*d^3)*
x)*e^(b^2*x^2 + 2*a*b*x + d*x^2 + a^2 + c))/(pi*(b^6*d^2 + 3*b^4*d^3 + 3*b^2*d^4 + d^5))

________________________________________________________________________________________

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**3*erfi(b*x+a),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \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^3*erfi(b*x+a),x, algorithm="giac")

[Out]

integrate(x^3*erfi(b*x + a)*e^(d*x^2 + c), x)