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

Optimal. Leaf size=342 \[ \frac{b e^{\frac{a^2 d}{b^2-d}+c} \text{Erf}\left (\frac{a b+x \left (b^2-d\right )}{\sqrt{b^2-d}}\right )}{2 d^2 \sqrt{b^2-d}}-\frac{a^2 b^3 e^{\frac{a^2 d}{b^2-d}+c} \text{Erf}\left (\frac{a b+x \left (b^2-d\right )}{\sqrt{b^2-d}}\right )}{2 d \left (b^2-d\right )^{5/2}}-\frac{b e^{\frac{a^2 d}{b^2-d}+c} \text{Erf}\left (\frac{a b+x \left (b^2-d\right )}{\sqrt{b^2-d}}\right )}{4 d \left (b^2-d\right )^{3/2}}-\frac{a b^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \sqrt{\pi } d \left (b^2-d\right )^2}+\frac{b x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \sqrt{\pi } d \left (b^2-d\right )}-\frac{e^{c+d x^2} \text{Erf}(a+b x)}{2 d^2}+\frac{x^2 e^{c+d x^2} \text{Erf}(a+b x)}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.514996, antiderivative size = 342, 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 = {6385, 6382, 2234, 2205, 2241, 2240} \[ \frac{b e^{\frac{a^2 d}{b^2-d}+c} \text{Erf}\left (\frac{a b+x \left (b^2-d\right )}{\sqrt{b^2-d}}\right )}{2 d^2 \sqrt{b^2-d}}-\frac{a^2 b^3 e^{\frac{a^2 d}{b^2-d}+c} \text{Erf}\left (\frac{a b+x \left (b^2-d\right )}{\sqrt{b^2-d}}\right )}{2 d \left (b^2-d\right )^{5/2}}-\frac{b e^{\frac{a^2 d}{b^2-d}+c} \text{Erf}\left (\frac{a b+x \left (b^2-d\right )}{\sqrt{b^2-d}}\right )}{4 d \left (b^2-d\right )^{3/2}}-\frac{a b^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \sqrt{\pi } d \left (b^2-d\right )^2}+\frac{b x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \sqrt{\pi } d \left (b^2-d\right )}-\frac{e^{c+d x^2} \text{Erf}(a+b x)}{2 d^2}+\frac{x^2 e^{c+d x^2} \text{Erf}(a+b x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x^2)*x^3*Erf[a + b*x],x]

[Out]

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

Rule 6385

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m - 1)*E^(c + d*x^2)*Erf
[a + b*x])/(2*d), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erf[a + b*x], x], x] - Dist[b/(d*Sqrt
[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 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]

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

Mathematica [A]  time = 4.59637, size = 240, normalized size = 0.7 \[ \frac{e^c \left (-\frac{b d e^{-a^2-2 a b x+x^2 \left (d-b^2\right )} \left (\sqrt{\pi } \sqrt{b^2-d} \left (\left (2 a^2+1\right ) b^2-d\right ) e^{\frac{\left (a b+x \left (b^2-d\right )\right )^2}{b^2-d}} \text{Erf}\left (\frac{a b+x \left (b^2-d\right )}{\sqrt{b^2-d}}\right )+2 \left (b^2-d\right ) \left (a b+x \left (d-b^2\right )\right )\right )}{\sqrt{\pi } \left (b^2-d\right )^3}+\frac{2 b e^{\frac{a^2 d}{b^2-d}} \text{Erfi}\left (\frac{x \left (d-b^2\right )-a b}{\sqrt{d-b^2}}\right )}{\sqrt{d-b^2}}+2 e^{d x^2} \left (d x^2-1\right ) \text{Erf}(a+b x)\right )}{4 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.319, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{d{x}^{2}+c}}{x}^{3}{\it Erf} \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*erf(b*x+a),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (d x^{2} e^{c} - e^{c}\right )} \operatorname{erf}\left (b x + a\right ) e^{\left (d x^{2}\right )}}{2 \, d^{2}} - \frac{-\frac{{\left (\frac{\sqrt{\pi }{\left (a b +{\left (b^{2} - d\right )} x\right )} a^{2} b^{2}{\left (\operatorname{erf}\left (\sqrt{\frac{{\left (a b +{\left (b^{2} - d\right )} x\right )}^{2}}{b^{2} - d}}\right ) - 1\right )}}{{\left (-b^{2} + d\right )}^{\frac{5}{2}} \sqrt{\frac{{\left (a b +{\left (b^{2} - d\right )} x\right )}^{2}}{b^{2} - d}}} - \frac{2 \, a b e^{\left (-\frac{{\left (a b +{\left (b^{2} - d\right )} x\right )}^{2}}{b^{2} - d}\right )}}{{\left (-b^{2} + d\right )}^{\frac{3}{2}}} - \frac{{\left (a b +{\left (b^{2} - d\right )} x\right )}^{3} \Gamma \left (\frac{3}{2}, \frac{{\left (a b +{\left (b^{2} - d\right )} x\right )}^{2}}{b^{2} - d}\right )}{{\left (-b^{2} + d\right )}^{\frac{5}{2}} \left (\frac{{\left (a b +{\left (b^{2} - d\right )} x\right )}^{2}}{b^{2} - d}\right )^{\frac{3}{2}}}\right )} b d e^{\left (\frac{a^{2} b^{2}}{b^{2} - d} - a^{2} + c\right )}}{2 \, \sqrt{-b^{2} + d}} - \frac{\sqrt{\pi } 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}}}{\sqrt{\pi } d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.81226, size = 549, normalized size = 1.61 \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{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 )} + 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{erf}\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*erf(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))*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))*erf(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))

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

[Out]

Timed out

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Giac [A]  time = 1.29738, size = 367, normalized size = 1.07 \begin{align*} \frac{{\left (d x^{2} - 1\right )} \operatorname{erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{2 \, d^{2}} - \frac{\frac{2 \, \pi 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 )}}{\sqrt{b^{2} - d}} - \frac{\sqrt{\pi }{\left (\frac{\sqrt{\pi }{\left (2 \, a^{2} b^{2} + b^{2} - d\right )} \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 )}}{\sqrt{b^{2} - d}} + 2 \,{\left ({\left (\frac{a b}{b^{2} - d} + x\right )} b^{2} - 2 \, a b -{\left (\frac{a b}{b^{2} - d} + x\right )} d\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x + d x^{2} - a^{2} + c\right )}\right )} b d}{b^{4} - 2 \, b^{2} d + d^{2}}}{4 \, \pi d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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