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

Optimal. Leaf size=155 \[ \frac{b e^c \text{Erf}\left (x \sqrt{b^2-d}\right )}{2 d^2 \sqrt{b^2-d}}-\frac{b e^c \text{Erf}\left (x \sqrt{b^2-d}\right )}{4 d \left (b^2-d\right )^{3/2}}+\frac{b x e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt{\pi } d \left (b^2-d\right )}-\frac{\text{Erf}(b x) e^{c+d x^2}}{2 d^2}+\frac{x^2 \text{Erf}(b x) e^{c+d x^2}}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.157916, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {6385, 6382, 2205, 2212} \[ \frac{b e^c \text{Erf}\left (x \sqrt{b^2-d}\right )}{2 d^2 \sqrt{b^2-d}}-\frac{b e^c \text{Erf}\left (x \sqrt{b^2-d}\right )}{4 d \left (b^2-d\right )^{3/2}}+\frac{b x e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt{\pi } d \left (b^2-d\right )}-\frac{\text{Erf}(b x) e^{c+d x^2}}{2 d^2}+\frac{x^2 \text{Erf}(b x) e^{c+d x^2}}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x^2)*x^3*Erf[b*x],x]

[Out]

(b*E^(c - (b^2 - d)*x^2)*x)/(2*(b^2 - d)*d*Sqrt[Pi]) - (E^(c + d*x^2)*Erf[b*x])/(2*d^2) + (E^(c + d*x^2)*x^2*E
rf[b*x])/(2*d) + (b*E^c*Erf[Sqrt[b^2 - d]*x])/(2*Sqrt[b^2 - d]*d^2) - (b*E^c*Erf[Sqrt[b^2 - d]*x])/(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 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 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rubi steps

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

Mathematica [A]  time = 0.278587, size = 99, normalized size = 0.64 \[ \frac{e^c \left (\frac{b \left (3 d-2 b^2\right ) \text{Erfi}\left (x \sqrt{d-b^2}\right )}{\left (d-b^2\right )^{3/2}}+\frac{2 b d x e^{x^2 \left (d-b^2\right )}}{\sqrt{\pi } \left (b^2-d\right )}+2 e^{d x^2} \left (d x^2-1\right ) \text{Erf}(b x)\right )}{4 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.35, size = 168, normalized size = 1.1 \begin{align*}{\frac{1}{b} \left ({\frac{{\it Erf} \left ( bx \right ){{\rm e}^{c}}}{{b}^{3}} \left ({\frac{{b}^{4}{x}^{2}{{\rm e}^{d{x}^{2}}}}{2\,d}}-{\frac{{b}^{4}{{\rm e}^{d{x}^{2}}}}{2\,{d}^{2}}} \right ) }-{\frac{{{\rm e}^{c}}}{{b}^{3}\sqrt{\pi }} \left ({\frac{{b}^{2}}{d} \left ({\frac{bx}{2}{{\rm e}^{ \left ( -1+{\frac{d}{{b}^{2}}} \right ){b}^{2}{x}^{2}}} \left ( -1+{\frac{d}{{b}^{2}}} \right ) ^{-1}}-{\frac{\sqrt{\pi }}{4}{\it Erf} \left ( \sqrt{1-{\frac{d}{{b}^{2}}}}bx \right ) \left ( -1+{\frac{d}{{b}^{2}}} \right ) ^{-1}{\frac{1}{\sqrt{1-{\frac{d}{{b}^{2}}}}}}} \right ) }-{\frac{{b}^{4}\sqrt{\pi }}{2\,{d}^{2}}{\it Erf} \left ( \sqrt{1-{\frac{d}{{b}^{2}}}}bx \right ){\frac{1}{\sqrt{1-{\frac{d}{{b}^{2}}}}}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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\right ) e^{\left (d x^{2}\right )}}{2 \, d^{2}} - \frac{-\frac{b d x^{3} e^{c} \Gamma \left (\frac{3}{2},{\left (b^{2} - d\right )} x^{2}\right )}{2 \, \left ({\left (b^{2} - d\right )} x^{2}\right )^{\frac{3}{2}}} - \frac{\sqrt{\pi } b \operatorname{erf}\left (\sqrt{b^{2} - d} x\right ) e^{c}}{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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 3.22417, size = 328, normalized size = 2.12 \begin{align*} \frac{\pi{\left (2 \, b^{3} - 3 \, b d\right )} \sqrt{b^{2} - d} \operatorname{erf}\left (\sqrt{b^{2} - d} x\right ) e^{c} + 2 \, \sqrt{\pi }{\left (b^{3} d - b d^{2}\right )} x e^{\left (-b^{2} x^{2} + d x^{2} + c\right )} + 2 \,{\left (\pi{\left (b^{4} d - 2 \, b^{2} d^{2} + d^{3}\right )} x^{2} - \pi{\left (b^{4} - 2 \, b^{2} d + d^{2}\right )}\right )} \operatorname{erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{4 \, \pi{\left (b^{4} d^{2} - 2 \, b^{2} d^{3} + d^{4}\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),x, algorithm="fricas")

[Out]

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

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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),x)

[Out]

Timed out

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Giac [A]  time = 1.31417, size = 169, normalized size = 1.09 \begin{align*} \frac{{\left (d x^{2} - 1\right )} \operatorname{erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{2 \, d^{2}} + \frac{\sqrt{\pi } b d{\left (\frac{2 \, x e^{\left (-b^{2} x^{2} + d x^{2} + c\right )}}{b^{2} - d} + \frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{b^{2} - d} x\right ) e^{c}}{{\left (b^{2} - d\right )}^{\frac{3}{2}}}\right )} - \frac{2 \, \pi b \operatorname{erf}\left (-\sqrt{b^{2} - d} x\right ) e^{c}}{\sqrt{b^{2} - d}}}{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),x, algorithm="giac")

[Out]

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