3.65 \(\int e^{c+b^2 x^2} x^3 \text {erf}(b x) \, dx\)

Optimal. Leaf size=79 \[ \frac {e^c x}{\sqrt {\pi } b^3}+\frac {x^2 e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}-\frac {e^{b^2 x^2+c} \text {erf}(b x)}{2 b^4}-\frac {e^c x^3}{3 \sqrt {\pi } b} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6385, 6382, 8, 12, 30} \[ \frac {x^2 e^{b^2 x^2+c} \text {Erf}(b x)}{2 b^2}-\frac {e^{b^2 x^2+c} \text {Erf}(b x)}{2 b^4}+\frac {e^c x}{\sqrt {\pi } b^3}-\frac {e^c x^3}{3 \sqrt {\pi } b} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + b^2*x^2)*x^3*Erf[b*x],x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[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 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]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 57, normalized size = 0.72 \[ \frac {e^c \left (-2 b^3 x^3+3 \sqrt {\pi } e^{b^2 x^2} \left (b^2 x^2-1\right ) \text {erf}(b x)+6 b x\right )}{6 \sqrt {\pi } b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^c*(6*b*x - 2*b^3*x^3 + 3*E^(b^2*x^2)*Sqrt[Pi]*(-1 + b^2*x^2)*Erf[b*x]))/(6*b^4*Sqrt[Pi])

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fricas [A]  time = 0.51, size = 55, normalized size = 0.70 \[ -\frac {3 \, {\left (\pi - \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} + 2 \, \sqrt {\pi } {\left (b^{3} x^{3} - 3 \, b x\right )} e^{c}}{6 \, \pi b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*(pi - pi*b^2*x^2)*erf(b*x)*e^(b^2*x^2 + c) + 2*sqrt(pi)*(b^3*x^3 - 3*b*x)*e^c)/(pi*b^4)

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giac [A]  time = 0.27, size = 71, normalized size = 0.90 \[ \frac {1}{2} \, {\left (\frac {{\left (b^{2} x^{2} + c - 1\right )} e^{\left (b^{2} x^{2} + c\right )}}{b^{4}} - \frac {c e^{\left (b^{2} x^{2} + c\right )}}{b^{4}}\right )} \operatorname {erf}\left (b x\right ) - \frac {b^{2} x^{3} e^{c} - 3 \, x e^{c}}{3 \, \sqrt {\pi } b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.17, size = 66, normalized size = 0.84 \[ \frac {\frac {\erf \left (b x \right ) {\mathrm e}^{c} \left (\frac {b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}}{2}-\frac {{\mathrm e}^{b^{2} x^{2}}}{2}\right )}{b^{3}}-\frac {{\mathrm e}^{c} \left (\frac {1}{3} b^{3} x^{3}-b x \right )}{\sqrt {\pi }\, b^{3}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.34, size = 59, normalized size = 0.75 \[ -\frac {2 \, b^{3} x^{3} e^{c} - 3 \, {\left (\sqrt {\pi } b^{2} x^{2} e^{c} - \sqrt {\pi } e^{c}\right )} \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - 6 \, b x e^{c}}{6 \, \sqrt {\pi } b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*b^3*x^3*e^c - 3*(sqrt(pi)*b^2*x^2*e^c - sqrt(pi)*e^c)*erf(b*x)*e^(b^2*x^2) - 6*b*x*e^c)/(sqrt(pi)*b^4)

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mupad [B]  time = 0.20, size = 65, normalized size = 0.82 \[ \frac {3\,x\,{\mathrm {e}}^c-b^2\,x^3\,{\mathrm {e}}^c}{3\,b^3\,\sqrt {\pi }}-\mathrm {erf}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^4}-\frac {x^2\,{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*exp(c + b^2*x^2)*erf(b*x),x)

[Out]

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

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sympy [A]  time = 131.39, size = 76, normalized size = 0.96 \[ \begin {cases} - \frac {x^{3} e^{c}}{3 \sqrt {\pi } b} + \frac {x^{2} e^{c} e^{b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{2 b^{2}} + \frac {x e^{c}}{\sqrt {\pi } b^{3}} - \frac {e^{c} e^{b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{2 b^{4}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b**2*x**2+c)*x**3*erf(b*x),x)

[Out]

Piecewise((-x**3*exp(c)/(3*sqrt(pi)*b) + x**2*exp(c)*exp(b**2*x**2)*erf(b*x)/(2*b**2) + x*exp(c)/(sqrt(pi)*b**
3) - exp(c)*exp(b**2*x**2)*erf(b*x)/(2*b**4), Ne(b, 0)), (0, True))

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