\(\int e^{c+b^2 x^2} x^4 \text {erfc}(b x) \, dx\) [173]

Optimal result

Integrand size = 19, antiderivative size = 138 \[ \int e^{c+b^2 x^2} x^4 \text {erfc}(b x) \, dx=-\frac {3 e^c x^2}{4 b^3 \sqrt {\pi }}+\frac {e^c x^4}{4 b \sqrt {\pi }}-\frac {3 e^{c+b^2 x^2} x \text {erfc}(b x)}{4 b^4}+\frac {e^{c+b^2 x^2} x^3 \text {erfc}(b x)}{2 b^2}+\frac {3 e^c \sqrt {\pi } \text {erfi}(b x)}{8 b^5}-\frac {3 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{4 b^3 \sqrt {\pi }} \]

[Out]

-3/4*exp(b^2*x^2+c)*x*erfc(b*x)/b^4+1/2*exp(b^2*x^2+c)*x^3*erfc(b*x)/b^2-3/4*exp(c)*x^2/b^3/Pi^(1/2)+1/4*exp(c
)*x^4/b/Pi^(1/2)-3/4*exp(c)*x^2*hypergeom([1, 1],[3/2, 2],b^2*x^2)/b^3/Pi^(1/2)+3/8*exp(c)*erfi(b*x)*Pi^(1/2)/
b^5

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6521, 6512, 2235, 6511, 12, 30} \[ \int e^{c+b^2 x^2} x^4 \text {erfc}(b x) \, dx=-\frac {3 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{4 \sqrt {\pi } b^3}+\frac {3 \sqrt {\pi } e^c \text {erfi}(b x)}{8 b^5}-\frac {3 e^c x^2}{4 \sqrt {\pi } b^3}+\frac {x^3 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}-\frac {3 x e^{b^2 x^2+c} \text {erfc}(b x)}{4 b^4}+\frac {e^c x^4}{4 \sqrt {\pi } b} \]

[In]

Int[E^(c + b^2*x^2)*x^4*Erfc[b*x],x]

[Out]

(-3*E^c*x^2)/(4*b^3*Sqrt[Pi]) + (E^c*x^4)/(4*b*Sqrt[Pi]) - (3*E^(c + b^2*x^2)*x*Erfc[b*x])/(4*b^4) + (E^(c + b
^2*x^2)*x^3*Erfc[b*x])/(2*b^2) + (3*E^c*Sqrt[Pi]*Erfi[b*x])/(8*b^5) - (3*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/
2, 2}, b^2*x^2])/(4*b^3*Sqrt[Pi])

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 2235

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 6511

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)], x_Symbol] :> Simp[b*E^c*(x^2/Sqrt[Pi])*HypergeometricPFQ[{1, 1},
{3/2, 2}, b^2*x^2], x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]

Rule 6512

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(b_.)*(x_)], x_Symbol] :> Int[E^(c + d*x^2), x] - Int[E^(c + d*x^2)*Erf[b*x]
, x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]

Rule 6521

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

Rubi steps \begin{align*} \text {integral}& = \frac {e^{c+b^2 x^2} x^3 \text {erfc}(b x)}{2 b^2}-\frac {3 \int e^{c+b^2 x^2} x^2 \text {erfc}(b x) \, dx}{2 b^2}+\frac {\int e^c x^3 \, dx}{b \sqrt {\pi }} \\ & = -\frac {3 e^{c+b^2 x^2} x \text {erfc}(b x)}{4 b^4}+\frac {e^{c+b^2 x^2} x^3 \text {erfc}(b x)}{2 b^2}+\frac {3 \int e^{c+b^2 x^2} \text {erfc}(b x) \, dx}{4 b^4}-\frac {3 \int e^c x \, dx}{2 b^3 \sqrt {\pi }}+\frac {e^c \int x^3 \, dx}{b \sqrt {\pi }} \\ & = \frac {e^c x^4}{4 b \sqrt {\pi }}-\frac {3 e^{c+b^2 x^2} x \text {erfc}(b x)}{4 b^4}+\frac {e^{c+b^2 x^2} x^3 \text {erfc}(b x)}{2 b^2}+\frac {3 \int e^{c+b^2 x^2} \, dx}{4 b^4}-\frac {3 \int e^{c+b^2 x^2} \text {erf}(b x) \, dx}{4 b^4}-\frac {\left (3 e^c\right ) \int x \, dx}{2 b^3 \sqrt {\pi }} \\ & = -\frac {3 e^c x^2}{4 b^3 \sqrt {\pi }}+\frac {e^c x^4}{4 b \sqrt {\pi }}-\frac {3 e^{c+b^2 x^2} x \text {erfc}(b x)}{4 b^4}+\frac {e^{c+b^2 x^2} x^3 \text {erfc}(b x)}{2 b^2}+\frac {3 e^c \sqrt {\pi } \text {erfi}(b x)}{8 b^5}-\frac {3 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{4 b^3 \sqrt {\pi }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.07 \[ \int e^{c+b^2 x^2} x^4 \text {erfc}(b x) \, dx=-\frac {e^c \left (6 b e^{b^2 x^2} \sqrt {\pi } x+6 b^2 x^2-4 b^3 e^{b^2 x^2} \sqrt {\pi } x^3-2 b^4 x^4+2 b e^{b^2 x^2} \sqrt {\pi } x \left (-3+2 b^2 x^2\right ) \text {erf}(b x)-3 \pi \text {erfi}(b x)+3 \pi \text {erf}(b x) \text {erfi}(b x)-6 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )\right )}{8 b^5 \sqrt {\pi }} \]

[In]

Integrate[E^(c + b^2*x^2)*x^4*Erfc[b*x],x]

[Out]

-1/8*(E^c*(6*b*E^(b^2*x^2)*Sqrt[Pi]*x + 6*b^2*x^2 - 4*b^3*E^(b^2*x^2)*Sqrt[Pi]*x^3 - 2*b^4*x^4 + 2*b*E^(b^2*x^
2)*Sqrt[Pi]*x*(-3 + 2*b^2*x^2)*Erf[b*x] - 3*Pi*Erfi[b*x] + 3*Pi*Erf[b*x]*Erfi[b*x] - 6*b^2*x^2*HypergeometricP
FQ[{1, 1}, {3/2, 2}, -(b^2*x^2)]))/(b^5*Sqrt[Pi])

Maple [F]

\[\int {\mathrm e}^{b^{2} x^{2}+c} x^{4} \operatorname {erfc}\left (b x \right )d x\]

[In]

int(exp(b^2*x^2+c)*x^4*erfc(b*x),x)

[Out]

int(exp(b^2*x^2+c)*x^4*erfc(b*x),x)

Fricas [F]

\[ \int e^{c+b^2 x^2} x^4 \text {erfc}(b x) \, dx=\int { x^{4} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \]

[In]

integrate(exp(b^2*x^2+c)*x^4*erfc(b*x),x, algorithm="fricas")

[Out]

integral(-(x^4*erf(b*x) - x^4)*e^(b^2*x^2 + c), x)

Sympy [F(-1)]

Timed out. \[ \int e^{c+b^2 x^2} x^4 \text {erfc}(b x) \, dx=\text {Timed out} \]

[In]

integrate(exp(b**2*x**2+c)*x**4*erfc(b*x),x)

[Out]

Timed out

Maxima [F]

\[ \int e^{c+b^2 x^2} x^4 \text {erfc}(b x) \, dx=\int { x^{4} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \]

[In]

integrate(exp(b^2*x^2+c)*x^4*erfc(b*x),x, algorithm="maxima")

[Out]

integrate(x^4*erfc(b*x)*e^(b^2*x^2 + c), x)

Giac [F]

\[ \int e^{c+b^2 x^2} x^4 \text {erfc}(b x) \, dx=\int { x^{4} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \]

[In]

integrate(exp(b^2*x^2+c)*x^4*erfc(b*x),x, algorithm="giac")

[Out]

integrate(x^4*erfc(b*x)*e^(b^2*x^2 + c), x)

Mupad [F(-1)]

Timed out. \[ \int e^{c+b^2 x^2} x^4 \text {erfc}(b x) \, dx=\int x^4\,{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfc}\left (b\,x\right ) \,d x \]

[In]

int(x^4*exp(c + b^2*x^2)*erfc(b*x),x)

[Out]

int(x^4*exp(c + b^2*x^2)*erfc(b*x), x)