3.104 \(\int x^5 \text{Erfc}(b x) \, dx\)

Optimal. Leaf size=96 \[ \frac{5 \text{Erf}(b x)}{16 b^6}-\frac{x^5 e^{-b^2 x^2}}{6 \sqrt{\pi } b}-\frac{5 x^3 e^{-b^2 x^2}}{12 \sqrt{\pi } b^3}-\frac{5 x e^{-b^2 x^2}}{8 \sqrt{\pi } b^5}+\frac{1}{6} x^6 \text{Erfc}(b x) \]

[Out]

(-5*x)/(8*b^5*E^(b^2*x^2)*Sqrt[Pi]) - (5*x^3)/(12*b^3*E^(b^2*x^2)*Sqrt[Pi]) - x^5/(6*b*E^(b^2*x^2)*Sqrt[Pi]) +
(5*Erf[b*x])/(16*b^6) + (x^6*Erfc[b*x])/6

________________________________________________________________________________________

Rubi [A]  time = 0.0897058, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6362, 2212, 2205} \[ \frac{5 \text{Erf}(b x)}{16 b^6}-\frac{x^5 e^{-b^2 x^2}}{6 \sqrt{\pi } b}-\frac{5 x^3 e^{-b^2 x^2}}{12 \sqrt{\pi } b^3}-\frac{5 x e^{-b^2 x^2}}{8 \sqrt{\pi } b^5}+\frac{1}{6} x^6 \text{Erfc}(b x) \]

Antiderivative was successfully verified.

[In]

Int[x^5*Erfc[b*x],x]

[Out]

(-5*x)/(8*b^5*E^(b^2*x^2)*Sqrt[Pi]) - (5*x^3)/(12*b^3*E^(b^2*x^2)*Sqrt[Pi]) - x^5/(6*b*E^(b^2*x^2)*Sqrt[Pi]) +
(5*Erf[b*x])/(16*b^6) + (x^6*Erfc[b*x])/6

Rule 6362

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

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])

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]

Rubi steps

\begin{align*} \int x^5 \text{erfc}(b x) \, dx &=\frac{1}{6} x^6 \text{erfc}(b x)+\frac{b \int e^{-b^2 x^2} x^6 \, dx}{3 \sqrt{\pi }}\\ &=-\frac{e^{-b^2 x^2} x^5}{6 b \sqrt{\pi }}+\frac{1}{6} x^6 \text{erfc}(b x)+\frac{5 \int e^{-b^2 x^2} x^4 \, dx}{6 b \sqrt{\pi }}\\ &=-\frac{5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt{\pi }}-\frac{e^{-b^2 x^2} x^5}{6 b \sqrt{\pi }}+\frac{1}{6} x^6 \text{erfc}(b x)+\frac{5 \int e^{-b^2 x^2} x^2 \, dx}{4 b^3 \sqrt{\pi }}\\ &=-\frac{5 e^{-b^2 x^2} x}{8 b^5 \sqrt{\pi }}-\frac{5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt{\pi }}-\frac{e^{-b^2 x^2} x^5}{6 b \sqrt{\pi }}+\frac{1}{6} x^6 \text{erfc}(b x)+\frac{5 \int e^{-b^2 x^2} \, dx}{8 b^5 \sqrt{\pi }}\\ &=-\frac{5 e^{-b^2 x^2} x}{8 b^5 \sqrt{\pi }}-\frac{5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt{\pi }}-\frac{e^{-b^2 x^2} x^5}{6 b \sqrt{\pi }}+\frac{5 \text{erf}(b x)}{16 b^6}+\frac{1}{6} x^6 \text{erfc}(b x)\\ \end{align*}

Mathematica [A]  time = 0.0581878, size = 62, normalized size = 0.65 \[ \frac{1}{48} \left (\frac{15 \text{Erf}(b x)}{b^6}-\frac{2 x e^{-b^2 x^2} \left (4 b^4 x^4+10 b^2 x^2+15\right )}{\sqrt{\pi } b^5}+8 x^6 \text{Erfc}(b x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*Erfc[b*x],x]

[Out]

((-2*x*(15 + 10*b^2*x^2 + 4*b^4*x^4))/(b^5*E^(b^2*x^2)*Sqrt[Pi]) + (15*Erf[b*x])/b^6 + 8*x^6*Erfc[b*x])/48

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 83, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{6}} \left ({\frac{{b}^{6}{x}^{6}{\it erfc} \left ( bx \right ) }{6}}+{\frac{1}{3\,\sqrt{\pi }} \left ( -{\frac{{b}^{5}{x}^{5}}{2\,{{\rm e}^{{b}^{2}{x}^{2}}}}}-{\frac{5\,{x}^{3}{b}^{3}}{4\,{{\rm e}^{{b}^{2}{x}^{2}}}}}-{\frac{15\,bx}{8\,{{\rm e}^{{b}^{2}{x}^{2}}}}}+{\frac{15\,\sqrt{\pi }{\it Erf} \left ( bx \right ) }{16}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*erfc(b*x),x)

[Out]

1/b^6*(1/6*b^6*x^6*erfc(b*x)+1/3/Pi^(1/2)*(-1/2/exp(b^2*x^2)*b^5*x^5-5/4*b^3*x^3/exp(b^2*x^2)-15/8*b*x/exp(b^2
*x^2)+15/16*Pi^(1/2)*erf(b*x)))

________________________________________________________________________________________

Maxima [A]  time = 1.00688, size = 85, normalized size = 0.89 \begin{align*} \frac{1}{6} \, x^{6} \operatorname{erfc}\left (b x\right ) - \frac{b{\left (\frac{2 \,{\left (4 \, b^{4} x^{5} + 10 \, b^{2} x^{3} + 15 \, x\right )} e^{\left (-b^{2} x^{2}\right )}}{b^{6}} - \frac{15 \, \sqrt{\pi } \operatorname{erf}\left (b x\right )}{b^{7}}\right )}}{48 \, \sqrt{\pi }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*erfc(b*x),x, algorithm="maxima")

[Out]

1/6*x^6*erfc(b*x) - 1/48*b*(2*(4*b^4*x^5 + 10*b^2*x^3 + 15*x)*e^(-b^2*x^2)/b^6 - 15*sqrt(pi)*erf(b*x)/b^7)/sqr
t(pi)

________________________________________________________________________________________

Fricas [A]  time = 2.40923, size = 167, normalized size = 1.74 \begin{align*} \frac{8 \, \pi b^{6} x^{6} - 2 \, \sqrt{\pi }{\left (4 \, b^{5} x^{5} + 10 \, b^{3} x^{3} + 15 \, b x\right )} e^{\left (-b^{2} x^{2}\right )} +{\left (15 \, \pi - 8 \, \pi b^{6} x^{6}\right )} \operatorname{erf}\left (b x\right )}{48 \, \pi b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*erfc(b*x),x, algorithm="fricas")

[Out]

1/48*(8*pi*b^6*x^6 - 2*sqrt(pi)*(4*b^5*x^5 + 10*b^3*x^3 + 15*b*x)*e^(-b^2*x^2) + (15*pi - 8*pi*b^6*x^6)*erf(b*
x))/(pi*b^6)

________________________________________________________________________________________

Sympy [A]  time = 4.05136, size = 92, normalized size = 0.96 \begin{align*} \begin{cases} \frac{x^{6} \operatorname{erfc}{\left (b x \right )}}{6} - \frac{x^{5} e^{- b^{2} x^{2}}}{6 \sqrt{\pi } b} - \frac{5 x^{3} e^{- b^{2} x^{2}}}{12 \sqrt{\pi } b^{3}} - \frac{5 x e^{- b^{2} x^{2}}}{8 \sqrt{\pi } b^{5}} - \frac{5 \operatorname{erfc}{\left (b x \right )}}{16 b^{6}} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*erfc(b*x),x)

[Out]

Piecewise((x**6*erfc(b*x)/6 - x**5*exp(-b**2*x**2)/(6*sqrt(pi)*b) - 5*x**3*exp(-b**2*x**2)/(12*sqrt(pi)*b**3)
- 5*x*exp(-b**2*x**2)/(8*sqrt(pi)*b**5) - 5*erfc(b*x)/(16*b**6), Ne(b, 0)), (x**6/6, True))

________________________________________________________________________________________

Giac [A]  time = 1.37257, size = 93, normalized size = 0.97 \begin{align*} -\frac{1}{6} \, x^{6} \operatorname{erf}\left (b x\right ) + \frac{1}{6} \, x^{6} - \frac{b{\left (\frac{2 \,{\left (4 \, b^{4} x^{5} + 10 \, b^{2} x^{3} + 15 \, x\right )} e^{\left (-b^{2} x^{2}\right )}}{b^{6}} + \frac{15 \, \sqrt{\pi } \operatorname{erf}\left (-b x\right )}{b^{7}}\right )}}{48 \, \sqrt{\pi }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*erfc(b*x),x, algorithm="giac")

[Out]

-1/6*x^6*erf(b*x) + 1/6*x^6 - 1/48*b*(2*(4*b^4*x^5 + 10*b^2*x^3 + 15*x)*e^(-b^2*x^2)/b^6 + 15*sqrt(pi)*erf(-b*
x)/b^7)/sqrt(pi)