\(\int e^{c+b^2 x^2} x^5 \text {erfc}(b x) \, dx\) [167]

Optimal result

Integrand size = 19, antiderivative size = 118 \[ \int e^{c+b^2 x^2} x^5 \text {erfc}(b x) \, dx=\frac {2 e^c x}{b^5 \sqrt {\pi }}-\frac {2 e^c x^3}{3 b^3 \sqrt {\pi }}+\frac {e^c x^5}{5 b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{b^6}-\frac {e^{c+b^2 x^2} x^2 \text {erfc}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erfc}(b x)}{2 b^2} \]

[Out]

exp(b^2*x^2+c)*erfc(b*x)/b^6-exp(b^2*x^2+c)*x^2*erfc(b*x)/b^4+1/2*exp(b^2*x^2+c)*x^4*erfc(b*x)/b^2+2*exp(c)*x/
b^5/Pi^(1/2)-2/3*exp(c)*x^3/b^3/Pi^(1/2)+1/5*exp(c)*x^5/b/Pi^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6521, 6518, 8, 12, 30} \[ \int e^{c+b^2 x^2} x^5 \text {erfc}(b x) \, dx=\frac {2 e^c x}{\sqrt {\pi } b^5}-\frac {2 e^c x^3}{3 \sqrt {\pi } b^3}+\frac {x^4 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{b^6}-\frac {x^2 e^{b^2 x^2+c} \text {erfc}(b x)}{b^4}+\frac {e^c x^5}{5 \sqrt {\pi } b} \]

[In]

Int[E^(c + b^2*x^2)*x^5*Erfc[b*x],x]

[Out]

(2*E^c*x)/(b^5*Sqrt[Pi]) - (2*E^c*x^3)/(3*b^3*Sqrt[Pi]) + (E^c*x^5)/(5*b*Sqrt[Pi]) + (E^(c + b^2*x^2)*Erfc[b*x
])/b^6 - (E^(c + b^2*x^2)*x^2*Erfc[b*x])/b^4 + (E^(c + b^2*x^2)*x^4*Erfc[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 6518

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Simp[E^(c + d*x^2)*(Erfc[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 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^4 \text {erfc}(b x)}{2 b^2}-\frac {2 \int e^{c+b^2 x^2} x^3 \text {erfc}(b x) \, dx}{b^2}+\frac {\int e^c x^4 \, dx}{b \sqrt {\pi }} \\ & = -\frac {e^{c+b^2 x^2} x^2 \text {erfc}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erfc}(b x)}{2 b^2}+\frac {2 \int e^{c+b^2 x^2} x \text {erfc}(b x) \, dx}{b^4}-\frac {2 \int e^c x^2 \, dx}{b^3 \sqrt {\pi }}+\frac {e^c \int x^4 \, dx}{b \sqrt {\pi }} \\ & = \frac {e^c x^5}{5 b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{b^6}-\frac {e^{c+b^2 x^2} x^2 \text {erfc}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erfc}(b x)}{2 b^2}+\frac {2 \int e^c \, dx}{b^5 \sqrt {\pi }}-\frac {\left (2 e^c\right ) \int x^2 \, dx}{b^3 \sqrt {\pi }} \\ & = \frac {2 e^c x}{b^5 \sqrt {\pi }}-\frac {2 e^c x^3}{3 b^3 \sqrt {\pi }}+\frac {e^c x^5}{5 b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{b^6}-\frac {e^{c+b^2 x^2} x^2 \text {erfc}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erfc}(b x)}{2 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.62 \[ \int e^{c+b^2 x^2} x^5 \text {erfc}(b x) \, dx=\frac {e^c \left (60 b x-20 b^3 x^3+6 b^5 x^5+15 e^{b^2 x^2} \sqrt {\pi } \left (2-2 b^2 x^2+b^4 x^4\right ) \text {erfc}(b x)\right )}{30 b^6 \sqrt {\pi }} \]

[In]

Integrate[E^(c + b^2*x^2)*x^5*Erfc[b*x],x]

[Out]

(E^c*(60*b*x - 20*b^3*x^3 + 6*b^5*x^5 + 15*E^(b^2*x^2)*Sqrt[Pi]*(2 - 2*b^2*x^2 + b^4*x^4)*Erfc[b*x]))/(30*b^6*
Sqrt[Pi])

Maple [A] (verified)

Time = 2.60 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.14

method	result	size
default	\(\frac {\frac {{\mathrm e}^{c} \left (\frac {{\mathrm e}^{b^{2} x^{2}} b^{4} x^{4}}{2}-b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}+{\mathrm e}^{b^{2} x^{2}}\right )}{b^{5}}-\frac {\operatorname {erf}\left (b x \right ) {\mathrm e}^{c} \left (\frac {{\mathrm e}^{b^{2} x^{2}} b^{4} x^{4}}{2}-b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}+{\mathrm e}^{b^{2} x^{2}}\right )}{b^{5}}+\frac {{\mathrm e}^{c} \left (\frac {1}{5} b^{5} x^{5}-\frac {2}{3} b^{3} x^{3}+2 b x \right )}{\sqrt {\pi }\, b^{5}}}{b}\)	\(135\)
parallelrisch	\(\frac {6 \,{\mathrm e}^{b^{2} x^{2}+c} {\mathrm e}^{-b^{2} x^{2}} x^{5} b^{5}+15 \,{\mathrm e}^{b^{2} x^{2}+c} x^{4} \operatorname {erfc}\left (b x \right ) b^{4} \sqrt {\pi }-20 \,{\mathrm e}^{b^{2} x^{2}+c} x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}-30 \,{\mathrm e}^{b^{2} x^{2}+c} x^{2} \operatorname {erfc}\left (b x \right ) b^{2} \sqrt {\pi }+60 \,{\mathrm e}^{b^{2} x^{2}+c} x \,{\mathrm e}^{-b^{2} x^{2}} b +30 \,{\mathrm e}^{b^{2} x^{2}+c} \operatorname {erfc}\left (b x \right ) \sqrt {\pi }}{30 b^{6} \sqrt {\pi }}\)	\(156\)

[In]

int(exp(b^2*x^2+c)*x^5*erfc(b*x),x,method=_RETURNVERBOSE)

[Out]

(1/b^5*exp(c)*(1/2*exp(b^2*x^2)*b^4*x^4-b^2*x^2*exp(b^2*x^2)+exp(b^2*x^2))-erf(b*x)/b^5*exp(c)*(1/2*exp(b^2*x^
2)*b^4*x^4-b^2*x^2*exp(b^2*x^2)+exp(b^2*x^2))+1/Pi^(1/2)/b^5*exp(c)*(1/5*b^5*x^5-2/3*b^3*x^3+2*b*x))/b

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.82 \[ \int e^{c+b^2 x^2} x^5 \text {erfc}(b x) \, dx=\frac {2 \, \sqrt {\pi } {\left (3 \, b^{5} x^{5} - 10 \, b^{3} x^{3} + 30 \, b x\right )} e^{c} + 15 \, {\left (2 \, \pi + \pi b^{4} x^{4} - 2 \, \pi b^{2} x^{2} - {\left (2 \, \pi + \pi b^{4} x^{4} - 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (b^{2} x^{2} + c\right )}}{30 \, \pi b^{6}} \]

[In]

integrate(exp(b^2*x^2+c)*x^5*erfc(b*x),x, algorithm="fricas")

[Out]

1/30*(2*sqrt(pi)*(3*b^5*x^5 - 10*b^3*x^3 + 30*b*x)*e^c + 15*(2*pi + pi*b^4*x^4 - 2*pi*b^2*x^2 - (2*pi + pi*b^4
*x^4 - 2*pi*b^2*x^2)*erf(b*x))*e^(b^2*x^2 + c))/(pi*b^6)

Sympy [A] (verification not implemented)

Time = 33.43 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.07 \[ \int e^{c+b^2 x^2} x^5 \text {erfc}(b x) \, dx=\begin {cases} \frac {x^{5} e^{c}}{5 \sqrt {\pi } b} + \frac {x^{4} e^{c} e^{b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{2 b^{2}} - \frac {2 x^{3} e^{c}}{3 \sqrt {\pi } b^{3}} - \frac {x^{2} e^{c} e^{b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{b^{4}} + \frac {2 x e^{c}}{\sqrt {\pi } b^{5}} + \frac {e^{c} e^{b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{b^{6}} & \text {for}\: b \neq 0 \\\frac {x^{6} e^{c}}{6} & \text {otherwise} \end {cases} \]

[In]

integrate(exp(b**2*x**2+c)*x**5*erfc(b*x),x)

[Out]

Piecewise((x**5*exp(c)/(5*sqrt(pi)*b) + x**4*exp(c)*exp(b**2*x**2)*erfc(b*x)/(2*b**2) - 2*x**3*exp(c)/(3*sqrt(
pi)*b**3) - x**2*exp(c)*exp(b**2*x**2)*erfc(b*x)/b**4 + 2*x*exp(c)/(sqrt(pi)*b**5) + exp(c)*exp(b**2*x**2)*erf
c(b*x)/b**6, Ne(b, 0)), (x**6*exp(c)/6, True))

Maxima [F]

\[ \int e^{c+b^2 x^2} x^5 \text {erfc}(b x) \, dx=\int { x^{5} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \]

[In]

integrate(exp(b^2*x^2+c)*x^5*erfc(b*x),x, algorithm="maxima")

[Out]

integrate(x^5*erfc(b*x)*e^(b^2*x^2 + c), x)

Giac [F]

\[ \int e^{c+b^2 x^2} x^5 \text {erfc}(b x) \, dx=\int { x^{5} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \]

[In]

integrate(exp(b^2*x^2+c)*x^5*erfc(b*x),x, algorithm="giac")

[Out]

integrate(x^5*erfc(b*x)*e^(b^2*x^2 + c), x)

Mupad [B] (verification not implemented)

Time = 4.97 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.80 \[ \int e^{c+b^2 x^2} x^5 \text {erfc}(b x) \, dx=\frac {{\mathrm {e}}^c\,\left (60\,b\,x-20\,b^3\,x^3+6\,b^5\,x^5+30\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )-30\,b^2\,x^2\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )+15\,b^4\,x^4\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )\right )}{30\,b^6\,\sqrt {\pi }} \]

[In]

int(x^5*exp(c + b^2*x^2)*erfc(b*x),x)

[Out]

(exp(c)*(60*b*x - 20*b^3*x^3 + 6*b^5*x^5 + 30*pi^(1/2)*exp(b^2*x^2)*erfc(b*x) - 30*b^2*x^2*pi^(1/2)*exp(b^2*x^
2)*erfc(b*x) + 15*b^4*x^4*pi^(1/2)*exp(b^2*x^2)*erfc(b*x)))/(30*b^6*pi^(1/2))