3.156 \(\int e^{c+d x^2} x^5 \text{Erfc}(b x) \, dx\)

Optimal. Leaf size=283 \[ -\frac{b e^c \text{Erf}\left (x \sqrt{b^2-d}\right )}{2 d^2 \left (b^2-d\right )^{3/2}}+\frac{b e^c \text{Erf}\left (x \sqrt{b^2-d}\right )}{d^3 \sqrt{b^2-d}}+\frac{b x e^{c-x^2 \left (b^2-d\right )}}{\sqrt{\pi } d^2 \left (b^2-d\right )}+\frac{3 b e^c \text{Erf}\left (x \sqrt{b^2-d}\right )}{8 d \left (b^2-d\right )^{5/2}}-\frac{b x^3 e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt{\pi } d \left (b^2-d\right )}-\frac{3 b x e^{c-x^2 \left (b^2-d\right )}}{4 \sqrt{\pi } d \left (b^2-d\right )^2}-\frac{x^2 \text{Erfc}(b x) e^{c+d x^2}}{d^2}+\frac{\text{Erfc}(b x) e^{c+d x^2}}{d^3}+\frac{x^4 \text{Erfc}(b x) e^{c+d x^2}}{2 d} \]

[Out]

(b*E^(c - (b^2 - d)*x^2)*x)/((b^2 - d)*d^2*Sqrt[Pi]) - (3*b*E^(c - (b^2 - d)*x^2)*x)/(4*(b^2 - d)^2*d*Sqrt[Pi]
) - (b*E^(c - (b^2 - d)*x^2)*x^3)/(2*(b^2 - d)*d*Sqrt[Pi]) + (b*E^c*Erf[Sqrt[b^2 - d]*x])/(Sqrt[b^2 - d]*d^3)
- (b*E^c*Erf[Sqrt[b^2 - d]*x])/(2*(b^2 - d)^(3/2)*d^2) + (3*b*E^c*Erf[Sqrt[b^2 - d]*x])/(8*(b^2 - d)^(5/2)*d)
+ (E^(c + d*x^2)*Erfc[b*x])/d^3 - (E^(c + d*x^2)*x^2*Erfc[b*x])/d^2 + (E^(c + d*x^2)*x^4*Erfc[b*x])/(2*d)

________________________________________________________________________________________

Rubi [A]  time = 0.367885, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {6386, 6383, 2205, 2212} \[ -\frac{b e^c \text{Erf}\left (x \sqrt{b^2-d}\right )}{2 d^2 \left (b^2-d\right )^{3/2}}+\frac{b e^c \text{Erf}\left (x \sqrt{b^2-d}\right )}{d^3 \sqrt{b^2-d}}+\frac{b x e^{c-x^2 \left (b^2-d\right )}}{\sqrt{\pi } d^2 \left (b^2-d\right )}+\frac{3 b e^c \text{Erf}\left (x \sqrt{b^2-d}\right )}{8 d \left (b^2-d\right )^{5/2}}-\frac{b x^3 e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt{\pi } d \left (b^2-d\right )}-\frac{3 b x e^{c-x^2 \left (b^2-d\right )}}{4 \sqrt{\pi } d \left (b^2-d\right )^2}-\frac{x^2 \text{Erfc}(b x) e^{c+d x^2}}{d^2}+\frac{\text{Erfc}(b x) e^{c+d x^2}}{d^3}+\frac{x^4 \text{Erfc}(b x) e^{c+d x^2}}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x^2)*x^5*Erfc[b*x],x]

[Out]

(b*E^(c - (b^2 - d)*x^2)*x)/((b^2 - d)*d^2*Sqrt[Pi]) - (3*b*E^(c - (b^2 - d)*x^2)*x)/(4*(b^2 - d)^2*d*Sqrt[Pi]
) - (b*E^(c - (b^2 - d)*x^2)*x^3)/(2*(b^2 - d)*d*Sqrt[Pi]) + (b*E^c*Erf[Sqrt[b^2 - d]*x])/(Sqrt[b^2 - d]*d^3)
- (b*E^c*Erf[Sqrt[b^2 - d]*x])/(2*(b^2 - d)^(3/2)*d^2) + (3*b*E^c*Erf[Sqrt[b^2 - d]*x])/(8*(b^2 - d)^(5/2)*d)
+ (E^(c + d*x^2)*Erfc[b*x])/d^3 - (E^(c + d*x^2)*x^2*Erfc[b*x])/d^2 + (E^(c + d*x^2)*x^4*Erfc[b*x])/(2*d)

Rule 6386

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
]

Rule 6383

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 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^5 \text{erfc}(b x) \, dx &=\frac{e^{c+d x^2} x^4 \text{erfc}(b x)}{2 d}-\frac{2 \int e^{c+d x^2} x^3 \text{erfc}(b x) \, dx}{d}+\frac{b \int e^{c-\left (b^2-d\right ) x^2} x^4 \, dx}{d \sqrt{\pi }}\\ &=-\frac{b e^{c-\left (b^2-d\right ) x^2} x^3}{2 \left (b^2-d\right ) d \sqrt{\pi }}-\frac{e^{c+d x^2} x^2 \text{erfc}(b x)}{d^2}+\frac{e^{c+d x^2} x^4 \text{erfc}(b x)}{2 d}+\frac{2 \int e^{c+d x^2} x \text{erfc}(b x) \, dx}{d^2}-\frac{(2 b) \int e^{c-\left (b^2-d\right ) x^2} x^2 \, dx}{d^2 \sqrt{\pi }}+\frac{(3 b) \int e^{c+\left (-b^2+d\right ) x^2} x^2 \, dx}{2 \left (b^2-d\right ) d \sqrt{\pi }}\\ &=\frac{b e^{c-\left (b^2-d\right ) x^2} x}{\left (b^2-d\right ) d^2 \sqrt{\pi }}-\frac{3 b e^{c-\left (b^2-d\right ) x^2} x}{4 \left (b^2-d\right )^2 d \sqrt{\pi }}-\frac{b e^{c-\left (b^2-d\right ) x^2} x^3}{2 \left (b^2-d\right ) d \sqrt{\pi }}+\frac{e^{c+d x^2} \text{erfc}(b x)}{d^3}-\frac{e^{c+d x^2} x^2 \text{erfc}(b x)}{d^2}+\frac{e^{c+d x^2} x^4 \text{erfc}(b x)}{2 d}+\frac{(2 b) \int e^{c-\left (b^2-d\right ) x^2} \, dx}{d^3 \sqrt{\pi }}-\frac{b \int e^{c+\left (-b^2+d\right ) x^2} \, dx}{\left (b^2-d\right ) d^2 \sqrt{\pi }}+\frac{(3 b) \int e^{c+\left (-b^2+d\right ) x^2} \, dx}{4 \left (b^2-d\right )^2 d \sqrt{\pi }}\\ &=\frac{b e^{c-\left (b^2-d\right ) x^2} x}{\left (b^2-d\right ) d^2 \sqrt{\pi }}-\frac{3 b e^{c-\left (b^2-d\right ) x^2} x}{4 \left (b^2-d\right )^2 d \sqrt{\pi }}-\frac{b e^{c-\left (b^2-d\right ) x^2} x^3}{2 \left (b^2-d\right ) d \sqrt{\pi }}+\frac{b e^c \text{erf}\left (\sqrt{b^2-d} x\right )}{\sqrt{b^2-d} d^3}-\frac{b e^c \text{erf}\left (\sqrt{b^2-d} x\right )}{2 \left (b^2-d\right )^{3/2} d^2}+\frac{3 b e^c \text{erf}\left (\sqrt{b^2-d} x\right )}{8 \left (b^2-d\right )^{5/2} d}+\frac{e^{c+d x^2} \text{erfc}(b x)}{d^3}-\frac{e^{c+d x^2} x^2 \text{erfc}(b x)}{d^2}+\frac{e^{c+d x^2} x^4 \text{erfc}(b x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.715748, size = 184, normalized size = 0.65 \[ \frac{e^c \left (-\frac{b d \left (4 b^2-7 d\right ) \text{Erf}\left (x \sqrt{b^2-d}\right )}{\left (b^2-d\right )^{5/2}}+\frac{8 b \text{Erfi}\left (x \sqrt{d-b^2}\right )}{\sqrt{d-b^2}}-\frac{2 b d x e^{x^2 \left (d-b^2\right )} \left (2 b^2 \left (d x^2-2\right )+d \left (7-2 d x^2\right )\right )}{\sqrt{\pi } \left (b^2-d\right )^2}-4 e^{d x^2} \left (d^2 x^4-2 d x^2+2\right ) \text{Erf}(b x)+4 e^{d x^2} \left (d^2 x^4-2 d x^2+2\right )\right )}{8 d^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(E^c*(4*E^(d*x^2)*(2 - 2*d*x^2 + d^2*x^4) - (2*b*d*E^((-b^2 + d)*x^2)*x*(d*(7 - 2*d*x^2) + 2*b^2*(-2 + d*x^2))
)/((b^2 - d)^2*Sqrt[Pi]) - 4*E^(d*x^2)*(2 - 2*d*x^2 + d^2*x^4)*Erf[b*x] - (b*(4*b^2 - 7*d)*d*Erf[Sqrt[b^2 - d]
*x])/(b^2 - d)^(5/2) + (8*b*Erfi[Sqrt[-b^2 + d]*x])/Sqrt[-b^2 + d]))/(8*d^3)

________________________________________________________________________________________

Maple [A]  time = 0.205, size = 376, normalized size = 1.3 \begin{align*}{\frac{1}{b} \left ({\frac{{{\rm e}^{c}}}{{b}^{5}} \left ({\frac{{{\rm e}^{d{x}^{2}}}{b}^{6}{x}^{4}}{2\,d}}-2\,{\frac{{b}^{2}}{d} \left ( 1/2\,{\frac{{b}^{4}{x}^{2}{{\rm e}^{d{x}^{2}}}}{d}}-1/2\,{\frac{{b}^{4}{{\rm e}^{d{x}^{2}}}}{{d}^{2}}} \right ) } \right ) }-{\frac{{\it Erf} \left ( bx \right ){{\rm e}^{c}}}{{b}^{5}} \left ({\frac{{{\rm e}^{d{x}^{2}}}{b}^{6}{x}^{4}}{2\,d}}-2\,{\frac{{b}^{2}}{d} \left ( 1/2\,{\frac{{b}^{4}{x}^{2}{{\rm e}^{d{x}^{2}}}}{d}}-1/2\,{\frac{{b}^{4}{{\rm e}^{d{x}^{2}}}}{{d}^{2}}} \right ) } \right ) }+{\frac{{{\rm e}^{c}}}{\sqrt{\pi }{b}^{5}} \left ({\frac{{b}^{2}}{d} \left ({\frac{{x}^{3}{b}^{3}}{2}{{\rm e}^{ \left ( -1+{\frac{d}{{b}^{2}}} \right ){b}^{2}{x}^{2}}} \left ( -1+{\frac{d}{{b}^{2}}} \right ) ^{-1}}-{\frac{3}{2} \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 ) \left ( -1+{\frac{d}{{b}^{2}}} \right ) ^{-1}} \right ) }+{\frac{{b}^{6}\sqrt{\pi }}{{d}^{3}}{\it Erf} \left ( \sqrt{1-{\frac{d}{{b}^{2}}}}bx \right ){\frac{1}{\sqrt{1-{\frac{d}{{b}^{2}}}}}}}-2\,{\frac{{b}^{4}}{{d}^{2}} \left ( 1/2\,{bx{{\rm e}^{ \left ( -1+{\frac{d}{{b}^{2}}} \right ){b}^{2}{x}^{2}}} \left ( -1+{\frac{d}{{b}^{2}}} \right ) ^{-1}}-1/4\,{\sqrt{\pi }{\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 ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(d*x^2+c)*x^5*erfc(b*x),x)

[Out]

(1/b^5*exp(c)*(1/2*exp(d*x^2)*b^6*x^4/d-2/d*b^2*(1/2/d*b^4*x^2*exp(d*x^2)-1/2/d^2*b^4*exp(d*x^2)))-erf(b*x)/b^
5*exp(c)*(1/2*exp(d*x^2)*b^6*x^4/d-2/d*b^2*(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^5*e
xp(c)*(1/d*b^2*(1/2/(-1+d/b^2)*b^3*x^3*exp((-1+d/b^2)*b^2*x^2)-3/2/(-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/d^3*b^6*Pi^(1/2)/(1-d/b^2)^(1
/2)*erf((1-d/b^2)^(1/2)*b*x)-2/d^2*b^4*(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))))/b

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \operatorname{erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.19624, size = 734, normalized size = 2.59 \begin{align*} \frac{\pi{\left (8 \, b^{5} - 20 \, b^{3} d + 15 \, b d^{2}\right )} \sqrt{b^{2} - d} \operatorname{erf}\left (\sqrt{b^{2} - d} x\right ) e^{c} - 2 \, \sqrt{\pi }{\left (2 \,{\left (b^{5} d^{2} - 2 \, b^{3} d^{3} + b d^{4}\right )} x^{3} -{\left (4 \, b^{5} d - 11 \, b^{3} d^{2} + 7 \, b d^{3}\right )} x\right )} e^{\left (-b^{2} x^{2} + d x^{2} + c\right )} + 4 \,{\left (\pi{\left (b^{6} d^{2} - 3 \, b^{4} d^{3} + 3 \, b^{2} d^{4} - d^{5}\right )} x^{4} - 2 \, \pi{\left (b^{6} d - 3 \, b^{4} d^{2} + 3 \, b^{2} d^{3} - d^{4}\right )} x^{2} + 2 \, \pi{\left (b^{6} - 3 \, b^{4} d + 3 \, b^{2} d^{2} - d^{3}\right )} -{\left (\pi{\left (b^{6} d^{2} - 3 \, b^{4} d^{3} + 3 \, b^{2} d^{4} - d^{5}\right )} x^{4} - 2 \, \pi{\left (b^{6} d - 3 \, b^{4} d^{2} + 3 \, b^{2} d^{3} - d^{4}\right )} x^{2} + 2 \, \pi{\left (b^{6} - 3 \, b^{4} d + 3 \, b^{2} d^{2} - d^{3}\right )}\right )} \operatorname{erf}\left (b x\right )\right )} e^{\left (d x^{2} + c\right )}}{8 \, \pi{\left (b^{6} d^{3} - 3 \, b^{4} d^{4} + 3 \, b^{2} d^{5} - d^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(pi*(8*b^5 - 20*b^3*d + 15*b*d^2)*sqrt(b^2 - d)*erf(sqrt(b^2 - d)*x)*e^c - 2*sqrt(pi)*(2*(b^5*d^2 - 2*b^3*
d^3 + b*d^4)*x^3 - (4*b^5*d - 11*b^3*d^2 + 7*b*d^3)*x)*e^(-b^2*x^2 + d*x^2 + c) + 4*(pi*(b^6*d^2 - 3*b^4*d^3 +
3*b^2*d^4 - d^5)*x^4 - 2*pi*(b^6*d - 3*b^4*d^2 + 3*b^2*d^3 - d^4)*x^2 + 2*pi*(b^6 - 3*b^4*d + 3*b^2*d^2 - d^3
) - (pi*(b^6*d^2 - 3*b^4*d^3 + 3*b^2*d^4 - d^5)*x^4 - 2*pi*(b^6*d - 3*b^4*d^2 + 3*b^2*d^3 - d^4)*x^2 + 2*pi*(b
^6 - 3*b^4*d + 3*b^2*d^2 - d^3))*erf(b*x))*e^(d*x^2 + c))/(pi*(b^6*d^3 - 3*b^4*d^4 + 3*b^2*d^5 - d^6))

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \operatorname{erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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