\(\int e^{c+d x^2} x^3 \text {erfc}(a+b x) \, dx\) [189]

Optimal result

Integrand size = 19, antiderivative size = 342 \[ \int e^{c+d x^2} x^3 \text {erfc}(a+b x) \, dx=\frac {a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}-\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {b e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \sqrt {b^2-d} d^2}+\frac {a^2 b^3 e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{5/2} d}+\frac {b e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2} d}-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erfc}(a+b x)}{2 d} \] Output:

1/2*a*b^2*exp(-a^2+c-2*a*b*x-(b^2-d)*x^2)/(b^2-d)^2/d/Pi^(1/2)-1/2*b*exp(- 
a^2+c-2*a*b*x-(b^2-d)*x^2)*x/(b^2-d)/d/Pi^(1/2)-1/2*b*exp(c+a^2*d/(b^2-d)) 
*erf((a*b+(b^2-d)*x)/(b^2-d)^(1/2))/(b^2-d)^(1/2)/d^2+1/2*a^2*b^3*exp(c+a^ 
2*d/(b^2-d))*erf((a*b+(b^2-d)*x)/(b^2-d)^(1/2))/(b^2-d)^(5/2)/d+1/4*b*exp( 
c+a^2*d/(b^2-d))*erf((a*b+(b^2-d)*x)/(b^2-d)^(1/2))/(b^2-d)^(3/2)/d-1/2*ex 
p(d*x^2+c)*erfc(b*x+a)/d^2+1/2*exp(d*x^2+c)*x^2*erfc(b*x+a)/d
 

Mathematica [A] (verified)

Time = 3.25 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.75 \[ \int e^{c+d x^2} x^3 \text {erfc}(a+b x) \, dx=-\frac {e^c \left (-2 e^{d x^2} \left (-1+d x^2\right )+2 e^{d x^2} \left (-1+d x^2\right ) \text {erf}(a+b x)-\frac {b d e^{-a^2-2 a b x+\left (-b^2+d\right ) x^2} \left (2 \left (b^2-d\right ) \left (a b+\left (-b^2+d\right ) x\right )+\sqrt {b^2-d} \left (\left (1+2 a^2\right ) b^2-d\right ) e^{\frac {\left (a b+\left (b^2-d\right ) x\right )^2}{b^2-d}} \sqrt {\pi } \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )\right )}{\left (b^2-d\right )^3 \sqrt {\pi }}+\frac {2 b e^{\frac {a^2 d}{b^2-d}} \text {erfi}\left (\frac {-a b+\left (-b^2+d\right ) x}{\sqrt {-b^2+d}}\right )}{\sqrt {-b^2+d}}\right )}{4 d^2} \] Input:

Integrate[E^(c + d*x^2)*x^3*Erfc[a + b*x],x]
 

Output:

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

Rubi [A] (verified)

Time = 1.66 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {6940, 2671, 2664, 2634, 2670, 2664, 2634, 6937, 2664, 2634}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^3 e^{c+d x^2} \text {erfc}(a+b x) \, dx\)

\(\Big \downarrow \) 6940

\(\displaystyle \frac {b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} x^2dx}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} x \text {erfc}(a+b x)dx}{d}+\frac {x^2 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2671

\(\displaystyle \frac {b \left (\frac {\int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}dx}{2 \left (b^2-d\right )}-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} xdx}{b^2-d}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} x \text {erfc}(a+b x)dx}{d}+\frac {x^2 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2664

\(\displaystyle \frac {b \left (-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} xdx}{b^2-d}+\frac {e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \int e^{-\frac {\left (a b+\left (b^2-d\right ) x\right )^2}{b^2-d}}dx}{2 \left (b^2-d\right )}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} x \text {erfc}(a+b x)dx}{d}+\frac {x^2 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {b \left (-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} xdx}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} x \text {erfc}(a+b x)dx}{d}+\frac {x^2 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2670

\(\displaystyle \frac {b \left (-\frac {a b \left (-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}dx}{b^2-d}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} x \text {erfc}(a+b x)dx}{d}+\frac {x^2 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2664

\(\displaystyle \frac {b \left (-\frac {a b \left (-\frac {a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \int e^{-\frac {\left (a b+\left (b^2-d\right ) x\right )^2}{b^2-d}}dx}{b^2-d}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} x \text {erfc}(a+b x)dx}{d}+\frac {x^2 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2634

\(\displaystyle -\frac {\int e^{d x^2+c} x \text {erfc}(a+b x)dx}{d}+\frac {b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^2 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 6937

\(\displaystyle -\frac {\frac {b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}dx}{\sqrt {\pi } d}+\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d}}{d}+\frac {b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^2 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2664

\(\displaystyle -\frac {\frac {b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \int e^{-\frac {\left (a b+\left (b^2-d\right ) x\right )^2}{b^2-d}}dx}{\sqrt {\pi } d}+\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d}}{d}+\frac {b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^2 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2634

\(\displaystyle -\frac {\frac {b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 d \sqrt {b^2-d}}+\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d}}{d}+\frac {b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^2 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

Input:

Int[E^(c + d*x^2)*x^3*Erfc[a + b*x],x]
 

Output:

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

Defintions of rubi rules used

rule 2634

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] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 2664

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[F^(a - b^2/ 
(4*c))   Int[F^((b + 2*c*x)^2/(4*c)), x], x] /; FreeQ[{F, a, b, c}, x]
 

rule 2670

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol 
] :> Simp[e*(F^(a + b*x + c*x^2)/(2*c*Log[F])), x] - Simp[(b*e - 2*c*d)/(2* 
c)   Int[F^(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ 
[b*e - 2*c*d, 0]
 

rule 2671

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*(F^(a + b*x + c*x^2)/(2*c*Log[F])), x] + 
 (-Simp[(b*e - 2*c*d)/(2*c)   Int[(d + e*x)^(m - 1)*F^(a + b*x + c*x^2), x] 
, x] - Simp[(m - 1)*(e^2/(2*c*Log[F]))   Int[(d + e*x)^(m - 2)*F^(a + b*x + 
 c*x^2), x], x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && 
GtQ[m, 1]
 

rule 6937

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Si 
mp[E^(c + d*x^2)*(Erfc[a + b*x]/(2*d)), x] + Simp[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 6940

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

Maple [F]

\[\int {\mathrm e}^{d \,x^{2}+c} x^{3} \operatorname {erfc}\left (b x +a \right )d x\]

Input:

int(exp(d*x^2+c)*x^3*erfc(b*x+a),x)
 

Output:

int(exp(d*x^2+c)*x^3*erfc(b*x+a),x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.96 \[ \int e^{c+d x^2} x^3 \text {erfc}(a+b x) \, dx=-\frac {\pi {\left (2 \, b^{5} - {\left (2 \, a^{2} + 5\right )} b^{3} d + 3 \, b d^{2}\right )} \sqrt {b^{2} - d} \operatorname {erf}\left (\frac {a b + {\left (b^{2} - d\right )} x}{\sqrt {b^{2} - d}}\right ) e^{\left (\frac {b^{2} c + {\left (a^{2} - c\right )} d}{b^{2} - d}\right )} - 2 \, \sqrt {\pi } {\left (a b^{4} d - a b^{2} d^{2} - {\left (b^{5} d - 2 \, b^{3} d^{2} + b d^{3}\right )} x\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x + d x^{2} - a^{2} + c\right )} - 2 \, {\left (\pi {\left (b^{6} d - 3 \, b^{4} d^{2} + 3 \, b^{2} d^{3} - d^{4}\right )} x^{2} - \pi {\left (b^{6} - 3 \, b^{4} d + 3 \, b^{2} d^{2} - d^{3}\right )} - {\left (\pi {\left (b^{6} d - 3 \, b^{4} d^{2} + 3 \, b^{2} d^{3} - d^{4}\right )} x^{2} - \pi {\left (b^{6} - 3 \, b^{4} d + 3 \, b^{2} d^{2} - d^{3}\right )}\right )} \operatorname {erf}\left (b x + a\right )\right )} e^{\left (d x^{2} + c\right )}}{4 \, \pi {\left (b^{6} d^{2} - 3 \, b^{4} d^{3} + 3 \, b^{2} d^{4} - d^{5}\right )}} \] Input:

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

Output:

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

Sympy [F(-1)]

Timed out. \[ \int e^{c+d x^2} x^3 \text {erfc}(a+b x) \, dx=\text {Timed out} \] Input:

integrate(exp(d*x**2+c)*x**3*erfc(b*x+a),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int e^{c+d x^2} x^3 \text {erfc}(a+b x) \, dx=\int { x^{3} \operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \] Input:

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

Output:

integrate(x^3*erfc(b*x + a)*e^(d*x^2 + c), x)
 

Giac [F]

\[ \int e^{c+d x^2} x^3 \text {erfc}(a+b x) \, dx=\int { x^{3} \operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \] Input:

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

Output:

integrate(x^3*erfc(b*x + a)*e^(d*x^2 + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{c+d x^2} x^3 \text {erfc}(a+b x) \, dx=\int x^3\,\mathrm {erfc}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c} \,d x \] Input:

int(x^3*erfc(a + b*x)*exp(c + d*x^2),x)
 

Output:

int(x^3*erfc(a + b*x)*exp(c + d*x^2), x)
 

Reduce [F]

\[ \int e^{c+d x^2} x^3 \text {erfc}(a+b x) \, dx=\frac {e^{c} \left (e^{d \,x^{2}} d \,x^{2}-e^{d \,x^{2}}-2 \left (\int e^{d \,x^{2}} \mathrm {erf}\left (b x +a \right ) x^{3}d x \right ) d^{2}\right )}{2 d^{2}} \] Input:

int(exp(d*x^2+c)*x^3*erfc(b*x+a),x)
                                                                                    
                                                                                    
 

Output:

(e**c*(e**(d*x**2)*d*x**2 - e**(d*x**2) - 2*int(e**(d*x**2)*erf(a + b*x)*x 
**3,x)*d**2))/(2*d**2)