3.1.86 \(\int e^{c+d x^2} x^3 \text {erf}(a+b x) \, dx\) [86]

3.1.86.1 Optimal result

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

output

-1/2*exp(d*x^2+c)*erf(b*x+a)/d^2+1/2*exp(d*x^2+c)*x^2*erf(b*x+a)/d-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*b*exp(c+a^2*d/(b^2-d))*erf((a*b+(b^2-d)*x)/(b^2-d)^(1/2))/d^2/( 
b^2-d)^(1/2)-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)
 

3.1.86.2 Mathematica [A] (verified)

Time = 3.08 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.70 \[ \int e^{c+d x^2} x^3 \text {erf}(a+b x) \, dx=\frac {e^c \left (2 e^{d x^2} \left (-1+d x^2\right ) \text {erf}(a+b x)-\frac {b d e^{-a^2-2 a b x+\left (-b^2+d\right ) x^2} \left (2 \left (b^2-d\right ) \left (a b+\left (-b^2+d\right ) x\right )+\sqrt {b^2-d} \left (\left (1+2 a^2\right ) b^2-d\right ) e^{\frac {\left (a b+\left (b^2-d\right ) x\right )^2}{b^2-d}} \sqrt {\pi } \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )\right )}{\left (b^2-d\right )^3 \sqrt {\pi }}+\frac {2 b e^{\frac {a^2 d}{b^2-d}} \text {erfi}\left (\frac {-a b+\left (-b^2+d\right ) x}{\sqrt {-b^2+d}}\right )}{\sqrt {-b^2+d}}\right )}{4 d^2} \]

input

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

output

(E^c*(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]))/(4*d^2)
 

3.1.86.3 Rubi [A] (verified)

Time = 1.68 (sec) , antiderivative size = 379, 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 = {6939, 2671, 2664, 2634, 2670, 2664, 2634, 6936, 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.

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

\(\Big \downarrow \) 6939

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

\(\Big \downarrow \) 2671

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

\(\Big \downarrow \) 2664

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

\(\Big \downarrow \) 2634

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

\(\Big \downarrow \) 2670

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

\(\Big \downarrow \) 2664

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

\(\Big \downarrow \) 2634

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

\(\Big \downarrow \) 6936

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

\(\Big \downarrow \) 2664

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

\(\Big \downarrow \) 2634

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

input

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

output

(E^(c + d*x^2)*x^2*Erf[a + b*x])/(2*d) - ((E^(c + d*x^2)*Erf[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))/d - (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])
 

3.1.86.3.1 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 6936

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Sim 
p[E^(c + d*x^2)*(Erf[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 6939

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

3.1.86.4 Maple [F]

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

input

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

output

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

3.1.86.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.78 \[ \int e^{c+d x^2} x^3 \text {erf}(a+b x) \, dx=\frac {\pi {\left (2 \, b^{5} - {\left (2 \, a^{2} + 5\right )} b^{3} d + 3 \, b d^{2}\right )} \sqrt {b^{2} - d} \operatorname {erf}\left (\frac {a b + {\left (b^{2} - d\right )} x}{\sqrt {b^{2} - d}}\right ) e^{\left (\frac {b^{2} c + {\left (a^{2} - c\right )} d}{b^{2} - d}\right )} + 2 \, {\left (\pi {\left (b^{6} d - 3 \, b^{4} d^{2} + 3 \, b^{2} d^{3} - d^{4}\right )} x^{2} - \pi {\left (b^{6} - 3 \, b^{4} d + 3 \, b^{2} d^{2} - d^{3}\right )}\right )} \operatorname {erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} - 2 \, \sqrt {\pi } {\left (a b^{4} d - a b^{2} d^{2} - {\left (b^{5} d - 2 \, b^{3} d^{2} + b d^{3}\right )} x\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x + d x^{2} - a^{2} + c\right )}}{4 \, \pi {\left (b^{6} d^{2} - 3 \, b^{4} d^{3} + 3 \, b^{2} d^{4} - d^{5}\right )}} \]

input

integrate(exp(d*x^2+c)*x^3*erf(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*(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) - 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))/(pi*(b^6* 
d^2 - 3*b^4*d^3 + 3*b^2*d^4 - d^5))
 

3.1.86.6 Sympy [F]

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

input

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

output

exp(c)*Integral(x**3*exp(d*x**2)*erf(a + b*x), x)
 

3.1.86.7 Maxima [F]

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

input

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

output

1/2*(d*x^2*e^c - e^c)*erf(b*x + a)*e^(d*x^2)/d^2 - integrate((b*d*x^2*e^c 
- b*e^c)*e^(-b^2*x^2 - 2*a*b*x + d*x^2 - a^2), x)/(sqrt(pi)*d^2)
 

3.1.86.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.84 \[ \int e^{c+d x^2} x^3 \text {erf}(a+b x) \, dx=\frac {1}{2} \, {\left (\frac {{\left (d x^{2} + c - 1\right )} e^{\left (d x^{2} + c\right )}}{d^{2}} - \frac {c e^{\left (d x^{2} + c\right )}}{d^{2}}\right )} \operatorname {erf}\left (b x + a\right ) - \frac {\frac {2 \, \sqrt {\pi } b \operatorname {erf}\left (-\sqrt {b^{2} - d} {\left (\frac {a b}{b^{2} - d} + x\right )}\right ) e^{\left (\frac {b^{2} c + a^{2} d - c d}{b^{2} - d}\right )}}{\sqrt {b^{2} - d}} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, a^{2} b^{2} + b^{2} - d\right )} \operatorname {erf}\left (-\sqrt {b^{2} - d} {\left (\frac {a b}{b^{2} - d} + x\right )}\right ) e^{\left (\frac {b^{2} c + a^{2} d - c d}{b^{2} - d}\right )}}{\sqrt {b^{2} - d}} + 2 \, {\left ({\left (\frac {a b}{b^{2} - d} + x\right )} b^{2} - 2 \, a b - {\left (\frac {a b}{b^{2} - d} + x\right )} d\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x + d x^{2} - a^{2} + c\right )}\right )} b d}{b^{4} - 2 \, b^{2} d + d^{2}}}{4 \, \sqrt {\pi } d^{2}} \]

input

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

output

1/2*((d*x^2 + c - 1)*e^(d*x^2 + c)/d^2 - c*e^(d*x^2 + c)/d^2)*erf(b*x + a) 
 - 1/4*(2*sqrt(pi)*b*erf(-sqrt(b^2 - d)*(a*b/(b^2 - d) + x))*e^((b^2*c + a 
^2*d - c*d)/(b^2 - d))/sqrt(b^2 - d) - (sqrt(pi)*(2*a^2*b^2 + b^2 - d)*erf 
(-sqrt(b^2 - d)*(a*b/(b^2 - d) + x))*e^((b^2*c + a^2*d - c*d)/(b^2 - d))/s 
qrt(b^2 - d) + 2*((a*b/(b^2 - d) + x)*b^2 - 2*a*b - (a*b/(b^2 - d) + x)*d) 
*e^(-b^2*x^2 - 2*a*b*x + d*x^2 - a^2 + c))*b*d/(b^4 - 2*b^2*d + d^2))/(sqr 
t(pi)*d^2)
 

3.1.86.9 Mupad [B] (verification not implemented)

Time = 6.94 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.13 \[ \int e^{c+d x^2} x^3 \text {erf}(a+b x) \, dx=\frac {\mathrm {erfi}\left (\frac {a\,b-x\,\left (d-b^2\right )}{\sqrt {d-b^2}}\right )\,\left (b^3\,{\mathrm {e}}^{\frac {c\,d}{d-b^2}-\frac {a^2\,d}{d-b^2}-\frac {b^2\,c}{d-b^2}}+2\,a^2\,b^3\,{\mathrm {e}}^{\frac {c\,d}{d-b^2}-\frac {a^2\,d}{d-b^2}-\frac {b^2\,c}{d-b^2}}-b\,d\,{\mathrm {e}}^{\frac {c\,d}{d-b^2}-\frac {a^2\,d}{d-b^2}-\frac {b^2\,c}{d-b^2}}\right )}{4\,d\,{\left (d-b^2\right )}^{5/2}}-\frac {\frac {a\,b^2\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2+d\,x^2+c}}{2\,{\left (d-b^2\right )}^2}+\frac {b\,x\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2+d\,x^2+c}}{2\,\left (d-b^2\right )}}{d\,\sqrt {\pi }}-\mathrm {erf}\left (a+b\,x\right )\,\left (\frac {{\mathrm {e}}^{d\,x^2+c}}{2\,d^2}-\frac {x^2\,{\mathrm {e}}^{d\,x^2+c}}{2\,d}\right )+\frac {b\,\mathrm {erf}\left (\frac {a\,b\,1{}\mathrm {i}-x\,\left (d-b^2\right )\,1{}\mathrm {i}}{\sqrt {d-b^2}}\right )\,{\mathrm {e}}^{c-a^2-\frac {a^2\,b^2}{d-b^2}}\,1{}\mathrm {i}}{2\,d^2\,\sqrt {d-b^2}} \]

input

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

output

(erfi((a*b - x*(d - b^2))/(d - b^2)^(1/2))*(b^3*exp((c*d)/(d - b^2) - (a^2 
*d)/(d - b^2) - (b^2*c)/(d - b^2)) + 2*a^2*b^3*exp((c*d)/(d - b^2) - (a^2* 
d)/(d - b^2) - (b^2*c)/(d - b^2)) - b*d*exp((c*d)/(d - b^2) - (a^2*d)/(d - 
 b^2) - (b^2*c)/(d - b^2))))/(4*d*(d - b^2)^(5/2)) - ((a*b^2*exp(c + d*x^2 
 - a^2 - b^2*x^2 - 2*a*b*x))/(2*(d - b^2)^2) + (b*x*exp(c + d*x^2 - a^2 - 
b^2*x^2 - 2*a*b*x))/(2*(d - b^2)))/(d*pi^(1/2)) - erf(a + b*x)*(exp(c + d* 
x^2)/(2*d^2) - (x^2*exp(c + d*x^2))/(2*d)) + (b*erf((a*b*1i - x*(d - b^2)* 
1i)/(d - b^2)^(1/2))*exp(c - a^2 - (a^2*b^2)/(d - b^2))*1i)/(2*d^2*(d - b^ 
2)^(1/2))