\(\int e^{c+d x^2} x^3 \text {erfi}(a+b x) \, dx\) [294]

Optimal result

Integrand size = 19, antiderivative size = 304 \[ \int e^{c+d x^2} x^3 \text {erfi}(a+b x) \, dx=\frac {a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^2 \sqrt {\pi }}-\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right ) \sqrt {\pi }}-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erfi}(a+b x)}{2 d}-\frac {a^2 b^3 e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{2 d \left (b^2+d\right )^{5/2}}+\frac {b e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{4 d \left (b^2+d\right )^{3/2}}+\frac {b e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{2 d^2 \sqrt {b^2+d}} \] Output:

1/2*a*b^2*exp(a^2+c+2*a*b*x+(b^2+d)*x^2)/d/(b^2+d)^2/Pi^(1/2)-1/2*b*exp(a^ 
2+c+2*a*b*x+(b^2+d)*x^2)*x/d/(b^2+d)/Pi^(1/2)-1/2*exp(d*x^2+c)*erfi(b*x+a) 
/d^2+1/2*exp(d*x^2+c)*x^2*erfi(b*x+a)/d-1/2*a^2*b^3*exp(c+a^2*d/(b^2+d))*e 
rfi((a*b+(b^2+d)*x)/(b^2+d)^(1/2))/d/(b^2+d)^(5/2)+1/4*b*exp(c+a^2*d/(b^2+ 
d))*erfi((a*b+(b^2+d)*x)/(b^2+d)^(1/2))/d/(b^2+d)^(3/2)+1/2*b*exp(c+a^2*d/ 
(b^2+d))*erfi((a*b+(b^2+d)*x)/(b^2+d)^(1/2))/d^2/(b^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 1.44 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.68 \[ \int e^{c+d x^2} x^3 \text {erfi}(a+b x) \, dx=\frac {e^c \left (2 e^{d x^2} \left (-1+d x^2\right ) \text {erfi}(a+b x)+\frac {2 b e^{\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{\sqrt {b^2+d}}-\frac {b d e^{\frac {a^2 d}{b^2+d}} \left (2 \left (b^2+d\right ) e^{\frac {\left (a b+\left (b^2+d\right ) x\right )^2}{b^2+d}} \left (-a b+\left (b^2+d\right ) x\right )+\left (\left (-1+2 a^2\right ) b^2-d\right ) \sqrt {b^2+d} \sqrt {\pi } \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )\right )}{\left (b^2+d\right )^3 \sqrt {\pi }}\right )}{4 d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.62 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {6941, 2671, 2664, 2633, 2670, 2664, 2633, 6938, 2664, 2633}

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.

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

\(\Big \downarrow \) 6941

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

\(\Big \downarrow \) 2671

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

\(\Big \downarrow \) 2664

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2670

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

\(\Big \downarrow \) 2664

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 6938

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

\(\Big \downarrow \) 2664

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

\(\Big \downarrow \) 2633

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[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 6938

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Si 
mp[E^(c + d*x^2)*(Erfi[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 6941

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

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.86 \[ \int e^{c+d x^2} x^3 \text {erfi}(a+b x) \, dx=-\frac {\pi {\left (2 \, b^{5} - {\left (2 \, a^{2} - 5\right )} b^{3} d + 3 \, b d^{2}\right )} \sqrt {-b^{2} - d} \operatorname {erf}\left (\frac {{\left (a b + {\left (b^{2} + d\right )} x\right )} \sqrt {-b^{2} - d}}{b^{2} + d}\right ) e^{\left (\frac {b^{2} c + {\left (a^{2} + c\right )} d}{b^{2} + d}\right )} - 2 \, {\left (\pi {\left (b^{6} d + 3 \, b^{4} d^{2} + 3 \, b^{2} d^{3} + d^{4}\right )} x^{2} - \pi {\left (b^{6} + 3 \, b^{4} d + 3 \, b^{2} d^{2} + d^{3}\right )}\right )} \operatorname {erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} - 2 \, \sqrt {\pi } {\left (a b^{4} d + a b^{2} d^{2} - {\left (b^{5} d + 2 \, b^{3} d^{2} + b d^{3}\right )} x\right )} e^{\left (b^{2} x^{2} + 2 \, a b x + d x^{2} + a^{2} + c\right )}}{4 \, \pi {\left (b^{6} d^{2} + 3 \, b^{4} d^{3} + 3 \, b^{2} d^{4} + d^{5}\right )}} \] Input:

integrate(exp(d*x^2+c)*x^3*erfi(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)/(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))*erfi(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))
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 4.82 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.11 \[ \int e^{c+d x^2} x^3 \text {erfi}(a+b x) \, dx=\frac {\mathrm {erfi}\left (\frac {a\,b+x\,\left (b^2+d\right )}{\sqrt {b^2+d}}\right )\,\left (b^3\,{\mathrm {e}}^{\frac {c\,d}{b^2+d}+\frac {a^2\,d}{b^2+d}+\frac {b^2\,c}{b^2+d}}-2\,a^2\,b^3\,{\mathrm {e}}^{\frac {c\,d}{b^2+d}+\frac {a^2\,d}{b^2+d}+\frac {b^2\,c}{b^2+d}}+b\,d\,{\mathrm {e}}^{\frac {c\,d}{b^2+d}+\frac {a^2\,d}{b^2+d}+\frac {b^2\,c}{b^2+d}}\right )}{4\,d\,{\left (b^2+d\right )}^{5/2}}-\frac {\frac {b\,x\,{\mathrm {e}}^{a^2+2\,a\,b\,x+b^2\,x^2+d\,x^2+c}}{2\,\left (b^2+d\right )}-\frac {a\,b^2\,{\mathrm {e}}^{a^2+2\,a\,b\,x+b^2\,x^2+d\,x^2+c}}{2\,{\left (b^2+d\right )}^2}}{d\,\sqrt {\pi }}-\mathrm {erfi}\left (a+b\,x\right )\,\left (\frac {{\mathrm {e}}^{d\,x^2+c}}{2\,d^2}-\frac {x^2\,{\mathrm {e}}^{d\,x^2+c}}{2\,d}\right )-\frac {b\,{\mathrm {e}}^{c+a^2-\frac {a^2\,b^2}{b^2+d}}\,\mathrm {erf}\left (\frac {a\,b\,1{}\mathrm {i}+x\,\left (b^2+d\right )\,1{}\mathrm {i}}{\sqrt {b^2+d}}\right )\,1{}\mathrm {i}}{2\,d^2\,\sqrt {b^2+d}} \] Input:

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

Reduce [F]

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

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

Output:

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