\(\int (c+d x)^2 \text {erfi}(a+b x) \, dx\) [222]

Optimal result

Integrand size = 14, antiderivative size = 186 \[ \int (c+d x)^2 \text {erfi}(a+b x) \, dx=\frac {d^2 e^{(a+b x)^2}}{3 b^3 \sqrt {\pi }}-\frac {(b c-a d)^2 e^{(a+b x)^2}}{b^3 \sqrt {\pi }}-\frac {d (b c-a d) e^{(a+b x)^2} (a+b x)}{b^3 \sqrt {\pi }}-\frac {d^2 e^{(a+b x)^2} (a+b x)^2}{3 b^3 \sqrt {\pi }}+\frac {d (b c-a d) \text {erfi}(a+b x)}{2 b^3}-\frac {(b c-a d)^3 \text {erfi}(a+b x)}{3 b^3 d}+\frac {(c+d x)^3 \text {erfi}(a+b x)}{3 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.76 \[ \int (c+d x)^2 \text {erfi}(a+b x) \, dx=\frac {-2 e^{(a+b x)^2} \left (\left (-1+a^2\right ) d^2-a b d (3 c+d x)+b^2 \left (3 c^2+3 c d x+d^2 x^2\right )\right )+\sqrt {\pi } \left (3 b c d-6 a^2 b c d+2 a^3 d^2+a \left (6 b^2 c^2-3 d^2\right )+2 b^3 x \left (3 c^2+3 c d x+d^2 x^2\right )\right ) \text {erfi}(a+b x)}{6 b^3 \sqrt {\pi }} \] Input:

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

Output:

(-2*E^(a + b*x)^2*((-1 + a^2)*d^2 - a*b*d*(3*c + d*x) + b^2*(3*c^2 + 3*c*d 
*x + d^2*x^2)) + Sqrt[Pi]*(3*b*c*d - 6*a^2*b*c*d + 2*a^3*d^2 + a*(6*b^2*c^ 
2 - 3*d^2) + 2*b^3*x*(3*c^2 + 3*c*d*x + d^2*x^2))*Erfi[a + b*x])/(6*b^3*Sq 
rt[Pi])
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6917, 2656, 2009}

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.

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

\(\Big \downarrow \) 6917

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

\(\Big \downarrow \) 2656

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2656

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(Px_), x_Symbol] :> Int[ 
ExpandLinearProduct[F^(a + b*(c + d*x)^n), Px, c, d, x], x] /; FreeQ[{F, a, 
 b, c, d, n}, x] && PolynomialQ[Px, x]
 

rule 6917

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

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.39

method	result	size
parallelrisch	\(\frac {2 d^{2} \operatorname {erfi}\left (b x +a \right ) x^{3} \sqrt {\pi }\, b^{3}+6 c d \,x^{2} \operatorname {erfi}\left (b x +a \right ) \sqrt {\pi }\, b^{3}+6 c^{2} x \,\operatorname {erfi}\left (b x +a \right ) \sqrt {\pi }\, b^{3}+2 \sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right ) a^{3} d^{2}-6 \sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right ) a^{2} b c d +6 \sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right ) a \,b^{2} c^{2}-2 d^{2} {\mathrm e}^{\left (b x +a \right )^{2}} x^{2} b^{2}+2 x \,{\mathrm e}^{\left (b x +a \right )^{2}} a b \,d^{2}-6 x \,{\mathrm e}^{\left (b x +a \right )^{2}} b^{2} c d -3 \sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right ) a \,d^{2}+3 \sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right ) b c d -2 \,{\mathrm e}^{\left (b x +a \right )^{2}} a^{2} d^{2}+6 \,{\mathrm e}^{\left (b x +a \right )^{2}} a b c d -6 \,{\mathrm e}^{\left (b x +a \right )^{2}} b^{2} c^{2}+2 \,{\mathrm e}^{\left (b x +a \right )^{2}} d^{2}}{6 \sqrt {\pi }\, b^{3}}\)	\(258\)
derivativedivides	\(\frac {-\frac {d^{2} \operatorname {erfi}\left (b x +a \right ) a^{3}}{3 b^{2}}+\frac {d \,\operatorname {erfi}\left (b x +a \right ) a^{2} c}{b}+\frac {d^{2} \operatorname {erfi}\left (b x +a \right ) a^{2} \left (b x +a \right )}{b^{2}}-\operatorname {erfi}\left (b x +a \right ) a \,c^{2}-\frac {2 d \,\operatorname {erfi}\left (b x +a \right ) a c \left (b x +a \right )}{b}-\frac {d^{2} \operatorname {erfi}\left (b x +a \right ) a \left (b x +a \right )^{2}}{b^{2}}+\frac {b \,\operatorname {erfi}\left (b x +a \right ) c^{3}}{3 d}+\operatorname {erfi}\left (b x +a \right ) c^{2} \left (b x +a \right )+\frac {d \,\operatorname {erfi}\left (b x +a \right ) c \left (b x +a \right )^{2}}{b}+\frac {d^{2} \operatorname {erfi}\left (b x +a \right ) \left (b x +a \right )^{3}}{3 b^{2}}+\frac {\frac {a^{3} d^{3} \sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right )}{3}-\frac {b^{3} c^{3} \sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right )}{3}-\frac {2 d^{3} \left (\frac {{\mathrm e}^{\left (b x +a \right )^{2}} \left (b x +a \right )^{2}}{2}-\frac {{\mathrm e}^{\left (b x +a \right )^{2}}}{2}\right )}{3}+2 a \,d^{3} \left (\frac {\left (b x +a \right ) {\mathrm e}^{\left (b x +a \right )^{2}}}{2}-\frac {\sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right )}{4}\right )-a^{2} d^{3} {\mathrm e}^{\left (b x +a \right )^{2}}+a \,b^{2} c^{2} d \sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right )-a^{2} b c \,d^{2} \sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right )-2 b c \,d^{2} \left (\frac {\left (b x +a \right ) {\mathrm e}^{\left (b x +a \right )^{2}}}{2}-\frac {\sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right )}{4}\right )-b^{2} c^{2} d \,{\mathrm e}^{\left (b x +a \right )^{2}}+2 a b c \,d^{2} {\mathrm e}^{\left (b x +a \right )^{2}}}{\sqrt {\pi }\, b^{2} d}}{b}\)	\(415\)
default	\(\frac {-\frac {d^{2} \operatorname {erfi}\left (b x +a \right ) a^{3}}{3 b^{2}}+\frac {d \,\operatorname {erfi}\left (b x +a \right ) a^{2} c}{b}+\frac {d^{2} \operatorname {erfi}\left (b x +a \right ) a^{2} \left (b x +a \right )}{b^{2}}-\operatorname {erfi}\left (b x +a \right ) a \,c^{2}-\frac {2 d \,\operatorname {erfi}\left (b x +a \right ) a c \left (b x +a \right )}{b}-\frac {d^{2} \operatorname {erfi}\left (b x +a \right ) a \left (b x +a \right )^{2}}{b^{2}}+\frac {b \,\operatorname {erfi}\left (b x +a \right ) c^{3}}{3 d}+\operatorname {erfi}\left (b x +a \right ) c^{2} \left (b x +a \right )+\frac {d \,\operatorname {erfi}\left (b x +a \right ) c \left (b x +a \right )^{2}}{b}+\frac {d^{2} \operatorname {erfi}\left (b x +a \right ) \left (b x +a \right )^{3}}{3 b^{2}}+\frac {\frac {a^{3} d^{3} \sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right )}{3}-\frac {b^{3} c^{3} \sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right )}{3}-\frac {2 d^{3} \left (\frac {{\mathrm e}^{\left (b x +a \right )^{2}} \left (b x +a \right )^{2}}{2}-\frac {{\mathrm e}^{\left (b x +a \right )^{2}}}{2}\right )}{3}+2 a \,d^{3} \left (\frac {\left (b x +a \right ) {\mathrm e}^{\left (b x +a \right )^{2}}}{2}-\frac {\sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right )}{4}\right )-a^{2} d^{3} {\mathrm e}^{\left (b x +a \right )^{2}}+a \,b^{2} c^{2} d \sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right )-a^{2} b c \,d^{2} \sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right )-2 b c \,d^{2} \left (\frac {\left (b x +a \right ) {\mathrm e}^{\left (b x +a \right )^{2}}}{2}-\frac {\sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right )}{4}\right )-b^{2} c^{2} d \,{\mathrm e}^{\left (b x +a \right )^{2}}+2 a b c \,d^{2} {\mathrm e}^{\left (b x +a \right )^{2}}}{\sqrt {\pi }\, b^{2} d}}{b}\)	\(415\)
parts	\(\frac {\operatorname {erfi}\left (b x +a \right ) d^{2} x^{3}}{3}+\operatorname {erfi}\left (b x +a \right ) d c \,x^{2}+\operatorname {erfi}\left (b x +a \right ) c^{2} x +\frac {\operatorname {erfi}\left (b x +a \right ) c^{3}}{3 d}-\frac {2 b \left (-\frac {i {\mathrm e}^{a^{2}} c^{3} \sqrt {\pi }\, {\mathrm e}^{-a^{2}} \operatorname {erf}\left (i b x +i a \right )}{2 b}+{\mathrm e}^{a^{2}} d^{3} \left (\frac {x^{2} {\mathrm e}^{b^{2} x^{2}+2 b x a}}{2 b^{2}}-\frac {a \left (\frac {x \,{\mathrm e}^{b^{2} x^{2}+2 b x a}}{2 b^{2}}-\frac {a \left (\frac {{\mathrm e}^{b^{2} x^{2}+2 b x a}}{2 b^{2}}+\frac {i a \sqrt {\pi }\, {\mathrm e}^{-a^{2}} \operatorname {erf}\left (i b x +i a \right )}{2 b^{2}}\right )}{b}+\frac {i \sqrt {\pi }\, {\mathrm e}^{-a^{2}} \operatorname {erf}\left (i b x +i a \right )}{4 b^{3}}\right )}{b}-\frac {\frac {{\mathrm e}^{b^{2} x^{2}+2 b x a}}{2 b^{2}}+\frac {i a \sqrt {\pi }\, {\mathrm e}^{-a^{2}} \operatorname {erf}\left (i b x +i a \right )}{2 b^{2}}}{b^{2}}\right )+3 \,{\mathrm e}^{a^{2}} c \,d^{2} \left (\frac {x \,{\mathrm e}^{b^{2} x^{2}+2 b x a}}{2 b^{2}}-\frac {a \left (\frac {{\mathrm e}^{b^{2} x^{2}+2 b x a}}{2 b^{2}}+\frac {i a \sqrt {\pi }\, {\mathrm e}^{-a^{2}} \operatorname {erf}\left (i b x +i a \right )}{2 b^{2}}\right )}{b}+\frac {i \sqrt {\pi }\, {\mathrm e}^{-a^{2}} \operatorname {erf}\left (i b x +i a \right )}{4 b^{3}}\right )+3 \,{\mathrm e}^{a^{2}} d \,c^{2} \left (\frac {{\mathrm e}^{b^{2} x^{2}+2 b x a}}{2 b^{2}}+\frac {i a \sqrt {\pi }\, {\mathrm e}^{-a^{2}} \operatorname {erf}\left (i b x +i a \right )}{2 b^{2}}\right )\right )}{3 d \sqrt {\pi }}\)	\(452\)

Input:

int((d*x+c)^2*erfi(b*x+a),x,method=_RETURNVERBOSE)
 

Output:

1/6*(2*d^2*erfi(b*x+a)*x^3*Pi^(1/2)*b^3+6*c*d*x^2*erfi(b*x+a)*Pi^(1/2)*b^3 
+6*c^2*x*erfi(b*x+a)*Pi^(1/2)*b^3+2*Pi^(1/2)*erfi(b*x+a)*a^3*d^2-6*Pi^(1/2 
)*erfi(b*x+a)*a^2*b*c*d+6*Pi^(1/2)*erfi(b*x+a)*a*b^2*c^2-2*d^2*exp((b*x+a) 
^2)*x^2*b^2+2*x*exp((b*x+a)^2)*a*b*d^2-6*x*exp((b*x+a)^2)*b^2*c*d-3*Pi^(1/ 
2)*erfi(b*x+a)*a*d^2+3*Pi^(1/2)*erfi(b*x+a)*b*c*d-2*exp((b*x+a)^2)*a^2*d^2 
+6*exp((b*x+a)^2)*a*b*c*d-6*exp((b*x+a)^2)*b^2*c^2+2*exp((b*x+a)^2)*d^2)/P 
i^(1/2)/b^3
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.87 \[ \int (c+d x)^2 \text {erfi}(a+b x) \, dx=-\frac {2 \, \sqrt {\pi } {\left (b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} - 3 \, a b c d + {\left (a^{2} - 1\right )} d^{2} + {\left (3 \, b^{2} c d - a b d^{2}\right )} x\right )} e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} - {\left (2 \, \pi b^{3} d^{2} x^{3} + 6 \, \pi b^{3} c d x^{2} + 6 \, \pi b^{3} c^{2} x + \pi {\left (6 \, a b^{2} c^{2} - 3 \, {\left (2 \, a^{2} - 1\right )} b c d + {\left (2 \, a^{3} - 3 \, a\right )} d^{2}\right )}\right )} \operatorname {erfi}\left (b x + a\right )}{6 \, \pi b^{3}} \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (165) = 330\).

Time = 0.94 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.14 \[ \int (c+d x)^2 \text {erfi}(a+b x) \, dx=\begin {cases} \frac {a^{3} d^{2} \operatorname {erfi}{\left (a + b x \right )}}{3 b^{3}} - \frac {a^{2} c d \operatorname {erfi}{\left (a + b x \right )}}{b^{2}} - \frac {a^{2} d^{2} e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{3 \sqrt {\pi } b^{3}} + \frac {a c^{2} \operatorname {erfi}{\left (a + b x \right )}}{b} + \frac {a c d e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{\sqrt {\pi } b^{2}} + \frac {a d^{2} x e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{3 \sqrt {\pi } b^{2}} - \frac {a d^{2} \operatorname {erfi}{\left (a + b x \right )}}{2 b^{3}} + c^{2} x \operatorname {erfi}{\left (a + b x \right )} + c d x^{2} \operatorname {erfi}{\left (a + b x \right )} + \frac {d^{2} x^{3} \operatorname {erfi}{\left (a + b x \right )}}{3} - \frac {c^{2} e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{\sqrt {\pi } b} - \frac {c d x e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{\sqrt {\pi } b} - \frac {d^{2} x^{2} e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{3 \sqrt {\pi } b} + \frac {c d \operatorname {erfi}{\left (a + b x \right )}}{2 b^{2}} + \frac {d^{2} e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{3 \sqrt {\pi } b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \operatorname {erfi}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a**3*d**2*erfi(a + b*x)/(3*b**3) - a**2*c*d*erfi(a + b*x)/b**2 
- a**2*d**2*exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(3*sqrt(pi)*b**3) + a*c* 
*2*erfi(a + b*x)/b + a*c*d*exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(sqrt(pi) 
*b**2) + a*d**2*x*exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(3*sqrt(pi)*b**2) 
- a*d**2*erfi(a + b*x)/(2*b**3) + c**2*x*erfi(a + b*x) + c*d*x**2*erfi(a + 
 b*x) + d**2*x**3*erfi(a + b*x)/3 - c**2*exp(a**2)*exp(b**2*x**2)*exp(2*a* 
b*x)/(sqrt(pi)*b) - c*d*x*exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(sqrt(pi)* 
b) - d**2*x**2*exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(3*sqrt(pi)*b) + c*d* 
erfi(a + b*x)/(2*b**2) + d**2*exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(3*sqr 
t(pi)*b**3), Ne(b, 0)), ((c**2*x + c*d*x**2 + d**2*x**3/3)*erfi(a), True))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 4.01 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.02 \[ \int (c+d x)^2 \text {erfi}(a+b x) \, dx=\frac {\frac {{\mathrm {e}}^{a^2+2\,a\,b\,x+b^2\,x^2}\,\left (-a^2\,d^2+3\,a\,b\,c\,d-3\,b^2\,c^2+d^2\right )}{b^3}+\frac {x\,{\mathrm {e}}^{a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a\,d^2-3\,b\,c\,d\right )}{b^2}-\frac {d^2\,x^2\,{\mathrm {e}}^{a^2+2\,a\,b\,x+b^2\,x^2}}{b}}{3\,\sqrt {\pi }}+\mathrm {erfi}\left (a+b\,x\right )\,\left (c^2\,x+c\,d\,x^2+\frac {d^2\,x^3}{3}\right )+\frac {\mathrm {erfi}\left (a+b\,x\right )\,\left (2\,a^3\,d^2-6\,a^2\,b\,c\,d+6\,a\,b^2\,c^2-3\,a\,d^2+3\,b\,c\,d\right )}{6\,b^3} \] Input:

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

Output:

((exp(a^2 + b^2*x^2 + 2*a*b*x)*(d^2 - a^2*d^2 - 3*b^2*c^2 + 3*a*b*c*d))/b^ 
3 + (x*exp(a^2 + b^2*x^2 + 2*a*b*x)*(a*d^2 - 3*b*c*d))/b^2 - (d^2*x^2*exp( 
a^2 + b^2*x^2 + 2*a*b*x))/b)/(3*pi^(1/2)) + erfi(a + b*x)*(c^2*x + (d^2*x^ 
3)/3 + c*d*x^2) + (erfi(a + b*x)*(2*a^3*d^2 - 3*a*d^2 + 6*a*b^2*c^2 + 3*b* 
c*d - 6*a^2*b*c*d))/(6*b^3)
 

Reduce [F]

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

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

Output:

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