\(\int (c+d x)^2 \text {erfc}(a+b x) \, dx\) [119]

Optimal result

Integrand size = 14, antiderivative size = 194 \[ \int (c+d x)^2 \text {erfc}(a+b x) \, dx=-\frac {d^2 e^{-(a+b x)^2}}{3 b^3 \sqrt {\pi }}-\frac {(b c-a d)^2 e^{-(a+b x)^2}}{b^3 \sqrt {\pi }}-\frac {d (b c-a d) e^{-(a+b x)^2} (a+b x)}{b^3 \sqrt {\pi }}-\frac {d^2 e^{-(a+b x)^2} (a+b x)^2}{3 b^3 \sqrt {\pi }}+\frac {d (b c-a d) \text {erf}(a+b x)}{2 b^3}+\frac {(b c-a d)^3 \text {erf}(a+b x)}{3 b^3 d}+\frac {(c+d x)^3 \text {erfc}(a+b x)}{3 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6497, 2258, 2236, 2240, 2243} \[ \int (c+d x)^2 \text {erfc}(a+b x) \, dx=\frac {(b c-a d)^3 \text {erf}(a+b x)}{3 b^3 d}+\frac {d (b c-a d) \text {erf}(a+b x)}{2 b^3}-\frac {e^{-(a+b x)^2} (b c-a d)^2}{\sqrt {\pi } b^3}-\frac {d e^{-(a+b x)^2} (a+b x) (b c-a d)}{\sqrt {\pi } b^3}-\frac {d^2 e^{-(a+b x)^2} (a+b x)^2}{3 \sqrt {\pi } b^3}-\frac {d^2 e^{-(a+b x)^2}}{3 \sqrt {\pi } b^3}+\frac {(c+d x)^3 \text {erfc}(a+b x)}{3 d} \]

[In]

Int[(c + d*x)^2*Erfc[a + b*x],x]

[Out]

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

Rule 2236

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 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2243

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])

Rule 2258

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

Rule 6497

Int[Erfc[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Erfc[a + b*x]/(d
*(m + 1))), x] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^3 \text {erfc}(a+b x)}{3 d}+\frac {(2 b) \int e^{-(a+b x)^2} (c+d x)^3 \, dx}{3 d \sqrt {\pi }} \\ & = \frac {(c+d x)^3 \text {erfc}(a+b x)}{3 d}+\frac {(2 b) \int \left (\frac {(b c-a d)^3 e^{-(a+b x)^2}}{b^3}+\frac {3 d (b c-a d)^2 e^{-(a+b x)^2} (a+b x)}{b^3}+\frac {3 d^2 (b c-a d) e^{-(a+b x)^2} (a+b x)^2}{b^3}+\frac {d^3 e^{-(a+b x)^2} (a+b x)^3}{b^3}\right ) \, dx}{3 d \sqrt {\pi }} \\ & = \frac {(c+d x)^3 \text {erfc}(a+b x)}{3 d}+\frac {\left (2 d^2\right ) \int e^{-(a+b x)^2} (a+b x)^3 \, dx}{3 b^2 \sqrt {\pi }}+\frac {(2 d (b c-a d)) \int e^{-(a+b x)^2} (a+b x)^2 \, dx}{b^2 \sqrt {\pi }}+\frac {\left (2 (b c-a d)^2\right ) \int e^{-(a+b x)^2} (a+b x) \, dx}{b^2 \sqrt {\pi }}+\frac {\left (2 (b c-a d)^3\right ) \int e^{-(a+b x)^2} \, dx}{3 b^2 d \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 {(b c-a d)^3 \text {erf}(a+b x)}{3 b^3 d}+\frac {(c+d x)^3 \text {erfc}(a+b x)}{3 d}+\frac {\left (2 d^2\right ) \int e^{-(a+b x)^2} (a+b x) \, dx}{3 b^2 \sqrt {\pi }}+\frac {(d (b c-a d)) \int e^{-(a+b x)^2} \, dx}{b^2 \sqrt {\pi }} \\ & = -\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 {erf}(a+b x)}{2 b^3}+\frac {(b c-a d)^3 \text {erf}(a+b x)}{3 b^3 d}+\frac {(c+d x)^3 \text {erfc}(a+b x)}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.82 \[ \int (c+d x)^2 \text {erfc}(a+b x) \, dx=\frac {-\left (\left (-3 b c d-6 a^2 b c d+2 a^3 d^2+3 a \left (2 b^2 c^2+d^2\right )\right ) \text {erf}(a+b x)\right )+\frac {2 e^{-(a+b x)^2} \left (-\left (\left (1+a^2\right ) d^2\right )+a b d (3 c+d x)-b^2 \left (3 c^2+3 c d x+d^2 x^2\right )+b^3 e^{(a+b x)^2} \sqrt {\pi } x \left (3 c^2+3 c d x+d^2 x^2\right ) \text {erfc}(a+b x)\right )}{\sqrt {\pi }}}{6 b^3} \]

[In]

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

[Out]

(-((-3*b*c*d - 6*a^2*b*c*d + 2*a^3*d^2 + 3*a*(2*b^2*c^2 + d^2))*Erf[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) + b^3*E^(a + b*x)^2*Sqrt[Pi]*x*(3*c^2 + 3*c*d*x + d^2*x^2)*Erfc[a
+ b*x]))/(E^(a + b*x)^2*Sqrt[Pi]))/(6*b^3)

Maple [A] (verified)

Time = 0.91 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.40

method	result	size
parallelrisch	\(\frac {2 d^{2} \operatorname {erfc}\left (b x +a \right ) x^{3} \sqrt {\pi }\, b^{3}+6 c d \,\operatorname {erfc}\left (b x +a \right ) x^{2} \sqrt {\pi }\, b^{3}+6 c^{2} x \,\operatorname {erfc}\left (b x +a \right ) \sqrt {\pi }\, b^{3}+2 \sqrt {\pi }\, \operatorname {erfc}\left (b x +a \right ) a^{3} d^{2}-6 \sqrt {\pi }\, \operatorname {erfc}\left (b x +a \right ) a^{2} b c d +6 \sqrt {\pi }\, \operatorname {erfc}\left (b x +a \right ) a \,b^{2} c^{2}-2 d^{2} x^{2} {\mathrm e}^{-\left (b x +a \right )^{2}} b^{2}+2 x \,{\mathrm e}^{-\left (b x +a \right )^{2}} a b \,d^{2}-6 x \,{\mathrm e}^{-\left (b x +a \right )^{2}} b^{2} c d +3 \sqrt {\pi }\, \operatorname {erfc}\left (b x +a \right ) a \,d^{2}-3 \sqrt {\pi }\, \operatorname {erfc}\left (b x +a \right ) b c d -2 \,{\mathrm e}^{-\left (b x +a \right )^{2}} a^{2} d^{2}+6 \,{\mathrm e}^{-\left (b x +a \right )^{2}} a b c d -6 \,{\mathrm e}^{-\left (b x +a \right )^{2}} b^{2} c^{2}-2 \,{\mathrm e}^{-\left (b x +a \right )^{2}} d^{2}}{6 \sqrt {\pi }\, b^{3}}\)	\(272\)
parts	\(\frac {\operatorname {erfc}\left (b x +a \right ) d^{2} x^{3}}{3}+\operatorname {erfc}\left (b x +a \right ) d c \,x^{2}+\operatorname {erfc}\left (b x +a \right ) c^{2} x +\frac {\operatorname {erfc}\left (b x +a \right ) c^{3}}{3 d}+\frac {2 b \left (\frac {c^{3} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2 b}+{\mathrm e}^{-a^{2}} d^{3} \left (-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \left (-\frac {x \,{\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \left (-\frac {{\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{2 b^{2}}\right )}{b}+\frac {\sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{4 b^{3}}\right )}{b}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{2 b^{2}}}{b^{2}}\right )+3 \,{\mathrm e}^{-a^{2}} c \,d^{2} \left (-\frac {x \,{\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \left (-\frac {{\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{2 b^{2}}\right )}{b}+\frac {\sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{4 b^{3}}\right )+3 \,{\mathrm e}^{-a^{2}} d \,c^{2} \left (-\frac {{\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{2 b^{2}}\right )\right )}{3 d \sqrt {\pi }}\)	\(400\)
derivativedivides	\(\frac {-\frac {d^{2} \operatorname {erfc}\left (b x +a \right ) a^{3}}{3 b^{2}}+\frac {d \,\operatorname {erfc}\left (b x +a \right ) a^{2} c}{b}+\frac {d^{2} \operatorname {erfc}\left (b x +a \right ) a^{2} \left (b x +a \right )}{b^{2}}-\operatorname {erfc}\left (b x +a \right ) a \,c^{2}-\frac {2 d \,\operatorname {erfc}\left (b x +a \right ) a c \left (b x +a \right )}{b}-\frac {d^{2} \operatorname {erfc}\left (b x +a \right ) a \left (b x +a \right )^{2}}{b^{2}}+\frac {b \,\operatorname {erfc}\left (b x +a \right ) c^{3}}{3 d}+\operatorname {erfc}\left (b x +a \right ) c^{2} \left (b x +a \right )+\frac {d \,\operatorname {erfc}\left (b x +a \right ) c \left (b x +a \right )^{2}}{b}+\frac {d^{2} \operatorname {erfc}\left (b x +a \right ) \left (b x +a \right )^{3}}{3 b^{2}}-\frac {2 \left (\frac {a^{3} d^{3} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2}-\frac {b^{3} c^{3} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2}-d^{3} \left (-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}} \left (b x +a \right )^{2}}{2}-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{2}\right )+3 a \,d^{3} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{4}\right )+\frac {3 a^{2} d^{3} {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {3 a \,b^{2} c^{2} d \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2}-\frac {3 a^{2} b c \,d^{2} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2}-3 b c \,d^{2} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{4}\right )+\frac {3 b^{2} c^{2} d \,{\mathrm e}^{-\left (b x +a \right )^{2}}}{2}-3 a b c \,d^{2} {\mathrm e}^{-\left (b x +a \right )^{2}}\right )}{3 \sqrt {\pi }\, b^{2} d}}{b}\)	\(429\)
default	\(\frac {-\frac {d^{2} \operatorname {erfc}\left (b x +a \right ) a^{3}}{3 b^{2}}+\frac {d \,\operatorname {erfc}\left (b x +a \right ) a^{2} c}{b}+\frac {d^{2} \operatorname {erfc}\left (b x +a \right ) a^{2} \left (b x +a \right )}{b^{2}}-\operatorname {erfc}\left (b x +a \right ) a \,c^{2}-\frac {2 d \,\operatorname {erfc}\left (b x +a \right ) a c \left (b x +a \right )}{b}-\frac {d^{2} \operatorname {erfc}\left (b x +a \right ) a \left (b x +a \right )^{2}}{b^{2}}+\frac {b \,\operatorname {erfc}\left (b x +a \right ) c^{3}}{3 d}+\operatorname {erfc}\left (b x +a \right ) c^{2} \left (b x +a \right )+\frac {d \,\operatorname {erfc}\left (b x +a \right ) c \left (b x +a \right )^{2}}{b}+\frac {d^{2} \operatorname {erfc}\left (b x +a \right ) \left (b x +a \right )^{3}}{3 b^{2}}-\frac {2 \left (\frac {a^{3} d^{3} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2}-\frac {b^{3} c^{3} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2}-d^{3} \left (-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}} \left (b x +a \right )^{2}}{2}-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{2}\right )+3 a \,d^{3} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{4}\right )+\frac {3 a^{2} d^{3} {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {3 a \,b^{2} c^{2} d \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2}-\frac {3 a^{2} b c \,d^{2} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2}-3 b c \,d^{2} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{4}\right )+\frac {3 b^{2} c^{2} d \,{\mathrm e}^{-\left (b x +a \right )^{2}}}{2}-3 a b c \,d^{2} {\mathrm e}^{-\left (b x +a \right )^{2}}\right )}{3 \sqrt {\pi }\, b^{2} d}}{b}\)	\(429\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.02 \[ \int (c+d x)^2 \text {erfc}(a+b x) \, dx=\frac {2 \, \pi b^{3} d^{2} x^{3} + 6 \, \pi b^{3} c d x^{2} + 6 \, \pi b^{3} c^{2} x - 2 \, \sqrt {\pi } {\left (b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} - 3 \, a b c d + {\left (a^{2} + 1\right )} d^{2} + {\left (3 \, b^{2} c d - a b d^{2}\right )} x\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )} - {\left (2 \, \pi b^{3} d^{2} x^{3} + 6 \, \pi b^{3} c d x^{2} + 6 \, \pi b^{3} c^{2} x + \pi {\left (6 \, a b^{2} c^{2} - 3 \, {\left (2 \, a^{2} + 1\right )} b c d + {\left (2 \, a^{3} + 3 \, a\right )} d^{2}\right )}\right )} \operatorname {erf}\left (b x + a\right )}{6 \, \pi b^{3}} \]

[In]

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

[Out]

1/6*(2*pi*b^3*d^2*x^3 + 6*pi*b^3*c*d*x^2 + 6*pi*b^3*c^2*x - 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))*erf(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 = 0.88 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.05 \[ \int (c+d x)^2 \text {erfc}(a+b x) \, dx=\begin {cases} \frac {a^{3} d^{2} \operatorname {erfc}{\left (a + b x \right )}}{3 b^{3}} - \frac {a^{2} c d \operatorname {erfc}{\left (a + b x \right )}}{b^{2}} - \frac {a^{2} d^{2} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{3 \sqrt {\pi } b^{3}} + \frac {a c^{2} \operatorname {erfc}{\left (a + b x \right )}}{b} + \frac {a c d e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b^{2}} + \frac {a d^{2} x e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{3 \sqrt {\pi } b^{2}} + \frac {a d^{2} \operatorname {erfc}{\left (a + b x \right )}}{2 b^{3}} + c^{2} x \operatorname {erfc}{\left (a + b x \right )} + c d x^{2} \operatorname {erfc}{\left (a + b x \right )} + \frac {d^{2} x^{3} \operatorname {erfc}{\left (a + b x \right )}}{3} - \frac {c^{2} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b} - \frac {c d x e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b} - \frac {d^{2} x^{2} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{3 \sqrt {\pi } b} - \frac {c d \operatorname {erfc}{\left (a + b x \right )}}{2 b^{2}} - \frac {d^{2} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{3 \sqrt {\pi } b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \operatorname {erfc}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**3*d**2*erfc(a + b*x)/(3*b**3) - a**2*c*d*erfc(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*erfc(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*erfc(a + b*x)/(2
*b**3) + c**2*x*erfc(a + b*x) + c*d*x**2*erfc(a + b*x) + d**2*x**3*erfc(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*ex
p(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*x)/(3*sqrt(pi)*b) - c*d*erfc(a + b*x)/(2*b**2) - d**2*exp(-a**2)*exp(-b**2
*x**2)*exp(-2*a*b*x)/(3*sqrt(pi)*b**3), Ne(b, 0)), ((c**2*x + c*d*x**2 + d**2*x**3/3)*erfc(a), True))

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.44 \[ \int (c+d x)^2 \text {erfc}(a+b x) \, dx=\frac {1}{3} \, d^{2} x^{3} + c d x^{2} - {\left (x \operatorname {erf}\left (b x + a\right ) - \frac {\frac {\sqrt {\pi } a \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi }}\right )} c^{2} - \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {erf}\left (b x + a\right ) + \frac {\frac {\sqrt {\pi } {\left (2 \, a^{2} + 1\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} + \frac {2 \, {\left (b {\left (x + \frac {a}{b}\right )} - 2 \, a\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi } b}\right )} c d - \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {erf}\left (b x + a\right ) - \frac {\frac {\sqrt {\pi } {\left (2 \, a^{3} + 3 \, a\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {2 \, {\left (b^{2} {\left (x + \frac {a}{b}\right )}^{2} - 3 \, a b {\left (x + \frac {a}{b}\right )} + 3 \, a^{2} + 1\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi } b^{2}}\right )} d^{2} + c^{2} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.97 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.13 \[ \int (c+d x)^2 \text {erfc}(a+b x) \, dx=\frac {d^2\,x^3\,\mathrm {erfc}\left (a+b\,x\right )}{3}-\frac {{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (\frac {b^2\,c^2}{\sqrt {\pi }}-\frac {a\,d\,b\,c}{\sqrt {\pi }}+\frac {a^2\,d^2+d^2}{3\,\sqrt {\pi }}\right )}{b^3}+\frac {\mathrm {erfc}\left (a+b\,x\right )\,\left (\frac {a\,d^2}{2}-b\,\left (c\,d\,a^2+\frac {c\,d}{2}\right )+\frac {a^3\,d^2}{3}+a\,b^2\,c^2\right )}{b^3}+c^2\,x\,\mathrm {erfc}\left (a+b\,x\right )+c\,d\,x^2\,\mathrm {erfc}\left (a+b\,x\right )+\frac {x\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (a\,d^2-3\,b\,c\,d\right )}{3\,b^2\,\sqrt {\pi }}-\frac {d^2\,x^2\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}}{3\,b\,\sqrt {\pi }} \]

[In]

int(erfc(a + b*x)*(c + d*x)^2,x)

[Out]

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