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

Optimal. Leaf size=194 \[ \frac{(b c-a d)^3 \text{Erf}(a+b x)}{3 b^3 d}+\frac{d (b c-a d) \text{Erf}(a+b x)}{2 b^3}-\frac{e^{-(a+b x)^2} (b c-a d)^2}{\sqrt{\pi } b^3}-\frac{d e^{-(a+b x)^2} (a+b x) (b c-a d)}{\sqrt{\pi } b^3}-\frac{d^2 e^{-(a+b x)^2} (a+b x)^2}{3 \sqrt{\pi } b^3}-\frac{d^2 e^{-(a+b x)^2}}{3 \sqrt{\pi } b^3}+\frac{(c+d x)^3 \text{Erfc}(a+b x)}{3 d} \]

[Out]

-d^2/(3*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)

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Rubi [A]  time = 0.180591, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {6362, 2226, 2205, 2209, 2212} \[ \frac{(b c-a d)^3 \text{Erf}(a+b x)}{3 b^3 d}+\frac{d (b c-a d) \text{Erf}(a+b x)}{2 b^3}-\frac{e^{-(a+b x)^2} (b c-a d)^2}{\sqrt{\pi } b^3}-\frac{d e^{-(a+b x)^2} (a+b x) (b c-a d)}{\sqrt{\pi } b^3}-\frac{d^2 e^{-(a+b x)^2} (a+b x)^2}{3 \sqrt{\pi } b^3}-\frac{d^2 e^{-(a+b x)^2}}{3 \sqrt{\pi } b^3}+\frac{(c+d x)^3 \text{Erfc}(a+b x)}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2*Erfc[a + b*x],x]

[Out]

-d^2/(3*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 6362

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]

Rule 2226

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 2205

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 2209

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 2212

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])

Rubi steps

\begin{align*} \int (c+d x)^2 \text{erfc}(a+b x) \, dx &=\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]  time = 0.28104, size = 159, normalized size = 0.82 \[ \frac{\frac{2 e^{-(a+b x)^2} \left (-\left (a^2+1\right ) d^2+\sqrt{\pi } b^3 x e^{(a+b x)^2} \left (3 c^2+3 c d x+d^2 x^2\right ) \text{Erfc}(a+b x)+a b d (3 c+d x)-b^2 \left (3 c^2+3 c d x+d^2 x^2\right )\right )}{\sqrt{\pi }}-\left (-6 a^2 b c d+2 a^3 d^2+3 a \left (2 b^2 c^2+d^2\right )-3 b c d\right ) \text{Erf}(a+b x)}{6 b^3} \]

Antiderivative was successfully verified.

[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)

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Maple [B]  time = 0.049, size = 428, normalized size = 2.2 \begin{align*}{\frac{1}{b} \left ({\frac{{d}^{2}{\it erfc} \left ( bx+a \right ) \left ( bx+a \right ) ^{3}}{3\,{b}^{2}}}-{\frac{{d}^{2}{\it erfc} \left ( bx+a \right ) \left ( bx+a \right ) ^{2}a}{{b}^{2}}}+{\frac{d{\it erfc} \left ( bx+a \right ) \left ( bx+a \right ) ^{2}c}{b}}+{\frac{{d}^{2}{\it erfc} \left ( bx+a \right ) \left ( bx+a \right ){a}^{2}}{{b}^{2}}}-2\,{\frac{d{\it erfc} \left ( bx+a \right ) \left ( bx+a \right ) ac}{b}}+{\it erfc} \left ( bx+a \right ) \left ( bx+a \right ){c}^{2}-{\frac{{d}^{2}{\it erfc} \left ( bx+a \right ){a}^{3}}{3\,{b}^{2}}}+{\frac{d{\it erfc} \left ( bx+a \right ){a}^{2}c}{b}}-{\it erfc} \left ( bx+a \right ) a{c}^{2}+{\frac{b{\it erfc} \left ( bx+a \right ){c}^{3}}{3\,d}}+{\frac{2}{3\,{b}^{2}\sqrt{\pi }d} \left ({\frac{{b}^{3}{c}^{3}\sqrt{\pi }{\it Erf} \left ( bx+a \right ) }{2}}+{d}^{3} \left ( -{\frac{ \left ( bx+a \right ) ^{2}}{2\,{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}}-{\frac{1}{2\,{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}} \right ) -{\frac{{a}^{3}{d}^{3}\sqrt{\pi }{\it Erf} \left ( bx+a \right ) }{2}}-{\frac{3\,{a}^{2}{d}^{3}}{2\,{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}}-3\,a{d}^{3} \left ( -1/2\,{\frac{bx+a}{{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}}+1/4\,\sqrt{\pi }{\it Erf} \left ( bx+a \right ) \right ) -{\frac{3\,{b}^{2}{c}^{2}d}{2\,{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}}+3\,bc{d}^{2} \left ( -1/2\,{\frac{bx+a}{{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}}+1/4\,\sqrt{\pi }{\it Erf} \left ( bx+a \right ) \right ) -{\frac{3\,a{b}^{2}{c}^{2}d\sqrt{\pi }{\it Erf} \left ( bx+a \right ) }{2}}+{\frac{3\,{a}^{2}bc{d}^{2}\sqrt{\pi }{\it Erf} \left ( bx+a \right ) }{2}}+3\,{\frac{abc{d}^{2}}{{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{2} \operatorname{erfc}\left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.15712, size = 435, normalized size = 2.24 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 6.16949, size = 398, normalized size = 2.05 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.52206, size = 378, normalized size = 1.95 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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