3.17 \(\int (c+d x) \text{Erf}(a+b x) \, dx\)

Optimal. Leaf size=118 \[ -\frac{(b c-a d)^2 \text{Erf}(a+b x)}{2 b^2 d}+\frac{e^{-(a+b x)^2} (b c-a d)}{\sqrt{\pi } b^2}-\frac{d \text{Erf}(a+b x)}{4 b^2}+\frac{d e^{-(a+b x)^2} (a+b x)}{2 \sqrt{\pi } b^2}+\frac{(c+d x)^2 \text{Erf}(a+b x)}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.119941, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6361, 2226, 2205, 2209, 2212} \[ -\frac{(b c-a d)^2 \text{Erf}(a+b x)}{2 b^2 d}+\frac{e^{-(a+b x)^2} (b c-a d)}{\sqrt{\pi } b^2}-\frac{d \text{Erf}(a+b x)}{4 b^2}+\frac{d e^{-(a+b x)^2} (a+b x)}{2 \sqrt{\pi } b^2}+\frac{(c+d x)^2 \text{Erf}(a+b x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)*Erf[a + b*x],x]

[Out]

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

Rule 6361

Int[Erf[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*Erf[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) \text{erf}(a+b x) \, dx &=\frac{(c+d x)^2 \text{erf}(a+b x)}{2 d}-\frac{b \int e^{-(a+b x)^2} (c+d x)^2 \, dx}{d \sqrt{\pi }}\\ &=\frac{(c+d x)^2 \text{erf}(a+b x)}{2 d}-\frac{b \int \left (\frac{(b c-a d)^2 e^{-(a+b x)^2}}{b^2}+\frac{2 d (b c-a d) e^{-(a+b x)^2} (a+b x)}{b^2}+\frac{d^2 e^{-(a+b x)^2} (a+b x)^2}{b^2}\right ) \, dx}{d \sqrt{\pi }}\\ &=\frac{(c+d x)^2 \text{erf}(a+b x)}{2 d}-\frac{d \int e^{-(a+b x)^2} (a+b x)^2 \, dx}{b \sqrt{\pi }}-\frac{(2 (b c-a d)) \int e^{-(a+b x)^2} (a+b x) \, dx}{b \sqrt{\pi }}-\frac{(b c-a d)^2 \int e^{-(a+b x)^2} \, dx}{b d \sqrt{\pi }}\\ &=\frac{(b c-a d) e^{-(a+b x)^2}}{b^2 \sqrt{\pi }}+\frac{d e^{-(a+b x)^2} (a+b x)}{2 b^2 \sqrt{\pi }}-\frac{(b c-a d)^2 \text{erf}(a+b x)}{2 b^2 d}+\frac{(c+d x)^2 \text{erf}(a+b x)}{2 d}-\frac{d \int e^{-(a+b x)^2} \, dx}{2 b \sqrt{\pi }}\\ &=\frac{(b c-a d) e^{-(a+b x)^2}}{b^2 \sqrt{\pi }}+\frac{d e^{-(a+b x)^2} (a+b x)}{2 b^2 \sqrt{\pi }}-\frac{d \text{erf}(a+b x)}{4 b^2}-\frac{(b c-a d)^2 \text{erf}(a+b x)}{2 b^2 d}+\frac{(c+d x)^2 \text{erf}(a+b x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0850256, size = 88, normalized size = 0.75 \[ \frac{e^{-(a+b x)^2} \left (-\sqrt{\pi } e^{(a+b x)^2} \text{Erf}(a+b x) \left (2 a^2 d-4 a b c-4 b^2 c x-2 b^2 d x^2+d\right )-2 a d+4 b c+2 b d x\right )}{4 \sqrt{\pi } b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)*Erf[a + b*x],x]

[Out]

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

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Maple [A]  time = 0.049, size = 111, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ({\frac{{\it Erf} \left ( bx+a \right ) }{b} \left ({\frac{d \left ( bx+a \right ) ^{2}}{2}}-ad \left ( bx+a \right ) +bc \left ( bx+a \right ) \right ) }-{\frac{1}{\sqrt{\pi }b} \left ( d \left ( -{\frac{bx+a}{2\,{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}}+{\frac{\sqrt{\pi }{\it Erf} \left ( bx+a \right ) }{4}} \right ) +{\frac{ad}{{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}}-{\frac{bc}{{{\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)*erf(b*x+a),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \,{\left (d x^{2} + 2 \, c x\right )} \operatorname{erf}\left (b x + a\right ) - \frac{-\frac{{\left (\frac{\sqrt{\pi }{\left (b^{2} x + a b\right )} a b{\left (\operatorname{erf}\left (\sqrt{\frac{{\left (b^{2} x + a b\right )}^{2}}{b^{2}}}\right ) - 1\right )}}{\left (-b^{2}\right )^{\frac{3}{2}} \sqrt{\frac{{\left (b^{2} x + a b\right )}^{2}}{b^{2}}}} + \frac{b^{2} e^{\left (-\frac{{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}}{\left (-b^{2}\right )^{\frac{3}{2}}}\right )} b c}{\sqrt{-b^{2}}} - \frac{{\left (\frac{\sqrt{\pi }{\left (b^{2} x + a b\right )} a^{2} b^{2}{\left (\operatorname{erf}\left (\sqrt{\frac{{\left (b^{2} x + a b\right )}^{2}}{b^{2}}}\right ) - 1\right )}}{\left (-b^{2}\right )^{\frac{5}{2}} \sqrt{\frac{{\left (b^{2} x + a b\right )}^{2}}{b^{2}}}} + \frac{2 \, a b^{3} e^{\left (-\frac{{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}}{\left (-b^{2}\right )^{\frac{5}{2}}} - \frac{{\left (b^{2} x + a b\right )}^{3} \Gamma \left (\frac{3}{2}, \frac{{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{\left (-b^{2}\right )^{\frac{5}{2}} \left (\frac{{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )^{\frac{3}{2}}}\right )} b d}{2 \, \sqrt{-b^{2}}}}{\sqrt{\pi }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.58843, size = 211, normalized size = 1.79 \begin{align*} \frac{2 \, \sqrt{\pi }{\left (b d x + 2 \, b c - a d\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )} +{\left (2 \, \pi b^{2} d x^{2} + 4 \, \pi b^{2} c x + \pi{\left (4 \, a b c -{\left (2 \, a^{2} + 1\right )} d\right )}\right )} \operatorname{erf}\left (b x + a\right )}{4 \, \pi b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*erf(b*x+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.57094, size = 178, normalized size = 1.51 \begin{align*} \begin{cases} - \frac{a^{2} d \operatorname{erf}{\left (a + b x \right )}}{2 b^{2}} + \frac{a c \operatorname{erf}{\left (a + b x \right )}}{b} - \frac{a d e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{2 \sqrt{\pi } b^{2}} + c x \operatorname{erf}{\left (a + b x \right )} + \frac{d x^{2} \operatorname{erf}{\left (a + b x \right )}}{2} + \frac{c e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt{\pi } b} + \frac{d x e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{2 \sqrt{\pi } b} - \frac{d \operatorname{erf}{\left (a + b x \right )}}{4 b^{2}} & \text{for}\: b \neq 0 \\\left (c x + \frac{d x^{2}}{2}\right ) \operatorname{erf}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*d*erf(a + b*x)/(2*b**2) + a*c*erf(a + b*x)/b - a*d*exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*x)/(
2*sqrt(pi)*b**2) + c*x*erf(a + b*x) + d*x**2*erf(a + b*x)/2 + c*exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*x)/(sqrt
(pi)*b) + d*x*exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*x)/(2*sqrt(pi)*b) - d*erf(a + b*x)/(4*b**2), Ne(b, 0)), ((
c*x + d*x**2/2)*erf(a), True))

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Giac [A]  time = 1.33218, size = 201, normalized size = 1.7 \begin{align*} \frac{1}{2} \,{\left (d x^{2} + 2 \, c x\right )} \operatorname{erf}\left (b x + a\right ) - \frac{4 \, \sqrt{\pi }{\left (\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}\right )} c - \frac{\sqrt{\pi }{\left (\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}\right )} d}{b}}{4 \, \pi } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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