\(\int (c+d x) \text {erf}(a+b x) \, dx\) [17]

Optimal result

Integrand size = 12, antiderivative size = 118 \[ \int (c+d x) \text {erf}(a+b x) \, dx=\frac {(b c-a d) e^{-(a+b x)^2}}{b^2 \sqrt {\pi }}+\frac {d e^{-(a+b x)^2} (a+b x)}{2 b^2 \sqrt {\pi }}-\frac {d \text {erf}(a+b x)}{4 b^2}-\frac {(b c-a d)^2 \text {erf}(a+b x)}{2 b^2 d}+\frac {(c+d x)^2 \text {erf}(a+b x)}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6496, 2258, 2236, 2240, 2243} \[ \int (c+d x) \text {erf}(a+b x) \, dx=-\frac {(b c-a d)^2 \text {erf}(a+b x)}{2 b^2 d}+\frac {e^{-(a+b x)^2} (b c-a d)}{\sqrt {\pi } b^2}-\frac {d \text {erf}(a+b x)}{4 b^2}+\frac {d e^{-(a+b x)^2} (a+b x)}{2 \sqrt {\pi } b^2}+\frac {(c+d x)^2 \text {erf}(a+b x)}{2 d} \]

[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 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 6496

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.75 \[ \int (c+d x) \text {erf}(a+b x) \, dx=\frac {e^{-(a+b x)^2} \left (4 b c-2 a d+2 b d x-e^{(a+b x)^2} \sqrt {\pi } \left (-4 a b c+d+2 a^2 d-4 b^2 c x-2 b^2 d x^2\right ) \text {erf}(a+b x)\right )}{4 b^2 \sqrt {\pi }} \]

[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])

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {erf}\left (b x +a \right ) \left (d a \left (b x +a \right )-c b \left (b x +a \right )-\frac {d \left (b x +a \right )^{2}}{2}\right )}{b}+\frac {-d \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{4}\right )+{\mathrm e}^{-\left (b x +a \right )^{2}} b c -{\mathrm e}^{-\left (b x +a \right )^{2}} a d}{\sqrt {\pi }\, b}}{b}\)	\(112\)
default	\(\frac {-\frac {\operatorname {erf}\left (b x +a \right ) \left (d a \left (b x +a \right )-c b \left (b x +a \right )-\frac {d \left (b x +a \right )^{2}}{2}\right )}{b}+\frac {-d \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{4}\right )+{\mathrm e}^{-\left (b x +a \right )^{2}} b c -{\mathrm e}^{-\left (b x +a \right )^{2}} a d}{\sqrt {\pi }\, b}}{b}\)	\(112\)
parallelrisch	\(\frac {2 d \,x^{2} \operatorname {erf}\left (b x +a \right ) \sqrt {\pi }\, b^{2}+4 x \,\operatorname {erf}\left (b x +a \right ) c \sqrt {\pi }\, b^{2}-2 \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right ) a^{2} d +4 \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right ) a b c +2 \,{\mathrm e}^{-\left (b x +a \right )^{2}} b d x -d \,\operatorname {erf}\left (b x +a \right ) \sqrt {\pi }-2 \,{\mathrm e}^{-\left (b x +a \right )^{2}} a d +4 \,{\mathrm e}^{-\left (b x +a \right )^{2}} b c}{4 \sqrt {\pi }\, b^{2}}\)	\(128\)
parts	\(\frac {\operatorname {erf}\left (b x +a \right ) d \,x^{2}}{2}+\operatorname {erf}\left (b x +a \right ) c x -\frac {b \left ({\mathrm e}^{-a^{2}} d \left (-\frac {x \,{\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \left (-\frac {{\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{2 b^{2}}\right )}{b}+\frac {\sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{4 b^{3}}\right )+2 \,{\mathrm e}^{-a^{2}} c \left (-\frac {{\mathrm e}^{-b^{2} x^{2}-2 a b x}}{2 b^{2}}-\frac {a \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{2 b^{2}}\right )\right )}{\sqrt {\pi }}\)	\(173\)

[In]

int((d*x+c)*erf(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.77 \[ \int (c+d x) \text {erf}(a+b x) \, dx=\frac {2 \, \sqrt {\pi } {\left (b d x + 2 \, b c - a d\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )} + {\left (2 \, \pi b^{2} d x^{2} + 4 \, \pi b^{2} c x + \pi {\left (4 \, a b c - {\left (2 \, a^{2} + 1\right )} d\right )}\right )} \operatorname {erf}\left (b x + a\right )}{4 \, \pi b^{2}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.51 \[ \int (c+d x) \text {erf}(a+b x) \, dx=\begin {cases} - \frac {a^{2} d \operatorname {erf}{\left (a + b x \right )}}{2 b^{2}} + \frac {a c \operatorname {erf}{\left (a + b x \right )}}{b} - \frac {a d e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{2 \sqrt {\pi } b^{2}} + c x \operatorname {erf}{\left (a + b x \right )} + \frac {d x^{2} \operatorname {erf}{\left (a + b x \right )}}{2} + \frac {c e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b} + \frac {d x e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{2 \sqrt {\pi } b} - \frac {d \operatorname {erf}{\left (a + b x \right )}}{4 b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \operatorname {erf}{\left (a \right )} & \text {otherwise} \end {cases} \]

[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))

Maxima [F]

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

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.26 \[ \int (c+d x) \text {erf}(a+b x) \, dx=\frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )} \operatorname {erf}\left (b x + a\right ) - \frac {4 \, \sqrt {\pi } {\left (\frac {\sqrt {\pi } a \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}\right )} c - \frac {\sqrt {\pi } {\left (\frac {\sqrt {\pi } {\left (2 \, a^{2} + 1\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} + \frac {2 \, {\left (b {\left (x + \frac {a}{b}\right )} - 2 \, a\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}\right )} d}{b}}{4 \, \pi } \]

[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

Mupad [B] (verification not implemented)

Time = 5.85 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.07 \[ \int (c+d x) \text {erf}(a+b x) \, dx=\mathrm {erf}\left (a+b\,x\right )\,\left (\frac {d\,x^2}{2}+c\,x\right )-\frac {\frac {{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (\frac {a\,d}{2}-b\,c\right )}{b^2}-\frac {d\,x\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}}{2\,b}}{\sqrt {\pi }}+\frac {\sqrt {\pi }\,\mathrm {erfi}\left (a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )\,\left (\frac {2\,d\,a^2+d}{2\,\sqrt {\pi }}-\frac {2\,a\,b\,c}{\sqrt {\pi }}\right )\,1{}\mathrm {i}}{2\,b^2} \]

[In]

int(erf(a + b*x)*(c + d*x),x)

[Out]

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