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

Optimal. Leaf size=192 \[ -\frac {(b c-a d)^3 \text {erf}(a+b x)}{3 b^3 d}-\frac {d (b c-a d) \text {erf}(a+b x)}{2 b^3}+\frac {e^{-(a+b x)^2} (b c-a d)^2}{\sqrt {\pi } b^3}+\frac {d e^{-(a+b x)^2} (a+b x) (b c-a d)}{\sqrt {\pi } b^3}+\frac {d^2 e^{-(a+b x)^2} (a+b x)^2}{3 \sqrt {\pi } b^3}+\frac {d^2 e^{-(a+b x)^2}}{3 \sqrt {\pi } b^3}+\frac {(c+d x)^3 \text {erf}(a+b x)}{3 d} \]

[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*erf(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]  time = 0.20, antiderivative size = 192, normalized size of antiderivative = 1.00, 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 = {6361, 2226, 2205, 2209, 2212} \[ -\frac {(b c-a d)^3 \text {Erf}(a+b x)}{3 b^3 d}-\frac {d (b c-a d) \text {Erf}(a+b x)}{2 b^3}+\frac {e^{-(a+b x)^2} (b c-a d)^2}{\sqrt {\pi } b^3}+\frac {d e^{-(a+b x)^2} (a+b x) (b c-a d)}{\sqrt {\pi } b^3}+\frac {d^2 e^{-(a+b x)^2} (a+b x)^2}{3 \sqrt {\pi } b^3}+\frac {d^2 e^{-(a+b x)^2}}{3 \sqrt {\pi } b^3}+\frac {(c+d x)^3 \text {Erf}(a+b x)}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2*Erf[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*Erf[a + b*x])/(3*d)

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])

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 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]

Rubi steps

\begin {align*} \int (c+d x)^2 \text {erf}(a+b x) \, dx &=\frac {(c+d x)^3 \text {erf}(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 {erf}(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 {erf}(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 {erf}(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 {erf}(a+b x)}{3 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.23, size = 138, normalized size = 0.72 \[ \frac {\frac {2 e^{-(a+b x)^2} \left (\left (a^2+1\right ) d^2-a b d (3 c+d x)+b^2 \left (3 c^2+3 c d x+d^2 x^2\right )\right )}{\sqrt {\pi }}+\text {erf}(a+b x) \left (2 a^3 d^2-6 a^2 b c d+3 a \left (2 b^2 c^2+d^2\right )+2 b^3 x \left (3 c^2+3 c d x+d^2 x^2\right )-3 b c d\right )}{6 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2*Erf[a + b*x],x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.80, size = 163, normalized size = 0.85 \[ \frac {2 \, \sqrt {\pi } {\left (b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} - 3 \, a b c d + {\left (a^{2} + 1\right )} d^{2} + {\left (3 \, b^{2} c d - a b d^{2}\right )} x\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )} + {\left (2 \, \pi b^{3} d^{2} x^{3} + 6 \, \pi b^{3} c d x^{2} + 6 \, \pi b^{3} c^{2} x + \pi {\left (6 \, a b^{2} c^{2} - 3 \, {\left (2 \, a^{2} + 1\right )} b c d + {\left (2 \, a^{3} + 3 \, a\right )} d^{2}\right )}\right )} \operatorname {erf}\left (b x + a\right )}{6 \, \pi b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))*erf(b*x + a))/(pi*b^3)

________________________________________________________________________________________

giac [A]  time = 0.43, size = 270, normalized size = 1.41 \[ \frac {{\left (d x + c\right )}^{3} \operatorname {erf}\left (b x + a\right )}{3 \, d} + \frac {2 \, \pi c^{3} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right ) - 6 \, \sqrt {\pi } {\left (\frac {\sqrt {\pi } a \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}\right )} c^{2} d + \frac {3 \, \sqrt {\pi } {\left (\frac {\sqrt {\pi } {\left (2 \, a^{2} + 1\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} + \frac {2 \, {\left (b {\left (x + \frac {a}{b}\right )} - 2 \, a\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}\right )} c d^{2}}{b} - \frac {\sqrt {\pi } {\left (\frac {\sqrt {\pi } {\left (2 \, a^{3} + 3 \, a\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {2 \, {\left (b^{2} {\left (x + \frac {a}{b}\right )}^{2} - 3 \, a b {\left (x + \frac {a}{b}\right )} + 3 \, a^{2} + 1\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}\right )} d^{3}}{b^{2}}}{6 \, \pi d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 283, normalized size = 1.47 \[ \frac {\frac {\erf \left (b x +a \right ) \left (\left (b x +a \right ) d -a d +b c \right )^{3}}{3 b^{2} d}-\frac {2 \left (d^{3} \left (-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}} \left (b x +a \right )^{2}}{2}-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{2}\right )+\frac {b^{3} c^{3} \sqrt {\pi }\, \erf \left (b x +a \right )}{2}-\frac {a^{3} d^{3} \sqrt {\pi }\, \erf \left (b x +a \right )}{2}-\frac {3 a^{2} d^{3} {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}-3 a \,d^{3} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \erf \left (b x +a \right )}{4}\right )-\frac {3 b^{2} c^{2} d \,{\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+3 b c \,d^{2} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \erf \left (b x +a \right )}{4}\right )-\frac {3 a \,b^{2} c^{2} d \sqrt {\pi }\, \erf \left (b x +a \right )}{2}+\frac {3 a^{2} b c \,d^{2} \sqrt {\pi }\, \erf \left (b x +a \right )}{2}+3 a b c \,d^{2} {\mathrm e}^{-\left (b x +a \right )^{2}}\right )}{3 \sqrt {\pi }\, b^{2} d}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, {\left (d^{2} x^{3} + 3 \, c d x^{2} + 3 \, c^{2} x\right )} \operatorname {erf}\left (b x + a\right ) - \frac {-\frac {3 \, {\left (\frac {\sqrt {\pi } {\left (b^{2} x + a b\right )} a b^{2} {\left (\operatorname {erf}\left (\frac {\sqrt {{\left (b^{2} x + a b\right )}^{2}}}{b}\right ) - 1\right )}}{\sqrt {{\left (b^{2} x + a b\right )}^{2}} \left (-b^{2}\right )^{\frac {3}{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^{2}}{\sqrt {-b^{2}}} - \frac {3 \, {\left (\frac {\sqrt {\pi } {\left (b^{2} x + a b\right )} a^{2} b^{3} {\left (\operatorname {erf}\left (\frac {\sqrt {{\left (b^{2} x + a b\right )}^{2}}}{b}\right ) - 1\right )}}{\sqrt {{\left (b^{2} x + a b\right )}^{2}} \left (-b^{2}\right )^{\frac {5}{2}}} - \frac {{\left (b^{2} x + a b\right )}^{3} b^{3} \Gamma \left (\frac {3}{2}, \frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{{\left ({\left (b^{2} x + a b\right )}^{2}\right )}^{\frac {3}{2}} \left (-b^{2}\right )^{\frac {5}{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}}}\right )} b c d}{\sqrt {-b^{2}}} - \frac {{\left (\frac {\sqrt {\pi } {\left (b^{2} x + a b\right )} a^{3} b^{4} {\left (\operatorname {erf}\left (\frac {\sqrt {{\left (b^{2} x + a b\right )}^{2}}}{b}\right ) - 1\right )}}{\sqrt {{\left (b^{2} x + a b\right )}^{2}} \left (-b^{2}\right )^{\frac {7}{2}}} - \frac {3 \, {\left (b^{2} x + a b\right )}^{3} a b^{4} \Gamma \left (\frac {3}{2}, \frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{{\left ({\left (b^{2} x + a b\right )}^{2}\right )}^{\frac {3}{2}} \left (-b^{2}\right )^{\frac {7}{2}}} + \frac {3 \, a^{2} b^{4} e^{\left (-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}}{\left (-b^{2}\right )^{\frac {7}{2}}} + \frac {b^{4} \Gamma \left (2, \frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{\left (-b^{2}\right )^{\frac {7}{2}}}\right )} b d^{2}}{\sqrt {-b^{2}}}}{3 \, \sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(d^2*x^3 + 3*c*d*x^2 + 3*c^2*x)*erf(b*x + a) - 1/3*integrate(2*(b*d^2*x^3 + 3*b*c*d*x^2 + 3*b*c^2*x)*e^(-b
^2*x^2 - 2*a*b*x - a^2), x)/sqrt(pi)

________________________________________________________________________________________

mupad [B]  time = 0.72, size = 204, normalized size = 1.06 \[ \frac {\frac {{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (a^2\,d^2-3\,a\,b\,c\,d+3\,b^2\,c^2+d^2\right )}{b^3}+\frac {d^2\,x^2\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}}{b}-\frac {x\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (a\,d^2-3\,b\,c\,d\right )}{b^2}}{3\,\sqrt {\pi }}+\mathrm {erf}\left (a+b\,x\right )\,\left (c^2\,x+c\,d\,x^2+\frac {d^2\,x^3}{3}\right )-\frac {\mathrm {erfi}\left (a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )\,\left (2\,a^3\,d^2-6\,a^2\,b\,c\,d+6\,a\,b^2\,c^2+3\,a\,d^2-3\,b\,c\,d\right )\,1{}\mathrm {i}}{6\,b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(a + b*x)*(c + d*x)^2,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 3.94, size = 398, normalized size = 2.07 \[ \begin {cases} \frac {a^{3} d^{2} \operatorname {erf}{\left (a + b x \right )}}{3 b^{3}} - \frac {a^{2} c d \operatorname {erf}{\left (a + b x \right )}}{b^{2}} + \frac {a^{2} d^{2} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{3 \sqrt {\pi } b^{3}} + \frac {a c^{2} \operatorname {erf}{\left (a + b x \right )}}{b} - \frac {a c d e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b^{2}} - \frac {a d^{2} x e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{3 \sqrt {\pi } b^{2}} + \frac {a d^{2} \operatorname {erf}{\left (a + b x \right )}}{2 b^{3}} + c^{2} x \operatorname {erf}{\left (a + b x \right )} + c d x^{2} \operatorname {erf}{\left (a + b x \right )} + \frac {d^{2} x^{3} \operatorname {erf}{\left (a + b x \right )}}{3} + \frac {c^{2} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b} + \frac {c d x e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b} + \frac {d^{2} x^{2} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{3 \sqrt {\pi } b} - \frac {c d \operatorname {erf}{\left (a + b x \right )}}{2 b^{2}} + \frac {d^{2} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{3 \sqrt {\pi } b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \operatorname {erf}{\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________