3.15 \(\int (c+d x)^3 \text {erf}(a+b x) \, dx\)

Optimal. Leaf size=289 \[ \frac {d^2 e^{-(a+b x)^2} (a+b x)^2 (b c-a d)}{\sqrt {\pi } b^4}+\frac {d^2 e^{-(a+b x)^2} (b c-a d)}{\sqrt {\pi } b^4}-\frac {(b c-a d)^4 \text {erf}(a+b x)}{4 b^4 d}-\frac {3 d (b c-a d)^2 \text {erf}(a+b x)}{4 b^4}+\frac {e^{-(a+b x)^2} (b c-a d)^3}{\sqrt {\pi } b^4}+\frac {3 d e^{-(a+b x)^2} (a+b x) (b c-a d)^2}{2 \sqrt {\pi } b^4}-\frac {3 d^3 \text {erf}(a+b x)}{16 b^4}+\frac {d^3 e^{-(a+b x)^2} (a+b x)^3}{4 \sqrt {\pi } b^4}+\frac {3 d^3 e^{-(a+b x)^2} (a+b x)}{8 \sqrt {\pi } b^4}+\frac {(c+d x)^4 \text {erf}(a+b x)}{4 d} \]

[Out]

-3/16*d^3*erf(b*x+a)/b^4-3/4*d*(-a*d+b*c)^2*erf(b*x+a)/b^4-1/4*(-a*d+b*c)^4*erf(b*x+a)/b^4/d+1/4*(d*x+c)^4*erf
(b*x+a)/d+d^2*(-a*d+b*c)/b^4/exp((b*x+a)^2)/Pi^(1/2)+(-a*d+b*c)^3/b^4/exp((b*x+a)^2)/Pi^(1/2)+3/8*d^3*(b*x+a)/
b^4/exp((b*x+a)^2)/Pi^(1/2)+3/2*d*(-a*d+b*c)^2*(b*x+a)/b^4/exp((b*x+a)^2)/Pi^(1/2)+d^2*(-a*d+b*c)*(b*x+a)^2/b^
4/exp((b*x+a)^2)/Pi^(1/2)+1/4*d^3*(b*x+a)^3/b^4/exp((b*x+a)^2)/Pi^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.32, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 12, 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} \[ \frac {d^2 e^{-(a+b x)^2} (a+b x)^2 (b c-a d)}{\sqrt {\pi } b^4}+\frac {d^2 e^{-(a+b x)^2} (b c-a d)}{\sqrt {\pi } b^4}-\frac {(b c-a d)^4 \text {Erf}(a+b x)}{4 b^4 d}-\frac {3 d (b c-a d)^2 \text {Erf}(a+b x)}{4 b^4}+\frac {e^{-(a+b x)^2} (b c-a d)^3}{\sqrt {\pi } b^4}+\frac {3 d e^{-(a+b x)^2} (a+b x) (b c-a d)^2}{2 \sqrt {\pi } b^4}-\frac {3 d^3 \text {Erf}(a+b x)}{16 b^4}+\frac {d^3 e^{-(a+b x)^2} (a+b x)^3}{4 \sqrt {\pi } b^4}+\frac {3 d^3 e^{-(a+b x)^2} (a+b x)}{8 \sqrt {\pi } b^4}+\frac {(c+d x)^4 \text {Erf}(a+b x)}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*Erf[a + b*x],x]

[Out]

(d^2*(b*c - a*d))/(b^4*E^(a + b*x)^2*Sqrt[Pi]) + (b*c - a*d)^3/(b^4*E^(a + b*x)^2*Sqrt[Pi]) + (3*d^3*(a + b*x)
)/(8*b^4*E^(a + b*x)^2*Sqrt[Pi]) + (3*d*(b*c - a*d)^2*(a + b*x))/(2*b^4*E^(a + b*x)^2*Sqrt[Pi]) + (d^2*(b*c -
a*d)*(a + b*x)^2)/(b^4*E^(a + b*x)^2*Sqrt[Pi]) + (d^3*(a + b*x)^3)/(4*b^4*E^(a + b*x)^2*Sqrt[Pi]) - (3*d^3*Erf
[a + b*x])/(16*b^4) - (3*d*(b*c - a*d)^2*Erf[a + b*x])/(4*b^4) - ((b*c - a*d)^4*Erf[a + b*x])/(4*b^4*d) + ((c
+ d*x)^4*Erf[a + b*x])/(4*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)^3 \text {erf}(a+b x) \, dx &=\frac {(c+d x)^4 \text {erf}(a+b x)}{4 d}-\frac {b \int e^{-(a+b x)^2} (c+d x)^4 \, dx}{2 d \sqrt {\pi }}\\ &=\frac {(c+d x)^4 \text {erf}(a+b x)}{4 d}-\frac {b \int \left (\frac {(b c-a d)^4 e^{-(a+b x)^2}}{b^4}+\frac {4 d (b c-a d)^3 e^{-(a+b x)^2} (a+b x)}{b^4}+\frac {6 d^2 (b c-a d)^2 e^{-(a+b x)^2} (a+b x)^2}{b^4}+\frac {4 d^3 (b c-a d) e^{-(a+b x)^2} (a+b x)^3}{b^4}+\frac {d^4 e^{-(a+b x)^2} (a+b x)^4}{b^4}\right ) \, dx}{2 d \sqrt {\pi }}\\ &=\frac {(c+d x)^4 \text {erf}(a+b x)}{4 d}-\frac {d^3 \int e^{-(a+b x)^2} (a+b x)^4 \, dx}{2 b^3 \sqrt {\pi }}-\frac {\left (2 d^2 (b c-a d)\right ) \int e^{-(a+b x)^2} (a+b x)^3 \, dx}{b^3 \sqrt {\pi }}-\frac {\left (3 d (b c-a d)^2\right ) \int e^{-(a+b x)^2} (a+b x)^2 \, dx}{b^3 \sqrt {\pi }}-\frac {\left (2 (b c-a d)^3\right ) \int e^{-(a+b x)^2} (a+b x) \, dx}{b^3 \sqrt {\pi }}-\frac {(b c-a d)^4 \int e^{-(a+b x)^2} \, dx}{2 b^3 d \sqrt {\pi }}\\ &=\frac {(b c-a d)^3 e^{-(a+b x)^2}}{b^4 \sqrt {\pi }}+\frac {3 d (b c-a d)^2 e^{-(a+b x)^2} (a+b x)}{2 b^4 \sqrt {\pi }}+\frac {d^2 (b c-a d) e^{-(a+b x)^2} (a+b x)^2}{b^4 \sqrt {\pi }}+\frac {d^3 e^{-(a+b x)^2} (a+b x)^3}{4 b^4 \sqrt {\pi }}-\frac {(b c-a d)^4 \text {erf}(a+b x)}{4 b^4 d}+\frac {(c+d x)^4 \text {erf}(a+b x)}{4 d}-\frac {\left (3 d^3\right ) \int e^{-(a+b x)^2} (a+b x)^2 \, dx}{4 b^3 \sqrt {\pi }}-\frac {\left (2 d^2 (b c-a d)\right ) \int e^{-(a+b x)^2} (a+b x) \, dx}{b^3 \sqrt {\pi }}-\frac {\left (3 d (b c-a d)^2\right ) \int e^{-(a+b x)^2} \, dx}{2 b^3 \sqrt {\pi }}\\ &=\frac {d^2 (b c-a d) e^{-(a+b x)^2}}{b^4 \sqrt {\pi }}+\frac {(b c-a d)^3 e^{-(a+b x)^2}}{b^4 \sqrt {\pi }}+\frac {3 d^3 e^{-(a+b x)^2} (a+b x)}{8 b^4 \sqrt {\pi }}+\frac {3 d (b c-a d)^2 e^{-(a+b x)^2} (a+b x)}{2 b^4 \sqrt {\pi }}+\frac {d^2 (b c-a d) e^{-(a+b x)^2} (a+b x)^2}{b^4 \sqrt {\pi }}+\frac {d^3 e^{-(a+b x)^2} (a+b x)^3}{4 b^4 \sqrt {\pi }}-\frac {3 d (b c-a d)^2 \text {erf}(a+b x)}{4 b^4}-\frac {(b c-a d)^4 \text {erf}(a+b x)}{4 b^4 d}+\frac {(c+d x)^4 \text {erf}(a+b x)}{4 d}-\frac {\left (3 d^3\right ) \int e^{-(a+b x)^2} \, dx}{8 b^3 \sqrt {\pi }}\\ &=\frac {d^2 (b c-a d) e^{-(a+b x)^2}}{b^4 \sqrt {\pi }}+\frac {(b c-a d)^3 e^{-(a+b x)^2}}{b^4 \sqrt {\pi }}+\frac {3 d^3 e^{-(a+b x)^2} (a+b x)}{8 b^4 \sqrt {\pi }}+\frac {3 d (b c-a d)^2 e^{-(a+b x)^2} (a+b x)}{2 b^4 \sqrt {\pi }}+\frac {d^2 (b c-a d) e^{-(a+b x)^2} (a+b x)^2}{b^4 \sqrt {\pi }}+\frac {d^3 e^{-(a+b x)^2} (a+b x)^3}{4 b^4 \sqrt {\pi }}-\frac {3 d^3 \text {erf}(a+b x)}{16 b^4}-\frac {3 d (b c-a d)^2 \text {erf}(a+b x)}{4 b^4}-\frac {(b c-a d)^4 \text {erf}(a+b x)}{4 b^4 d}+\frac {(c+d x)^4 \text {erf}(a+b x)}{4 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.31, size = 248, normalized size = 0.86 \[ \frac {e^{-(a+b x)^2} \left (2 b d^2 \left (8 \left (a^2+1\right ) c+\left (2 a^2+3\right ) d x\right )-2 a \left (2 a^2+5\right ) d^3-\sqrt {\pi } e^{(a+b x)^2} \text {erf}(a+b x) \left (4 a^4 d^3-16 a^3 b c d^2+12 a^2 \left (2 b^2 c^2 d+d^3\right )-8 a \left (2 b^3 c^3+3 b c d^2\right )-4 b^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+12 b^2 c^2 d+3 d^3\right )-4 a b^2 d \left (6 c^2+4 c d x+d^2 x^2\right )+4 b^3 \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )\right )}{16 \sqrt {\pi } b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3*Erf[a + b*x],x]

[Out]

(-2*a*(5 + 2*a^2)*d^3 + 2*b*d^2*(8*(1 + a^2)*c + (3 + 2*a^2)*d*x) - 4*a*b^2*d*(6*c^2 + 4*c*d*x + d^2*x^2) + 4*
b^3*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) - E^(a + b*x)^2*Sqrt[Pi]*(12*b^2*c^2*d - 16*a^3*b*c*d^2 + 3*d^
3 + 4*a^4*d^3 - 8*a*(2*b^3*c^3 + 3*b*c*d^2) + 12*a^2*(2*b^2*c^2*d + d^3) - 4*b^4*x*(4*c^3 + 6*c^2*d*x + 4*c*d^
2*x^2 + d^3*x^3))*Erf[a + b*x])/(16*b^4*E^(a + b*x)^2*Sqrt[Pi])

________________________________________________________________________________________

fricas [A]  time = 0.95, size = 265, normalized size = 0.92 \[ \frac {2 \, \sqrt {\pi } {\left (2 \, b^{3} d^{3} x^{3} + 8 \, b^{3} c^{3} - 12 \, a b^{2} c^{2} d + 8 \, {\left (a^{2} + 1\right )} b c d^{2} - {\left (2 \, a^{3} + 5 \, a\right )} d^{3} + 2 \, {\left (4 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + {\left (12 \, b^{3} c^{2} d - 8 \, a b^{2} c d^{2} + {\left (2 \, a^{2} + 3\right )} b d^{3}\right )} x\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )} + {\left (4 \, \pi b^{4} d^{3} x^{4} + 16 \, \pi b^{4} c d^{2} x^{3} + 24 \, \pi b^{4} c^{2} d x^{2} + 16 \, \pi b^{4} c^{3} x + \pi {\left (16 \, a b^{3} c^{3} - 12 \, {\left (2 \, a^{2} + 1\right )} b^{2} c^{2} d + 8 \, {\left (2 \, a^{3} + 3 \, a\right )} b c d^{2} - {\left (4 \, a^{4} + 12 \, a^{2} + 3\right )} d^{3}\right )}\right )} \operatorname {erf}\left (b x + a\right )}{16 \, \pi b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*erf(b*x+a),x, algorithm="fricas")

[Out]

1/16*(2*sqrt(pi)*(2*b^3*d^3*x^3 + 8*b^3*c^3 - 12*a*b^2*c^2*d + 8*(a^2 + 1)*b*c*d^2 - (2*a^3 + 5*a)*d^3 + 2*(4*
b^3*c*d^2 - a*b^2*d^3)*x^2 + (12*b^3*c^2*d - 8*a*b^2*c*d^2 + (2*a^2 + 3)*b*d^3)*x)*e^(-b^2*x^2 - 2*a*b*x - a^2
) + (4*pi*b^4*d^3*x^4 + 16*pi*b^4*c*d^2*x^3 + 24*pi*b^4*c^2*d*x^2 + 16*pi*b^4*c^3*x + pi*(16*a*b^3*c^3 - 12*(2
*a^2 + 1)*b^2*c^2*d + 8*(2*a^3 + 3*a)*b*c*d^2 - (4*a^4 + 12*a^2 + 3)*d^3))*erf(b*x + a))/(pi*b^4)

________________________________________________________________________________________

giac [A]  time = 0.33, size = 400, normalized size = 1.38 \[ \frac {{\left (d x + c\right )}^{4} \operatorname {erf}\left (b x + a\right )}{4 \, d} + \frac {4 \, \pi c^{4} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right ) - 16 \, \sqrt {\pi } {\left (\frac {\sqrt {\pi } a \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}\right )} c^{3} d + \frac {12 \, \sqrt {\pi } {\left (\frac {\sqrt {\pi } {\left (2 \, a^{2} + 1\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} + \frac {2 \, {\left (b {\left (x + \frac {a}{b}\right )} - 2 \, a\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}\right )} c^{2} d^{2}}{b} - \frac {8 \, \sqrt {\pi } {\left (\frac {\sqrt {\pi } {\left (2 \, a^{3} + 3 \, a\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {2 \, {\left (b^{2} {\left (x + \frac {a}{b}\right )}^{2} - 3 \, a b {\left (x + \frac {a}{b}\right )} + 3 \, a^{2} + 1\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}\right )} c d^{3}}{b^{2}} + \frac {\sqrt {\pi } {\left (\frac {\sqrt {\pi } {\left (4 \, a^{4} + 12 \, a^{2} + 3\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} + \frac {2 \, {\left (2 \, b^{3} {\left (x + \frac {a}{b}\right )}^{3} - 8 \, a b^{2} {\left (x + \frac {a}{b}\right )}^{2} + 12 \, a^{2} b {\left (x + \frac {a}{b}\right )} - 8 \, a^{3} + 3 \, b {\left (x + \frac {a}{b}\right )} - 8 \, a\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}\right )} d^{4}}{b^{3}}}{16 \, \pi d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*erf(b*x+a),x, algorithm="giac")

[Out]

1/4*(d*x + c)^4*erf(b*x + a)/d + 1/16*(4*pi*c^4*erf(-b*(x + a/b)) - 16*sqrt(pi)*(sqrt(pi)*a*erf(-b*(x + a/b))/
b - e^(-b^2*x^2 - 2*a*b*x - a^2)/b)*c^3*d + 12*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^2*d^2/b - 8*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)*c*d^3/b^2 + sqrt(pi)*(sqr
t(pi)*(4*a^4 + 12*a^2 + 3)*erf(-b*(x + a/b))/b + 2*(2*b^3*(x + a/b)^3 - 8*a*b^2*(x + a/b)^2 + 12*a^2*b*(x + a/
b) - 8*a^3 + 3*b*(x + a/b) - 8*a)*e^(-b^2*x^2 - 2*a*b*x - a^2)/b)*d^4/b^3)/(pi*d)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 466, normalized size = 1.61 \[ \frac {\frac {\erf \left (b x +a \right ) \left (\left (b x +a \right ) d -a d +b c \right )^{4}}{4 b^{3} d}-\frac {d^{4} \left (-\frac {\left (b x +a \right )^{3} {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}-\frac {3 \left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{4}+\frac {3 \sqrt {\pi }\, \erf \left (b x +a \right )}{8}\right )+\frac {a^{4} d^{4} \sqrt {\pi }\, \erf \left (b x +a \right )}{2}+\frac {b^{4} c^{4} \sqrt {\pi }\, \erf \left (b x +a \right )}{2}+2 a^{3} d^{4} {\mathrm e}^{-\left (b x +a \right )^{2}}+6 a^{2} d^{4} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \erf \left (b x +a \right )}{4}\right )-4 a \,d^{4} \left (-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}} \left (b x +a \right )^{2}}{2}-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{2}\right )-2 b^{3} c^{3} d \,{\mathrm e}^{-\left (b x +a \right )^{2}}+6 b^{2} c^{2} d^{2} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \erf \left (b x +a \right )}{4}\right )+4 b c \,d^{3} \left (-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}} \left (b x +a \right )^{2}}{2}-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{2}\right )-2 a \,b^{3} c^{3} d \sqrt {\pi }\, \erf \left (b x +a \right )+3 a^{2} b^{2} c^{2} d^{2} \sqrt {\pi }\, \erf \left (b x +a \right )-2 a^{3} b c \,d^{3} \sqrt {\pi }\, \erf \left (b x +a \right )+6 a \,b^{2} c^{2} d^{2} {\mathrm e}^{-\left (b x +a \right )^{2}}-6 a^{2} b c \,d^{3} {\mathrm e}^{-\left (b x +a \right )^{2}}-12 a b c \,d^{3} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \erf \left (b x +a \right )}{4}\right )}{2 \sqrt {\pi }\, b^{3} d}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3*erf(b*x+a),x)

[Out]

1/b*(1/4*erf(b*x+a)*((b*x+a)*d-a*d+b*c)^4/b^3/d-1/2/Pi^(1/2)/b^3/d*(d^4*(-1/2*(b*x+a)^3/exp((b*x+a)^2)-3/4*(b*
x+a)/exp((b*x+a)^2)+3/8*Pi^(1/2)*erf(b*x+a))+1/2*a^4*d^4*Pi^(1/2)*erf(b*x+a)+1/2*b^4*c^4*Pi^(1/2)*erf(b*x+a)+2
*a^3*d^4/exp((b*x+a)^2)+6*a^2*d^4*(-1/2*(b*x+a)/exp((b*x+a)^2)+1/4*Pi^(1/2)*erf(b*x+a))-4*a*d^4*(-1/2/exp((b*x
+a)^2)*(b*x+a)^2-1/2/exp((b*x+a)^2))-2*b^3*c^3*d/exp((b*x+a)^2)+6*b^2*c^2*d^2*(-1/2*(b*x+a)/exp((b*x+a)^2)+1/4
*Pi^(1/2)*erf(b*x+a))+4*b*c*d^3*(-1/2/exp((b*x+a)^2)*(b*x+a)^2-1/2/exp((b*x+a)^2))-2*a*b^3*c^3*d*Pi^(1/2)*erf(
b*x+a)+3*a^2*b^2*c^2*d^2*Pi^(1/2)*erf(b*x+a)-2*a^3*b*c*d^3*Pi^(1/2)*erf(b*x+a)+6*a*b^2*c^2*d^2/exp((b*x+a)^2)-
6*a^2*b*c*d^3/exp((b*x+a)^2)-12*a*b*c*d^3*(-1/2*(b*x+a)/exp((b*x+a)^2)+1/4*Pi^(1/2)*erf(b*x+a))))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, {\left (d^{3} x^{4} + 4 \, c d^{2} x^{3} + 6 \, c^{2} d x^{2} + 4 \, c^{3} x\right )} \operatorname {erf}\left (b x + a\right ) - \frac {-\frac {2 \, {\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^{3}}{\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^{2} d}{\sqrt {-b^{2}}} - \frac {2 \, {\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 c d^{2}}{\sqrt {-b^{2}}} - \frac {{\left (\frac {\sqrt {\pi } {\left (b^{2} x + a b\right )} a^{4} b^{5} {\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 {9}{2}}} - \frac {6 \, {\left (b^{2} x + a b\right )}^{3} a^{2} b^{5} \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 {9}{2}}} + \frac {4 \, a^{3} b^{5} e^{\left (-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}}{\left (-b^{2}\right )^{\frac {9}{2}}} - \frac {{\left (b^{2} x + a b\right )}^{5} b^{5} \Gamma \left (\frac {5}{2}, \frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{{\left ({\left (b^{2} x + a b\right )}^{2}\right )}^{\frac {5}{2}} \left (-b^{2}\right )^{\frac {9}{2}}} + \frac {4 \, a b^{5} \Gamma \left (2, \frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{\left (-b^{2}\right )^{\frac {9}{2}}}\right )} b d^{3}}{2 \, \sqrt {-b^{2}}}}{2 \, \sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*erf(b*x+a),x, algorithm="maxima")

[Out]

1/4*(d^3*x^4 + 4*c*d^2*x^3 + 6*c^2*d*x^2 + 4*c^3*x)*erf(b*x + a) - 1/2*integrate((b*d^3*x^4 + 4*b*c*d^2*x^3 +
6*b*c^2*d*x^2 + 4*b*c^3*x)*e^(-b^2*x^2 - 2*a*b*x - a^2), x)/sqrt(pi)

________________________________________________________________________________________

mupad [B]  time = 1.09, size = 337, normalized size = 1.17 \[ \mathrm {erf}\left (a+b\,x\right )\,\left (c^3\,x+\frac {3\,c^2\,d\,x^2}{2}+c\,d^2\,x^3+\frac {d^3\,x^4}{4}\right )-\frac {\frac {{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (\frac {5\,a\,d^3}{4}+\frac {a^3\,d^3}{2}-2\,b^3\,c^3-b\,\left (2\,c\,a^2\,d^2+2\,c\,d^2\right )+3\,a\,b^2\,c^2\,d\right )}{b^4}-\frac {x\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (2\,a^2\,d^3-8\,a\,b\,c\,d^2+12\,b^2\,c^2\,d+3\,d^3\right )}{4\,b^3}-\frac {d^3\,x^3\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}}{2\,b}+\frac {x^2\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (a\,d^3-4\,b\,c\,d^2\right )}{2\,b^2}}{2\,\sqrt {\pi }}+\frac {\mathrm {erfi}\left (a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )\,\left (4\,a^4\,d^3-16\,a^3\,b\,c\,d^2+24\,a^2\,b^2\,c^2\,d+12\,a^2\,d^3-16\,a\,b^3\,c^3-24\,a\,b\,c\,d^2+12\,b^2\,c^2\,d+3\,d^3\right )\,1{}\mathrm {i}}{16\,b^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(a + b*x)*(c + d*x)^3,x)

[Out]

erf(a + b*x)*(c^3*x + (d^3*x^4)/4 + (3*c^2*d*x^2)/2 + c*d^2*x^3) - ((exp(- a^2 - b^2*x^2 - 2*a*b*x)*((5*a*d^3)
/4 + (a^3*d^3)/2 - 2*b^3*c^3 - b*(2*c*d^2 + 2*a^2*c*d^2) + 3*a*b^2*c^2*d))/b^4 - (x*exp(- a^2 - b^2*x^2 - 2*a*
b*x)*(3*d^3 + 2*a^2*d^3 + 12*b^2*c^2*d - 8*a*b*c*d^2))/(4*b^3) - (d^3*x^3*exp(- a^2 - b^2*x^2 - 2*a*b*x))/(2*b
) + (x^2*exp(- a^2 - b^2*x^2 - 2*a*b*x)*(a*d^3 - 4*b*c*d^2))/(2*b^2))/(2*pi^(1/2)) + (erfi(a*1i + b*x*1i)*(3*d
^3 + 12*a^2*d^3 + 4*a^4*d^3 - 16*a*b^3*c^3 + 12*b^2*c^2*d + 24*a^2*b^2*c^2*d - 24*a*b*c*d^2 - 16*a^3*b*c*d^2)*
1i)/(16*b^4)

________________________________________________________________________________________

sympy [A]  time = 9.72, size = 746, normalized size = 2.58 \[ \begin {cases} - \frac {a^{4} d^{3} \operatorname {erf}{\left (a + b x \right )}}{4 b^{4}} + \frac {a^{3} c d^{2} \operatorname {erf}{\left (a + b x \right )}}{b^{3}} - \frac {a^{3} d^{3} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{4 \sqrt {\pi } b^{4}} - \frac {3 a^{2} c^{2} d \operatorname {erf}{\left (a + b x \right )}}{2 b^{2}} + \frac {a^{2} c d^{2} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b^{3}} + \frac {a^{2} d^{3} x e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{4 \sqrt {\pi } b^{3}} - \frac {3 a^{2} d^{3} \operatorname {erf}{\left (a + b x \right )}}{4 b^{4}} + \frac {a c^{3} \operatorname {erf}{\left (a + b x \right )}}{b} - \frac {3 a c^{2} d e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{2 \sqrt {\pi } b^{2}} - \frac {a c d^{2} x e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b^{2}} - \frac {a d^{3} x^{2} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{4 \sqrt {\pi } b^{2}} + \frac {3 a c d^{2} \operatorname {erf}{\left (a + b x \right )}}{2 b^{3}} - \frac {5 a d^{3} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{8 \sqrt {\pi } b^{4}} + c^{3} x \operatorname {erf}{\left (a + b x \right )} + \frac {3 c^{2} d x^{2} \operatorname {erf}{\left (a + b x \right )}}{2} + c d^{2} x^{3} \operatorname {erf}{\left (a + b x \right )} + \frac {d^{3} x^{4} \operatorname {erf}{\left (a + b x \right )}}{4} + \frac {c^{3} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b} + \frac {3 c^{2} d x e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{2 \sqrt {\pi } b} + \frac {c d^{2} x^{2} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b} + \frac {d^{3} x^{3} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{4 \sqrt {\pi } b} - \frac {3 c^{2} d \operatorname {erf}{\left (a + b x \right )}}{4 b^{2}} + \frac {c d^{2} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b^{3}} + \frac {3 d^{3} x e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{8 \sqrt {\pi } b^{3}} - \frac {3 d^{3} \operatorname {erf}{\left (a + b x \right )}}{16 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \operatorname {erf}{\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3*erf(b*x+a),x)

[Out]

Piecewise((-a**4*d**3*erf(a + b*x)/(4*b**4) + a**3*c*d**2*erf(a + b*x)/b**3 - a**3*d**3*exp(-a**2)*exp(-b**2*x
**2)*exp(-2*a*b*x)/(4*sqrt(pi)*b**4) - 3*a**2*c**2*d*erf(a + b*x)/(2*b**2) + a**2*c*d**2*exp(-a**2)*exp(-b**2*
x**2)*exp(-2*a*b*x)/(sqrt(pi)*b**3) + a**2*d**3*x*exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*x)/(4*sqrt(pi)*b**3) -
3*a**2*d**3*erf(a + b*x)/(4*b**4) + a*c**3*erf(a + b*x)/b - 3*a*c**2*d*exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*
x)/(2*sqrt(pi)*b**2) - a*c*d**2*x*exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*x)/(sqrt(pi)*b**2) - a*d**3*x**2*exp(-
a**2)*exp(-b**2*x**2)*exp(-2*a*b*x)/(4*sqrt(pi)*b**2) + 3*a*c*d**2*erf(a + b*x)/(2*b**3) - 5*a*d**3*exp(-a**2)
*exp(-b**2*x**2)*exp(-2*a*b*x)/(8*sqrt(pi)*b**4) + c**3*x*erf(a + b*x) + 3*c**2*d*x**2*erf(a + b*x)/2 + c*d**2
*x**3*erf(a + b*x) + d**3*x**4*erf(a + b*x)/4 + c**3*exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*x)/(sqrt(pi)*b) + 3
*c**2*d*x*exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*x)/(2*sqrt(pi)*b) + c*d**2*x**2*exp(-a**2)*exp(-b**2*x**2)*exp
(-2*a*b*x)/(sqrt(pi)*b) + d**3*x**3*exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*x)/(4*sqrt(pi)*b) - 3*c**2*d*erf(a +
b*x)/(4*b**2) + c*d**2*exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*x)/(sqrt(pi)*b**3) + 3*d**3*x*exp(-a**2)*exp(-b*
*2*x**2)*exp(-2*a*b*x)/(8*sqrt(pi)*b**3) - 3*d**3*erf(a + b*x)/(16*b**4), Ne(b, 0)), ((c**3*x + 3*c**2*d*x**2/
2 + c*d**2*x**3 + d**3*x**4/4)*erf(a), True))

________________________________________________________________________________________