3.2.18 \(\int (c+d x)^3 \text {Erfc}(a+b x) \, dx\) [118]

Optimal. Leaf size=292 \[ -\frac {d^2 (b c-a d) e^{-(a+b x)^2}}{b^4 \sqrt {\pi }}-\frac {(b c-a d)^3 e^{-(a+b x)^2}}{b^4 \sqrt {\pi }}-\frac {3 d^3 e^{-(a+b x)^2} (a+b x)}{8 b^4 \sqrt {\pi }}-\frac {3 d (b c-a d)^2 e^{-(a+b x)^2} (a+b x)}{2 b^4 \sqrt {\pi }}-\frac {d^2 (b c-a d) e^{-(a+b x)^2} (a+b x)^2}{b^4 \sqrt {\pi }}-\frac {d^3 e^{-(a+b x)^2} (a+b x)^3}{4 b^4 \sqrt {\pi }}+\frac {3 d^3 \text {Erf}(a+b x)}{16 b^4}+\frac {3 d (b c-a d)^2 \text {Erf}(a+b x)}{4 b^4}+\frac {(b c-a d)^4 \text {Erf}(a+b x)}{4 b^4 d}+\frac {(c+d x)^4 \text {Erfc}(a+b x)}{4 d} \]

[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*erfc
(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)

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Rubi [A]
time = 0.20, antiderivative size = 292, 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 = {6497, 2258, 2236, 2240, 2243} \begin {gather*} -\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 {Erfc}(a+b x)}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*Erfc[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*Erfc[a + b*x])/(4*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 6497

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]

Rubi steps

\begin {align*} \int (c+d x)^3 \text {erfc}(a+b x) \, dx &=\frac {(c+d x)^4 \text {erfc}(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 {erfc}(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 {erfc}(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 {erfc}(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 {erfc}(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 {erfc}(a+b x)}{4 d}\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 268, normalized size = 0.92 \begin {gather*} \frac {e^{-(a+b x)^2} \left (2 a \left (5+2 a^2\right ) d^3-2 b d^2 \left (8 \left (1+a^2\right ) c+\left (3+2 a^2\right ) d x\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 )+\left (-16 a^3 b c d^2+4 a^4 d^3-8 a \left (2 b^3 c^3+3 b c d^2\right )+12 a^2 \left (2 b^2 c^2 d+d^3\right )+3 \left (4 b^2 c^2 d+d^3\right )\right ) e^{(a+b x)^2} \sqrt {\pi } \text {Erf}(a+b x)+4 b^4 e^{(a+b x)^2} \sqrt {\pi } x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \text {Erfc}(a+b x)\right )}{16 b^4 \sqrt {\pi }} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3*Erfc[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) + (-16*a^3*b*c*d^2 + 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) + 3*(4*b^2*c^2*d + d^3))*E^(a + b*x)^2*Sqrt[Pi]*Erf[a + b*x] + 4*b^4*E^(a + b*x)^2*
Sqrt[Pi]*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Erfc[a + b*x])/(16*b^4*E^(a + b*x)^2*Sqrt[Pi])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(728\) vs. \(2(260)=520\).
time = 0.44, size = 729, normalized size = 2.50 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3*erfc(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.37, size = 314, normalized size = 1.08 \begin {gather*} \frac {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 - 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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(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 - 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)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 746 vs. \(2 (258) = 516\).
time = 2.34, size = 746, normalized size = 2.55 \begin {gather*} \begin {cases} - \frac {a^{4} d^{3} \operatorname {erfc}{\left (a + b x \right )}}{4 b^{4}} + \frac {a^{3} c d^{2} \operatorname {erfc}{\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 {erfc}{\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 {erfc}{\left (a + b x \right )}}{4 b^{4}} + \frac {a c^{3} \operatorname {erfc}{\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 {erfc}{\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 {erfc}{\left (a + b x \right )} + \frac {3 c^{2} d x^{2} \operatorname {erfc}{\left (a + b x \right )}}{2} + c d^{2} x^{3} \operatorname {erfc}{\left (a + b x \right )} + \frac {d^{3} x^{4} \operatorname {erfc}{\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 {erfc}{\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 {erfc}{\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 {erfc}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**4*d**3*erfc(a + b*x)/(4*b**4) + a**3*c*d**2*erfc(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*erfc(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*erfc(a + b*x)/(4*b**4) + a*c**3*erfc(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*erfc(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*erfc(a + b*x) + 3*c**2*d*x**2*erfc(a + b*x)/2
+ c*d**2*x**3*erfc(a + b*x) + d**3*x**4*erfc(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*erfc(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*erfc(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)*erfc(a), True))

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Giac [A]
time = 0.55, size = 435, normalized size = 1.49 \begin {gather*} \frac {1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac {3}{2} \, c^{2} 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^{3} - \frac {3}{4} \, {\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^{2} d - \frac {1}{2} \, {\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 )} c d^{2} - \frac {1}{16} \, {\left (4 \, x^{4} \operatorname {erf}\left (b x + a\right ) + \frac {\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}}{\sqrt {\pi } b^{3}}\right )} d^{3} + c^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.39, size = 352, normalized size = 1.21 \begin {gather*} \frac {d^3\,x^4\,\mathrm {erfc}\left (a+b\,x\right )}{4}-\frac {\mathrm {erfc}\left (a+b\,x\right )\,\left (b^2\,\left (\frac {3\,d\,a^2\,c^2}{2}+\frac {3\,d\,c^2}{4}\right )-b\,\left (c\,a^3\,d^2+\frac {3\,c\,a\,d^2}{2}\right )+\frac {3\,d^3}{16}+\frac {3\,a^2\,d^3}{4}+\frac {a^4\,d^3}{4}-a\,b^3\,c^3\right )}{b^4}+c^3\,x\,\mathrm {erfc}\left (a+b\,x\right )+\frac {{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (2\,a^3\,d^3-8\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+5\,a\,d^3-8\,b^3\,c^3-8\,b\,c\,d^2\right )}{8\,b^4\,\sqrt {\pi }}+\frac {3\,c^2\,d\,x^2\,\mathrm {erfc}\left (a+b\,x\right )}{2}+c\,d^2\,x^3\,\mathrm {erfc}\left (a+b\,x\right )-\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 )}{8\,b^3\,\sqrt {\pi }}-\frac {d^3\,x^3\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}}{4\,b\,\sqrt {\pi }}+\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 )}{4\,b^2\,\sqrt {\pi }} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfc(a + b*x)*(c + d*x)^3,x)

[Out]

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

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