3.2.38 \(\int (c+d x)^2 \text {Erfc}(a+b x)^2 \, dx\) [138]

Optimal. Leaf size=375 \[ \frac {d (b c-a d) e^{-2 (a+b x)^2}}{b^3 \pi }+\frac {d^2 e^{-2 (a+b x)^2} (a+b x)}{3 b^3 \pi }-\frac {(b c-a d)^2 \sqrt {\frac {2}{\pi }} \text {Erf}\left (\sqrt {2} (a+b x)\right )}{b^3}-\frac {5 d^2 \text {Erf}\left (\sqrt {2} (a+b x)\right )}{6 b^3 \sqrt {2 \pi }}-\frac {2 d^2 e^{-(a+b x)^2} \text {Erfc}(a+b x)}{3 b^3 \sqrt {\pi }}-\frac {2 (b c-a d)^2 e^{-(a+b x)^2} \text {Erfc}(a+b x)}{b^3 \sqrt {\pi }}-\frac {2 d (b c-a d) e^{-(a+b x)^2} (a+b x) \text {Erfc}(a+b x)}{b^3 \sqrt {\pi }}-\frac {2 d^2 e^{-(a+b x)^2} (a+b x)^2 \text {Erfc}(a+b x)}{3 b^3 \sqrt {\pi }}-\frac {d (b c-a d) \text {Erfc}(a+b x)^2}{2 b^3}+\frac {(b c-a d)^2 (a+b x) \text {Erfc}(a+b x)^2}{b^3}+\frac {d (b c-a d) (a+b x)^2 \text {Erfc}(a+b x)^2}{b^3}+\frac {d^2 (a+b x)^3 \text {Erfc}(a+b x)^2}{3 b^3} \]

[Out]

d*(-a*d+b*c)/b^3/exp(2*(b*x+a)^2)/Pi+1/3*d^2*(b*x+a)/b^3/exp(2*(b*x+a)^2)/Pi-1/2*d*(-a*d+b*c)*erfc(b*x+a)^2/b^
3+(-a*d+b*c)^2*(b*x+a)*erfc(b*x+a)^2/b^3+d*(-a*d+b*c)*(b*x+a)^2*erfc(b*x+a)^2/b^3+1/3*d^2*(b*x+a)^3*erfc(b*x+a
)^2/b^3-(-a*d+b*c)^2*erf((b*x+a)*2^(1/2))*2^(1/2)/Pi^(1/2)/b^3-2/3*d^2*erfc(b*x+a)/b^3/exp((b*x+a)^2)/Pi^(1/2)
-2*(-a*d+b*c)^2*erfc(b*x+a)/b^3/exp((b*x+a)^2)/Pi^(1/2)-2*d*(-a*d+b*c)*(b*x+a)*erfc(b*x+a)/b^3/exp((b*x+a)^2)/
Pi^(1/2)-2/3*d^2*(b*x+a)^2*erfc(b*x+a)/b^3/exp((b*x+a)^2)/Pi^(1/2)-5/12*d^2*erf((b*x+a)*2^(1/2))/b^3*2^(1/2)/P
i^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.27, antiderivative size = 375, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6503, 6488, 6518, 2236, 6500, 6521, 6509, 30, 2240, 2243} \begin {gather*} -\frac {\sqrt {\frac {2}{\pi }} (b c-a d)^2 \text {Erf}\left (\sqrt {2} (a+b x)\right )}{b^3}+\frac {d (a+b x)^2 (b c-a d) \text {Erfc}(a+b x)^2}{b^3}+\frac {(a+b x) (b c-a d)^2 \text {Erfc}(a+b x)^2}{b^3}-\frac {2 d e^{-(a+b x)^2} (a+b x) (b c-a d) \text {Erfc}(a+b x)}{\sqrt {\pi } b^3}-\frac {d (b c-a d) \text {Erfc}(a+b x)^2}{2 b^3}-\frac {2 e^{-(a+b x)^2} (b c-a d)^2 \text {Erfc}(a+b x)}{\sqrt {\pi } b^3}+\frac {d e^{-2 (a+b x)^2} (b c-a d)}{\pi b^3}-\frac {5 d^2 \text {Erf}\left (\sqrt {2} (a+b x)\right )}{6 \sqrt {2 \pi } b^3}+\frac {d^2 (a+b x)^3 \text {Erfc}(a+b x)^2}{3 b^3}-\frac {2 d^2 e^{-(a+b x)^2} (a+b x)^2 \text {Erfc}(a+b x)}{3 \sqrt {\pi } b^3}-\frac {2 d^2 e^{-(a+b x)^2} \text {Erfc}(a+b x)}{3 \sqrt {\pi } b^3}+\frac {d^2 e^{-2 (a+b x)^2} (a+b x)}{3 \pi b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2*Erfc[a + b*x]^2,x]

[Out]

(d*(b*c - a*d))/(b^3*E^(2*(a + b*x)^2)*Pi) + (d^2*(a + b*x))/(3*b^3*E^(2*(a + b*x)^2)*Pi) - ((b*c - a*d)^2*Sqr
t[2/Pi]*Erf[Sqrt[2]*(a + b*x)])/b^3 - (5*d^2*Erf[Sqrt[2]*(a + b*x)])/(6*b^3*Sqrt[2*Pi]) - (2*d^2*Erfc[a + b*x]
)/(3*b^3*E^(a + b*x)^2*Sqrt[Pi]) - (2*(b*c - a*d)^2*Erfc[a + b*x])/(b^3*E^(a + b*x)^2*Sqrt[Pi]) - (2*d*(b*c -
a*d)*(a + b*x)*Erfc[a + b*x])/(b^3*E^(a + b*x)^2*Sqrt[Pi]) - (2*d^2*(a + b*x)^2*Erfc[a + b*x])/(3*b^3*E^(a + b
*x)^2*Sqrt[Pi]) - (d*(b*c - a*d)*Erfc[a + b*x]^2)/(2*b^3) + ((b*c - a*d)^2*(a + b*x)*Erfc[a + b*x]^2)/b^3 + (d
*(b*c - a*d)*(a + b*x)^2*Erfc[a + b*x]^2)/b^3 + (d^2*(a + b*x)^3*Erfc[a + b*x]^2)/(3*b^3)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 6488

Int[Erfc[(a_.) + (b_.)*(x_)]^2, x_Symbol] :> Simp[(a + b*x)*(Erfc[a + b*x]^2/b), x] + Dist[4/Sqrt[Pi], Int[(a
+ b*x)*(Erfc[a + b*x]/E^(a + b*x)^2), x], x] /; FreeQ[{a, b}, x]

Rule 6500

Int[Erfc[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Erfc[b*x]^2/(m + 1)), x] + Dist[4*(b/(Sqrt[Pi]
*(m + 1))), Int[(x^(m + 1)*Erfc[b*x])/E^(b^2*x^2), x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])

Rule 6503

Int[Erfc[(a_) + (b_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[1/b^(m + 1), Subst[Int[ExpandInteg
rand[Erfc[x]^2, (b*c - a*d + d*x)^m, x], x], x, a + b*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]

Rule 6509

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[(-E^c)*(Sqrt[Pi]/(2*b)), Subst[Int[x^n,
x], x, Erfc[b*x]], x] /; FreeQ[{b, c, d, n}, x] && EqQ[d, -b^2]

Rule 6518

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Simp[E^(c + d*x^2)*(Erfc[a + b*x]/(2*
d)), x] + Dist[b/(d*Sqrt[Pi]), Int[E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x], x] /; FreeQ[{a, b, c, d}, x]

Rule 6521

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^(m - 1)*E^(c + d*x^2)*(Er
fc[a + b*x]/(2*d)), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erfc[a + b*x], x], x] + Dist[b/(d*S
qrt[Pi]), Int[x^(m - 1)*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x], x]) /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1
]

Rubi steps

\begin {align*} \int (c+d x)^2 \text {erfc}(a+b x)^2 \, dx &=\frac {\text {Subst}\left (\int \left (b^2 c^2 \left (1+\frac {a d (-2 b c+a d)}{b^2 c^2}\right ) \text {erfc}(x)^2+2 b c d \left (1-\frac {a d}{b c}\right ) x \text {erfc}(x)^2+d^2 x^2 \text {erfc}(x)^2\right ) \, dx,x,a+b x\right )}{b^3}\\ &=\frac {d^2 \text {Subst}\left (\int x^2 \text {erfc}(x)^2 \, dx,x,a+b x\right )}{b^3}+\frac {(2 d (b c-a d)) \text {Subst}\left (\int x \text {erfc}(x)^2 \, dx,x,a+b x\right )}{b^3}+\frac {(b c-a d)^2 \text {Subst}\left (\int \text {erfc}(x)^2 \, dx,x,a+b x\right )}{b^3}\\ &=\frac {(b c-a d)^2 (a+b x) \text {erfc}(a+b x)^2}{b^3}+\frac {d (b c-a d) (a+b x)^2 \text {erfc}(a+b x)^2}{b^3}+\frac {d^2 (a+b x)^3 \text {erfc}(a+b x)^2}{3 b^3}+\frac {\left (4 d^2\right ) \text {Subst}\left (\int e^{-x^2} x^3 \text {erfc}(x) \, dx,x,a+b x\right )}{3 b^3 \sqrt {\pi }}+\frac {(4 d (b c-a d)) \text {Subst}\left (\int e^{-x^2} x^2 \text {erfc}(x) \, dx,x,a+b x\right )}{b^3 \sqrt {\pi }}+\frac {\left (4 (b c-a d)^2\right ) \text {Subst}\left (\int e^{-x^2} x \text {erfc}(x) \, dx,x,a+b x\right )}{b^3 \sqrt {\pi }}\\ &=-\frac {2 (b c-a d)^2 e^{-(a+b x)^2} \text {erfc}(a+b x)}{b^3 \sqrt {\pi }}-\frac {2 d (b c-a d) e^{-(a+b x)^2} (a+b x) \text {erfc}(a+b x)}{b^3 \sqrt {\pi }}-\frac {2 d^2 e^{-(a+b x)^2} (a+b x)^2 \text {erfc}(a+b x)}{3 b^3 \sqrt {\pi }}+\frac {(b c-a d)^2 (a+b x) \text {erfc}(a+b x)^2}{b^3}+\frac {d (b c-a d) (a+b x)^2 \text {erfc}(a+b x)^2}{b^3}+\frac {d^2 (a+b x)^3 \text {erfc}(a+b x)^2}{3 b^3}-\frac {\left (4 d^2\right ) \text {Subst}\left (\int e^{-2 x^2} x^2 \, dx,x,a+b x\right )}{3 b^3 \pi }-\frac {(4 d (b c-a d)) \text {Subst}\left (\int e^{-2 x^2} x \, dx,x,a+b x\right )}{b^3 \pi }-\frac {\left (4 (b c-a d)^2\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,a+b x\right )}{b^3 \pi }+\frac {\left (4 d^2\right ) \text {Subst}\left (\int e^{-x^2} x \text {erfc}(x) \, dx,x,a+b x\right )}{3 b^3 \sqrt {\pi }}+\frac {(2 d (b c-a d)) \text {Subst}\left (\int e^{-x^2} \text {erfc}(x) \, dx,x,a+b x\right )}{b^3 \sqrt {\pi }}\\ &=\frac {d (b c-a d) e^{-2 (a+b x)^2}}{b^3 \pi }+\frac {d^2 e^{-2 (a+b x)^2} (a+b x)}{3 b^3 \pi }-\frac {(b c-a d)^2 \sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} (a+b x)\right )}{b^3}-\frac {2 d^2 e^{-(a+b x)^2} \text {erfc}(a+b x)}{3 b^3 \sqrt {\pi }}-\frac {2 (b c-a d)^2 e^{-(a+b x)^2} \text {erfc}(a+b x)}{b^3 \sqrt {\pi }}-\frac {2 d (b c-a d) e^{-(a+b x)^2} (a+b x) \text {erfc}(a+b x)}{b^3 \sqrt {\pi }}-\frac {2 d^2 e^{-(a+b x)^2} (a+b x)^2 \text {erfc}(a+b x)}{3 b^3 \sqrt {\pi }}+\frac {(b c-a d)^2 (a+b x) \text {erfc}(a+b x)^2}{b^3}+\frac {d (b c-a d) (a+b x)^2 \text {erfc}(a+b x)^2}{b^3}+\frac {d^2 (a+b x)^3 \text {erfc}(a+b x)^2}{3 b^3}-\frac {(d (b c-a d)) \text {Subst}(\int x \, dx,x,\text {erfc}(a+b x))}{b^3}-\frac {d^2 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,a+b x\right )}{3 b^3 \pi }-\frac {\left (4 d^2\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,a+b x\right )}{3 b^3 \pi }\\ &=\frac {d (b c-a d) e^{-2 (a+b x)^2}}{b^3 \pi }+\frac {d^2 e^{-2 (a+b x)^2} (a+b x)}{3 b^3 \pi }-\frac {d^2 \sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} (a+b x)\right )}{3 b^3}-\frac {(b c-a d)^2 \sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} (a+b x)\right )}{b^3}-\frac {d^2 \text {erf}\left (\sqrt {2} (a+b x)\right )}{6 b^3 \sqrt {2 \pi }}-\frac {2 d^2 e^{-(a+b x)^2} \text {erfc}(a+b x)}{3 b^3 \sqrt {\pi }}-\frac {2 (b c-a d)^2 e^{-(a+b x)^2} \text {erfc}(a+b x)}{b^3 \sqrt {\pi }}-\frac {2 d (b c-a d) e^{-(a+b x)^2} (a+b x) \text {erfc}(a+b x)}{b^3 \sqrt {\pi }}-\frac {2 d^2 e^{-(a+b x)^2} (a+b x)^2 \text {erfc}(a+b x)}{3 b^3 \sqrt {\pi }}-\frac {d (b c-a d) \text {erfc}(a+b x)^2}{2 b^3}+\frac {(b c-a d)^2 (a+b x) \text {erfc}(a+b x)^2}{b^3}+\frac {d (b c-a d) (a+b x)^2 \text {erfc}(a+b x)^2}{b^3}+\frac {d^2 (a+b x)^3 \text {erfc}(a+b x)^2}{3 b^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 3.20, size = 610, normalized size = 1.63 \begin {gather*} \frac {-12 b^2 \sqrt {\pi } (c+d x)^2 \left (\sqrt {2} \text {Erf}\left (\sqrt {2} (a+b x)\right )+\text {Erfc}(a+b x) \left (2 e^{-(a+b x)^2}-\sqrt {\pi } (a+b x) \text {Erfc}(a+b x)\right )\right )+6 b d (c+d x) \left (2 e^{-2 (a+b x)^2}+4 e^{-(a+b x)^2} \sqrt {\pi } (a+b x)-2 \pi (a+b x)^2-2 \pi \text {Erf}(a+b x)-4 e^{-(a+b x)^2} \sqrt {\pi } (a+b x) \text {Erf}(a+b x)+4 \pi (a+b x)^2 \text {Erf}(a+b x)+\pi \text {Erf}(a+b x)^2-2 \pi (a+b x)^2 \text {Erf}(a+b x)^2+4 a \sqrt {2 \pi } \text {Erf}\left (\sqrt {2} (a+b x)\right )+4 b \sqrt {2 \pi } x \text {Erf}\left (\sqrt {2} (a+b x)\right )+2 \pi (2+\text {Erfc}(-a-b x) \text {Erfc}(a+b x))-4 \sqrt {\pi } (a+b x) E_{\frac {1}{2}}\left ((a+b x)^2\right )\right )+d^2 \left (24 e^{-(a+b x)^2} \sqrt {\pi }-36 b \pi x+12 a^2 b \pi x+12 a b^2 \pi x^2+4 b^3 \pi x^3-8 e^{-2 (a+b x)^2} (a+b x)-8 e^{-(a+b x)^2} \sqrt {\pi } \left (1+(a+b x)^2\right )+12 a \pi \text {Erf}(a+b x)+12 b \pi x \text {Erf}(a+b x)-8 \pi (a+b x)^3 \text {Erf}(a+b x)+8 e^{-(a+b x)^2} \sqrt {\pi } \left (1+(a+b x)^2\right ) \text {Erf}(a+b x)+6 \pi (a+b x) \text {Erf}(a+b x)^2+4 \pi (a+b x)^3 \text {Erf}(a+b x)^2-5 \sqrt {2 \pi } \text {Erf}\left (\sqrt {2} (a+b x)\right )-12 a^2 \sqrt {2 \pi } \text {Erf}\left (\sqrt {2} (a+b x)\right )-12 b \sqrt {2 \pi } x (2 a+b x) \text {Erf}\left (\sqrt {2} (a+b x)\right )-12 \sqrt {\pi } E_{\frac {3}{2}}\left ((a+b x)^2\right )\right )}{12 b^3 \pi } \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2*Erfc[a + b*x]^2,x]

[Out]

(-12*b^2*Sqrt[Pi]*(c + d*x)^2*(Sqrt[2]*Erf[Sqrt[2]*(a + b*x)] + Erfc[a + b*x]*(2/E^(a + b*x)^2 - Sqrt[Pi]*(a +
b*x)*Erfc[a + b*x])) + 6*b*d*(c + d*x)*(2/E^(2*(a + b*x)^2) + (4*Sqrt[Pi]*(a + b*x))/E^(a + b*x)^2 - 2*Pi*(a
+ b*x)^2 - 2*Pi*Erf[a + b*x] - (4*Sqrt[Pi]*(a + b*x)*Erf[a + b*x])/E^(a + b*x)^2 + 4*Pi*(a + b*x)^2*Erf[a + b*
x] + Pi*Erf[a + b*x]^2 - 2*Pi*(a + b*x)^2*Erf[a + b*x]^2 + 4*a*Sqrt[2*Pi]*Erf[Sqrt[2]*(a + b*x)] + 4*b*Sqrt[2*
Pi]*x*Erf[Sqrt[2]*(a + b*x)] + 2*Pi*(2 + Erfc[-a - b*x]*Erfc[a + b*x]) - 4*Sqrt[Pi]*(a + b*x)*ExpIntegralE[1/2
, (a + b*x)^2]) + d^2*((24*Sqrt[Pi])/E^(a + b*x)^2 - 36*b*Pi*x + 12*a^2*b*Pi*x + 12*a*b^2*Pi*x^2 + 4*b^3*Pi*x^
3 - (8*(a + b*x))/E^(2*(a + b*x)^2) - (8*Sqrt[Pi]*(1 + (a + b*x)^2))/E^(a + b*x)^2 + 12*a*Pi*Erf[a + b*x] + 12
*b*Pi*x*Erf[a + b*x] - 8*Pi*(a + b*x)^3*Erf[a + b*x] + (8*Sqrt[Pi]*(1 + (a + b*x)^2)*Erf[a + b*x])/E^(a + b*x)
^2 + 6*Pi*(a + b*x)*Erf[a + b*x]^2 + 4*Pi*(a + b*x)^3*Erf[a + b*x]^2 - 5*Sqrt[2*Pi]*Erf[Sqrt[2]*(a + b*x)] - 1
2*a^2*Sqrt[2*Pi]*Erf[Sqrt[2]*(a + b*x)] - 12*b*Sqrt[2*Pi]*x*(2*a + b*x)*Erf[Sqrt[2]*(a + b*x)] - 12*Sqrt[Pi]*E
xpIntegralE[3/2, (a + b*x)^2]))/(12*b^3*Pi)

________________________________________________________________________________________

Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \left (d x +c \right )^{2} \mathrm {erfc}\left (b x +a \right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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)^2*erfc(b*x+a)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 472, normalized size = 1.26 \begin {gather*} \frac {4 \, \pi b^{4} d^{2} x^{3} + 12 \, \pi b^{4} c d x^{2} + 12 \, \pi b^{4} c^{2} x - \sqrt {2} \sqrt {\pi } {\left (12 \, b^{2} c^{2} - 24 \, a b c d + {\left (12 \, a^{2} + 5\right )} d^{2}\right )} \sqrt {b^{2}} \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - 4 \, \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 )} \sqrt {b^{2}} \operatorname {erf}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) + 2 \, {\left (2 \, \pi b^{4} d^{2} x^{3} + 6 \, \pi b^{4} c d x^{2} + 6 \, \pi b^{4} c^{2} x + \pi {\left (6 \, a b^{3} c^{2} - 3 \, {\left (2 \, a^{2} + 1\right )} b^{2} c d + {\left (2 \, a^{3} + 3 \, a\right )} b d^{2}\right )}\right )} \operatorname {erf}\left (b x + a\right )^{2} - 8 \, \sqrt {\pi } {\left (b^{3} d^{2} x^{2} + 3 \, b^{3} c^{2} - 3 \, a b^{2} c d + {\left (a^{2} + 1\right )} b d^{2} + {\left (3 \, b^{3} c d - a b^{2} d^{2}\right )} x - {\left (b^{3} d^{2} x^{2} + 3 \, b^{3} c^{2} - 3 \, a b^{2} c d + {\left (a^{2} + 1\right )} b d^{2} + {\left (3 \, b^{3} c d - a b^{2} d^{2}\right )} x\right )} \operatorname {erf}\left (b x + a\right )\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )} - 8 \, {\left (\pi b^{4} d^{2} x^{3} + 3 \, \pi b^{4} c d x^{2} + 3 \, \pi b^{4} c^{2} x\right )} \operatorname {erf}\left (b x + a\right ) + 4 \, {\left (b^{2} d^{2} x + 3 \, b^{2} c d - 2 \, a b d^{2}\right )} e^{\left (-2 \, b^{2} x^{2} - 4 \, a b x - 2 \, a^{2}\right )}}{12 \, \pi b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(4*pi*b^4*d^2*x^3 + 12*pi*b^4*c*d*x^2 + 12*pi*b^4*c^2*x - sqrt(2)*sqrt(pi)*(12*b^2*c^2 - 24*a*b*c*d + (12
*a^2 + 5)*d^2)*sqrt(b^2)*erf(sqrt(2)*sqrt(b^2)*(b*x + a)/b) - 4*pi*(6*a*b^2*c^2 - 3*(2*a^2 + 1)*b*c*d + (2*a^3
+ 3*a)*d^2)*sqrt(b^2)*erf(sqrt(b^2)*(b*x + a)/b) + 2*(2*pi*b^4*d^2*x^3 + 6*pi*b^4*c*d*x^2 + 6*pi*b^4*c^2*x +
pi*(6*a*b^3*c^2 - 3*(2*a^2 + 1)*b^2*c*d + (2*a^3 + 3*a)*b*d^2))*erf(b*x + a)^2 - 8*sqrt(pi)*(b^3*d^2*x^2 + 3*b
^3*c^2 - 3*a*b^2*c*d + (a^2 + 1)*b*d^2 + (3*b^3*c*d - a*b^2*d^2)*x - (b^3*d^2*x^2 + 3*b^3*c^2 - 3*a*b^2*c*d +
(a^2 + 1)*b*d^2 + (3*b^3*c*d - a*b^2*d^2)*x)*erf(b*x + a))*e^(-b^2*x^2 - 2*a*b*x - a^2) - 8*(pi*b^4*d^2*x^3 +
3*pi*b^4*c*d*x^2 + 3*pi*b^4*c^2*x)*erf(b*x + a) + 4*(b^2*d^2*x + 3*b^2*c*d - 2*a*b*d^2)*e^(-2*b^2*x^2 - 4*a*b*
x - 2*a^2))/(pi*b^4)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c + d x\right )^{2} \operatorname {erfc}^{2}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c + d*x)**2*erfc(a + b*x)**2, x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {erfc}\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfc(a + b*x)^2*(c + d*x)^2,x)

[Out]

int(erfc(a + b*x)^2*(c + d*x)^2, x)

________________________________________________________________________________________