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

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

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

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Rubi [A]  time = 0.421624, antiderivative size = 375, normalized size of antiderivative = 1., 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 = {6367, 6352, 6382, 2205, 6364, 6385, 6373, 30, 2209, 2212} \[ \frac{d (a+b x)^2 (b c-a d) \text{Erf}(a+b x)^2}{b^3}+\frac{(a+b x) (b c-a d)^2 \text{Erf}(a+b x)^2}{b^3}+\frac{2 d e^{-(a+b x)^2} (a+b x) (b c-a d) \text{Erf}(a+b x)}{\sqrt{\pi } b^3}-\frac{d (b c-a d) \text{Erf}(a+b x)^2}{2 b^3}+\frac{2 e^{-(a+b x)^2} (b c-a d)^2 \text{Erf}(a+b x)}{\sqrt{\pi } b^3}-\frac{\sqrt{\frac{2}{\pi }} (b c-a d)^2 \text{Erf}\left (\sqrt{2} (a+b x)\right )}{b^3}+\frac{d e^{-2 (a+b x)^2} (b c-a d)}{\pi b^3}+\frac{d^2 (a+b x)^3 \text{Erf}(a+b x)^2}{3 b^3}+\frac{2 d^2 e^{-(a+b x)^2} (a+b x)^2 \text{Erf}(a+b x)}{3 \sqrt{\pi } b^3}+\frac{2 d^2 e^{-(a+b x)^2} \text{Erf}(a+b x)}{3 \sqrt{\pi } b^3}-\frac{5 d^2 \text{Erf}\left (\sqrt{2} (a+b x)\right )}{6 \sqrt{2 \pi } b^3}+\frac{d^2 e^{-2 (a+b x)^2} (a+b x)}{3 \pi b^3} \]

Antiderivative was successfully verified.

[In]

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

Rule 6367

Int[Erf[(a_) + (b_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[1/b^(m + 1), Subst[Int[ExpandIntegr
and[Erf[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 6352

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

Rule 6382

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Simp[(E^(c + d*x^2)*Erf[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 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 6364

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

Rule 6385

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m - 1)*E^(c + d*x^2)*Erf
[a + b*x])/(2*d), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erf[a + b*x], x], x] - Dist[b/(d*Sqrt
[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]

Rule 6373

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

Rule 30

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

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])

Rubi steps

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

Mathematica [A]  time = 1.00768, size = 226, normalized size = 0.6 \[ \frac{2 \text{Erf}(a+b x)^2 \left (-6 a^2 b c d+2 a^3 d^2+3 a \left (2 b^2 c^2+d^2\right )+2 b^3 x \left (3 c^2+3 c d x+d^2 x^2\right )-3 b c d\right )+\frac{8 e^{-(a+b x)^2} \text{Erf}(a+b x) \left (\left (a^2+1\right ) d^2-a b d (3 c+d x)+b^2 \left (3 c^2+3 c d x+d^2 x^2\right )\right )}{\sqrt{\pi }}-\sqrt{\frac{2}{\pi }} \left (\left (12 a^2+5\right ) d^2-24 a b c d+12 b^2 c^2\right ) \text{Erf}\left (\sqrt{2} (a+b x)\right )+\frac{4 d e^{-2 (a+b x)^2} (-2 a d+3 b c+b d x)}{\pi }}{12 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*d*(3*b*c - 2*a*d + b*d*x))/(E^(2*(a + b*x)^2)*Pi) + (8*((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))*Erf[a + b*x])/(E^(a + b*x)^2*Sqrt[Pi]) + 2*(-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]^2 - (12*b^2*c^2 - 24*a*b*c*d + (5 + 12*a^2)*d^
2)*Sqrt[2/Pi]*Erf[Sqrt[2]*(a + b*x)])/(12*b^3)

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Maple [F]  time = 0.224, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{2} \left ({\it Erf} \left ( bx+a \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 )^{2} - \frac{4 \, \int{\left (b d^{2} x^{3} + 3 \, b c d x^{2} + 3 \, b c^{2} x\right )} \operatorname{erf}\left (b x + a\right ) e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}\,{d x}}{3 \, \sqrt{\pi }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.63072, size = 645, normalized size = 1.72 \begin{align*} -\frac{\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 ) - 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\right )} \operatorname{erf}\left (b x + a\right ) e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\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} - 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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(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) - 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)*erf(b*x
+ a)*e^(-b^2*x^2 - 2*a*b*x - a^2) - 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 - 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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{2} \operatorname{erf}\left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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