3.289 \(\int e^{c+b^2 x^2} x^4 \text{Erfi}(b x) \, dx\)

Optimal. Leaf size=121 \[ \frac{x^3 e^{b^2 x^2+c} \text{Erfi}(b x)}{2 b^2}-\frac{3 x e^{b^2 x^2+c} \text{Erfi}(b x)}{4 b^4}+\frac{3 \sqrt{\pi } e^c \text{Erfi}(b x)^2}{16 b^5}-\frac{x^2 e^{2 b^2 x^2+c}}{4 \sqrt{\pi } b^3}+\frac{e^{2 b^2 x^2+c}}{2 \sqrt{\pi } b^5} \]

[Out]

E^(c + 2*b^2*x^2)/(2*b^5*Sqrt[Pi]) - (E^(c + 2*b^2*x^2)*x^2)/(4*b^3*Sqrt[Pi]) - (3*E^(c + b^2*x^2)*x*Erfi[b*x]
)/(4*b^4) + (E^(c + b^2*x^2)*x^3*Erfi[b*x])/(2*b^2) + (3*E^c*Sqrt[Pi]*Erfi[b*x]^2)/(16*b^5)

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Rubi [A]  time = 0.162773, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {6387, 6375, 30, 2209, 2212} \[ \frac{x^3 e^{b^2 x^2+c} \text{Erfi}(b x)}{2 b^2}-\frac{3 x e^{b^2 x^2+c} \text{Erfi}(b x)}{4 b^4}+\frac{3 \sqrt{\pi } e^c \text{Erfi}(b x)^2}{16 b^5}-\frac{x^2 e^{2 b^2 x^2+c}}{4 \sqrt{\pi } b^3}+\frac{e^{2 b^2 x^2+c}}{2 \sqrt{\pi } b^5} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + b^2*x^2)*x^4*Erfi[b*x],x]

[Out]

E^(c + 2*b^2*x^2)/(2*b^5*Sqrt[Pi]) - (E^(c + 2*b^2*x^2)*x^2)/(4*b^3*Sqrt[Pi]) - (3*E^(c + b^2*x^2)*x*Erfi[b*x]
)/(4*b^4) + (E^(c + b^2*x^2)*x^3*Erfi[b*x])/(2*b^2) + (3*E^c*Sqrt[Pi]*Erfi[b*x]^2)/(16*b^5)

Rule 6387

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

Rule 6375

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[(E^c*Sqrt[Pi])/(2*b), Subst[Int[x^n, x]
, x, Erfi[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 e^{c+b^2 x^2} x^4 \text{erfi}(b x) \, dx &=\frac{e^{c+b^2 x^2} x^3 \text{erfi}(b x)}{2 b^2}-\frac{3 \int e^{c+b^2 x^2} x^2 \text{erfi}(b x) \, dx}{2 b^2}-\frac{\int e^{c+2 b^2 x^2} x^3 \, dx}{b \sqrt{\pi }}\\ &=-\frac{e^{c+2 b^2 x^2} x^2}{4 b^3 \sqrt{\pi }}-\frac{3 e^{c+b^2 x^2} x \text{erfi}(b x)}{4 b^4}+\frac{e^{c+b^2 x^2} x^3 \text{erfi}(b x)}{2 b^2}+\frac{3 \int e^{c+b^2 x^2} \text{erfi}(b x) \, dx}{4 b^4}+\frac{\int e^{c+2 b^2 x^2} x \, dx}{2 b^3 \sqrt{\pi }}+\frac{3 \int e^{c+2 b^2 x^2} x \, dx}{2 b^3 \sqrt{\pi }}\\ &=\frac{e^{c+2 b^2 x^2}}{2 b^5 \sqrt{\pi }}-\frac{e^{c+2 b^2 x^2} x^2}{4 b^3 \sqrt{\pi }}-\frac{3 e^{c+b^2 x^2} x \text{erfi}(b x)}{4 b^4}+\frac{e^{c+b^2 x^2} x^3 \text{erfi}(b x)}{2 b^2}+\frac{\left (3 e^c \sqrt{\pi }\right ) \operatorname{Subst}(\int x \, dx,x,\text{erfi}(b x))}{8 b^5}\\ &=\frac{e^{c+2 b^2 x^2}}{2 b^5 \sqrt{\pi }}-\frac{e^{c+2 b^2 x^2} x^2}{4 b^3 \sqrt{\pi }}-\frac{3 e^{c+b^2 x^2} x \text{erfi}(b x)}{4 b^4}+\frac{e^{c+b^2 x^2} x^3 \text{erfi}(b x)}{2 b^2}+\frac{3 e^c \sqrt{\pi } \text{erfi}(b x)^2}{16 b^5}\\ \end{align*}

Mathematica [A]  time = 0.0364366, size = 78, normalized size = 0.64 \[ \frac{e^c \left (4 \sqrt{\pi } b x e^{b^2 x^2} \left (2 b^2 x^2-3\right ) \text{Erfi}(b x)-4 e^{2 b^2 x^2} \left (b^2 x^2-2\right )+3 \pi \text{Erfi}(b x)^2\right )}{16 \sqrt{\pi } b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(c + b^2*x^2)*x^4*Erfi[b*x],x]

[Out]

(E^c*(-4*E^(2*b^2*x^2)*(-2 + b^2*x^2) + 4*b*E^(b^2*x^2)*Sqrt[Pi]*x*(-3 + 2*b^2*x^2)*Erfi[b*x] + 3*Pi*Erfi[b*x]
^2))/(16*b^5*Sqrt[Pi])

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Maple [F]  time = 0.112, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{{b}^{2}{x}^{2}+c}}{x}^{4}{\it erfi} \left ( bx \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(b^2*x^2+c)*x^4*erfi(b*x),x)

[Out]

int(exp(b^2*x^2+c)*x^4*erfi(b*x),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b^2*x^2+c)*x^4*erfi(b*x),x, algorithm="maxima")

[Out]

integrate(x^4*erfi(b*x)*e^(b^2*x^2 + c), x)

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Fricas [A]  time = 2.53352, size = 180, normalized size = 1.49 \begin{align*} \frac{{\left (4 \,{\left (2 \, \pi b^{3} x^{3} - 3 \, \pi b x\right )} \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} + \sqrt{\pi }{\left (3 \, \pi \operatorname{erfi}\left (b x\right )^{2} - 4 \,{\left (b^{2} x^{2} - 2\right )} e^{\left (2 \, b^{2} x^{2}\right )}\right )}\right )} e^{c}}{16 \, \pi b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b^2*x^2+c)*x^4*erfi(b*x),x, algorithm="fricas")

[Out]

1/16*(4*(2*pi*b^3*x^3 - 3*pi*b*x)*erfi(b*x)*e^(b^2*x^2) + sqrt(pi)*(3*pi*erfi(b*x)^2 - 4*(b^2*x^2 - 2)*e^(2*b^
2*x^2)))*e^c/(pi*b^5)

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Sympy [A]  time = 32.1354, size = 124, normalized size = 1.02 \begin{align*} \begin{cases} \frac{x^{3} e^{c} e^{b^{2} x^{2}} \operatorname{erfi}{\left (b x \right )}}{2 b^{2}} - \frac{x^{2} e^{c} e^{2 b^{2} x^{2}}}{4 \sqrt{\pi } b^{3}} - \frac{3 x e^{c} e^{b^{2} x^{2}} \operatorname{erfi}{\left (b x \right )}}{4 b^{4}} + \frac{e^{c} e^{2 b^{2} x^{2}}}{2 \sqrt{\pi } b^{5}} + \frac{3 \sqrt{\pi } e^{c} \operatorname{erfi}^{2}{\left (b x \right )}}{16 b^{5}} & \text{for}\: b \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b**2*x**2+c)*x**4*erfi(b*x),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b^2*x^2+c)*x^4*erfi(b*x),x, algorithm="giac")

[Out]

integrate(x^4*erfi(b*x)*e^(b^2*x^2 + c), x)