3.259 \(\int e^{c+d x^2} x^5 \text{Erfi}(b x) \, dx\)

Optimal. Leaf size=257 \[ -\frac{b e^c \text{Erfi}\left (x \sqrt{b^2+d}\right )}{d^3 \sqrt{b^2+d}}-\frac{b e^c \text{Erfi}\left (x \sqrt{b^2+d}\right )}{2 d^2 \left (b^2+d\right )^{3/2}}+\frac{b x e^{x^2 \left (b^2+d\right )+c}}{\sqrt{\pi } d^2 \left (b^2+d\right )}-\frac{3 b e^c \text{Erfi}\left (x \sqrt{b^2+d}\right )}{8 d \left (b^2+d\right )^{5/2}}-\frac{b x^3 e^{x^2 \left (b^2+d\right )+c}}{2 \sqrt{\pi } d \left (b^2+d\right )}+\frac{3 b x e^{x^2 \left (b^2+d\right )+c}}{4 \sqrt{\pi } d \left (b^2+d\right )^2}-\frac{x^2 \text{Erfi}(b x) e^{c+d x^2}}{d^2}+\frac{\text{Erfi}(b x) e^{c+d x^2}}{d^3}+\frac{x^4 \text{Erfi}(b x) e^{c+d x^2}}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.408909, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {6387, 6384, 2204, 2212} \[ -\frac{b e^c \text{Erfi}\left (x \sqrt{b^2+d}\right )}{d^3 \sqrt{b^2+d}}-\frac{b e^c \text{Erfi}\left (x \sqrt{b^2+d}\right )}{2 d^2 \left (b^2+d\right )^{3/2}}+\frac{b x e^{x^2 \left (b^2+d\right )+c}}{\sqrt{\pi } d^2 \left (b^2+d\right )}-\frac{3 b e^c \text{Erfi}\left (x \sqrt{b^2+d}\right )}{8 d \left (b^2+d\right )^{5/2}}-\frac{b x^3 e^{x^2 \left (b^2+d\right )+c}}{2 \sqrt{\pi } d \left (b^2+d\right )}+\frac{3 b x e^{x^2 \left (b^2+d\right )+c}}{4 \sqrt{\pi } d \left (b^2+d\right )^2}-\frac{x^2 \text{Erfi}(b x) e^{c+d x^2}}{d^2}+\frac{\text{Erfi}(b x) e^{c+d x^2}}{d^3}+\frac{x^4 \text{Erfi}(b x) e^{c+d x^2}}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x^2)*x^5*Erfi[b*x],x]

[Out]

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

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 6384

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Simp[(E^(c + d*x^2)*Erfi[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 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

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+d x^2} x^5 \text{erfi}(b x) \, dx &=\frac{e^{c+d x^2} x^4 \text{erfi}(b x)}{2 d}-\frac{2 \int e^{c+d x^2} x^3 \text{erfi}(b x) \, dx}{d}-\frac{b \int e^{c+\left (b^2+d\right ) x^2} x^4 \, dx}{d \sqrt{\pi }}\\ &=-\frac{b e^{c+\left (b^2+d\right ) x^2} x^3}{2 d \left (b^2+d\right ) \sqrt{\pi }}-\frac{e^{c+d x^2} x^2 \text{erfi}(b x)}{d^2}+\frac{e^{c+d x^2} x^4 \text{erfi}(b x)}{2 d}+\frac{2 \int e^{c+d x^2} x \text{erfi}(b x) \, dx}{d^2}+\frac{(2 b) \int e^{c+\left (b^2+d\right ) x^2} x^2 \, dx}{d^2 \sqrt{\pi }}+\frac{(3 b) \int e^{c+\left (b^2+d\right ) x^2} x^2 \, dx}{2 d \left (b^2+d\right ) \sqrt{\pi }}\\ &=\frac{3 b e^{c+\left (b^2+d\right ) x^2} x}{4 d \left (b^2+d\right )^2 \sqrt{\pi }}+\frac{b e^{c+\left (b^2+d\right ) x^2} x}{d^2 \left (b^2+d\right ) \sqrt{\pi }}-\frac{b e^{c+\left (b^2+d\right ) x^2} x^3}{2 d \left (b^2+d\right ) \sqrt{\pi }}+\frac{e^{c+d x^2} \text{erfi}(b x)}{d^3}-\frac{e^{c+d x^2} x^2 \text{erfi}(b x)}{d^2}+\frac{e^{c+d x^2} x^4 \text{erfi}(b x)}{2 d}-\frac{(2 b) \int e^{c+\left (b^2+d\right ) x^2} \, dx}{d^3 \sqrt{\pi }}-\frac{(3 b) \int e^{c+\left (b^2+d\right ) x^2} \, dx}{4 d \left (b^2+d\right )^2 \sqrt{\pi }}-\frac{b \int e^{c+\left (b^2+d\right ) x^2} \, dx}{d^2 \left (b^2+d\right ) \sqrt{\pi }}\\ &=\frac{3 b e^{c+\left (b^2+d\right ) x^2} x}{4 d \left (b^2+d\right )^2 \sqrt{\pi }}+\frac{b e^{c+\left (b^2+d\right ) x^2} x}{d^2 \left (b^2+d\right ) \sqrt{\pi }}-\frac{b e^{c+\left (b^2+d\right ) x^2} x^3}{2 d \left (b^2+d\right ) \sqrt{\pi }}+\frac{e^{c+d x^2} \text{erfi}(b x)}{d^3}-\frac{e^{c+d x^2} x^2 \text{erfi}(b x)}{d^2}+\frac{e^{c+d x^2} x^4 \text{erfi}(b x)}{2 d}-\frac{3 b e^c \text{erfi}\left (\sqrt{b^2+d} x\right )}{8 d \left (b^2+d\right )^{5/2}}-\frac{b e^c \text{erfi}\left (\sqrt{b^2+d} x\right )}{2 d^2 \left (b^2+d\right )^{3/2}}-\frac{b e^c \text{erfi}\left (\sqrt{b^2+d} x\right )}{d^3 \sqrt{b^2+d}}\\ \end{align*}

Mathematica [A]  time = 0.278333, size = 131, normalized size = 0.51 \[ \frac{e^c \left (-\frac{b \left (20 b^2 d+8 b^4+15 d^2\right ) \text{Erfi}\left (x \sqrt{b^2+d}\right )}{\left (b^2+d\right )^{5/2}}-\frac{2 b d x e^{x^2 \left (b^2+d\right )} \left (2 b^2 \left (d x^2-2\right )+d \left (2 d x^2-7\right )\right )}{\sqrt{\pi } \left (b^2+d\right )^2}+4 e^{d x^2} \left (d^2 x^4-2 d x^2+2\right ) \text{Erfi}(b x)\right )}{8 d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(c + d*x^2)*x^5*Erfi[b*x],x]

[Out]

(E^c*((-2*b*d*E^((b^2 + d)*x^2)*x*(2*b^2*(-2 + d*x^2) + d*(-7 + 2*d*x^2)))/((b^2 + d)^2*Sqrt[Pi]) + 4*E^(d*x^2
)*(2 - 2*d*x^2 + d^2*x^4)*Erfi[b*x] - (b*(8*b^4 + 20*b^2*d + 15*d^2)*Erfi[Sqrt[b^2 + d]*x])/(b^2 + d)^(5/2)))/
(8*d^3)

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Maple [F]  time = 0.109, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{d{x}^{2}+c}}{x}^{5}{\it erfi} \left ( bx \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(d*x^2+c)*x^5*erfi(b*x),x)

[Out]

int(exp(d*x^2+c)*x^5*erfi(b*x),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \operatorname{erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x^2+c)*x^5*erfi(b*x),x, algorithm="maxima")

[Out]

integrate(x^5*erfi(b*x)*e^(d*x^2 + c), x)

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Fricas [A]  time = 2.80757, size = 547, normalized size = 2.13 \begin{align*} \frac{\pi{\left (8 \, b^{5} + 20 \, b^{3} d + 15 \, b d^{2}\right )} \sqrt{-b^{2} - d} \operatorname{erf}\left (\sqrt{-b^{2} - d} x\right ) e^{c} + 4 \,{\left (\pi{\left (b^{6} d^{2} + 3 \, b^{4} d^{3} + 3 \, b^{2} d^{4} + d^{5}\right )} x^{4} - 2 \, \pi{\left (b^{6} d + 3 \, b^{4} d^{2} + 3 \, b^{2} d^{3} + d^{4}\right )} x^{2} + 2 \, \pi{\left (b^{6} + 3 \, b^{4} d + 3 \, b^{2} d^{2} + d^{3}\right )}\right )} \operatorname{erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )} - 2 \, \sqrt{\pi }{\left (2 \,{\left (b^{5} d^{2} + 2 \, b^{3} d^{3} + b d^{4}\right )} x^{3} -{\left (4 \, b^{5} d + 11 \, b^{3} d^{2} + 7 \, b d^{3}\right )} x\right )} e^{\left (b^{2} x^{2} + d x^{2} + c\right )}}{8 \, \pi{\left (b^{6} d^{3} + 3 \, b^{4} d^{4} + 3 \, b^{2} d^{5} + d^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x^2+c)*x^5*erfi(b*x),x, algorithm="fricas")

[Out]

1/8*(pi*(8*b^5 + 20*b^3*d + 15*b*d^2)*sqrt(-b^2 - d)*erf(sqrt(-b^2 - d)*x)*e^c + 4*(pi*(b^6*d^2 + 3*b^4*d^3 +
3*b^2*d^4 + d^5)*x^4 - 2*pi*(b^6*d + 3*b^4*d^2 + 3*b^2*d^3 + d^4)*x^2 + 2*pi*(b^6 + 3*b^4*d + 3*b^2*d^2 + d^3)
)*erfi(b*x)*e^(d*x^2 + c) - 2*sqrt(pi)*(2*(b^5*d^2 + 2*b^3*d^3 + b*d^4)*x^3 - (4*b^5*d + 11*b^3*d^2 + 7*b*d^3)
*x)*e^(b^2*x^2 + d*x^2 + c))/(pi*(b^6*d^3 + 3*b^4*d^4 + 3*b^2*d^5 + d^6))

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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(exp(d*x**2+c)*x**5*erfi(b*x),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \operatorname{erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x^2+c)*x^5*erfi(b*x),x, algorithm="giac")

[Out]

integrate(x^5*erfi(b*x)*e^(d*x^2 + c), x)