3.277 \(\int e^{-b^2 x^2} x^4 \text{Erfi}(b x) \, dx\)

Optimal. Leaf size=109 \[ \frac{3 x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-b^2 x^2\right )}{4 \sqrt{\pi } b^3}-\frac{x^3 e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^2}-\frac{3 x e^{-b^2 x^2} \text{Erfi}(b x)}{4 b^4}+\frac{3 x^2}{4 \sqrt{\pi } b^3}+\frac{x^4}{4 \sqrt{\pi } b} \]

[Out]

(3*x^2)/(4*b^3*Sqrt[Pi]) + x^4/(4*b*Sqrt[Pi]) - (3*x*Erfi[b*x])/(4*b^4*E^(b^2*x^2)) - (x^3*Erfi[b*x])/(2*b^2*E
^(b^2*x^2)) + (3*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, -(b^2*x^2)])/(4*b^3*Sqrt[Pi])

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Rubi [A]  time = 0.0987593, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6387, 6378, 30} \[ \frac{3 x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-b^2 x^2\right )}{4 \sqrt{\pi } b^3}-\frac{x^3 e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^2}-\frac{3 x e^{-b^2 x^2} \text{Erfi}(b x)}{4 b^4}+\frac{3 x^2}{4 \sqrt{\pi } b^3}+\frac{x^4}{4 \sqrt{\pi } b} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*Erfi[b*x])/E^(b^2*x^2),x]

[Out]

(3*x^2)/(4*b^3*Sqrt[Pi]) + x^4/(4*b*Sqrt[Pi]) - (3*x*Erfi[b*x])/(4*b^4*E^(b^2*x^2)) - (x^3*Erfi[b*x])/(2*b^2*E
^(b^2*x^2)) + (3*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, -(b^2*x^2)])/(4*b^3*Sqrt[Pi])

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 6378

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(b_.)*(x_)], x_Symbol] :> Simp[(b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}
, -(b^2*x^2)])/Sqrt[Pi], x] /; FreeQ[{b, c, d}, 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]

Rubi steps

\begin{align*} \int e^{-b^2 x^2} x^4 \text{erfi}(b x) \, dx &=-\frac{e^{-b^2 x^2} x^3 \text{erfi}(b x)}{2 b^2}+\frac{3 \int e^{-b^2 x^2} x^2 \text{erfi}(b x) \, dx}{2 b^2}+\frac{\int x^3 \, dx}{b \sqrt{\pi }}\\ &=\frac{x^4}{4 b \sqrt{\pi }}-\frac{3 e^{-b^2 x^2} x \text{erfi}(b x)}{4 b^4}-\frac{e^{-b^2 x^2} x^3 \text{erfi}(b x)}{2 b^2}+\frac{3 \int e^{-b^2 x^2} \text{erfi}(b x) \, dx}{4 b^4}+\frac{3 \int x \, dx}{2 b^3 \sqrt{\pi }}\\ &=\frac{3 x^2}{4 b^3 \sqrt{\pi }}+\frac{x^4}{4 b \sqrt{\pi }}-\frac{3 e^{-b^2 x^2} x \text{erfi}(b x)}{4 b^4}-\frac{e^{-b^2 x^2} x^3 \text{erfi}(b x)}{2 b^2}+\frac{3 x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-b^2 x^2\right )}{4 b^3 \sqrt{\pi }}\\ \end{align*}

Mathematica [A]  time = 0.0196826, size = 43, normalized size = 0.39 \[ \frac{x^2 \left (-\text{HypergeometricPFQ}\left (\{1,1\},\left \{-\frac{1}{2},2\right \},-b^2 x^2\right )+b^2 x^2+1\right )}{4 \sqrt{\pi } b^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x^2*(1 + b^2*x^2 - HypergeometricPFQ[{1, 1}, {-1/2, 2}, -(b^2*x^2)]))/(4*b^3*Sqrt[Pi])

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Maple [F]  time = 0.26, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{4}{\it erfi} \left ( bx \right ) }{{{\rm e}^{{b}^{2}{x}^{2}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^4*erfi(b*x)/exp(b^2*x^2),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}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{4} \operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^4*erfi(b*x)*e^(-b^2*x^2), x)

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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(x**4*erfi(b*x)/exp(b**2*x**2),x)

[Out]

Timed out

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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}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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