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

Optimal. Leaf size=109 \[ \frac {3 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{4 \sqrt {\pi } b^3}+\frac {3 x^2}{4 \sqrt {\pi } b^3}-\frac {x^3 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {3 x e^{-b^2 x^2} \text {erfi}(b x)}{4 b^4}+\frac {x^4}{4 \sqrt {\pi } b} \]

[Out]

-3/4*x*erfi(b*x)/b^4/exp(b^2*x^2)-1/2*x^3*erfi(b*x)/b^2/exp(b^2*x^2)+3/4*x^2/b^3/Pi^(1/2)+1/4*x^4/b/Pi^(1/2)+3
/4*x^2*HypergeometricPFQ([1, 1],[3/2, 2],-b^2*x^2)/b^3/Pi^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 109, normalized size of antiderivative = 1.00, 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 30

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

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 = 0.03, size = 43, normalized size = 0.39 \[ \frac {x^2 \left (-\, _2F_2\left (1,1;-\frac {1}{2},2;-b^2 x^2\right )+b^2 x^2+1\right )}{4 \sqrt {\pi } b^3} \]

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])

________________________________________________________________________________________

fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{4} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}, x\right ) \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \]

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)

________________________________________________________________________________________

maple [F]  time = 0.22, size = 0, normalized size = 0.00 \[ \int x^{4} \erfi \left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}\, dx \]

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)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^4*exp(-b^2*x^2)*erfi(b*x), x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

________________________________________________________________________________________