3.3.70 \(\int e^{-b^2 x^2} x^5 \text {Erfi}(b x) \, dx\) [270]

Optimal. Leaf size=107 \[ \frac {2 x}{b^5 \sqrt {\pi }}+\frac {2 x^3}{3 b^3 \sqrt {\pi }}+\frac {x^5}{5 b \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {Erfi}(b x)}{b^6}-\frac {e^{-b^2 x^2} x^2 \text {Erfi}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {Erfi}(b x)}{2 b^2} \]

[Out]

-erfi(b*x)/b^6/exp(b^2*x^2)-x^2*erfi(b*x)/b^4/exp(b^2*x^2)-1/2*x^4*erfi(b*x)/b^2/exp(b^2*x^2)+2*x/b^5/Pi^(1/2)
+2/3*x^3/b^3/Pi^(1/2)+1/5*x^5/b/Pi^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6522, 6519, 8, 30} \begin {gather*} \frac {2 x}{\sqrt {\pi } b^5}+\frac {2 x^3}{3 \sqrt {\pi } b^3}-\frac {x^4 e^{-b^2 x^2} \text {Erfi}(b x)}{2 b^2}-\frac {e^{-b^2 x^2} \text {Erfi}(b x)}{b^6}-\frac {x^2 e^{-b^2 x^2} \text {Erfi}(b x)}{b^4}+\frac {x^5}{5 \sqrt {\pi } b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^5*Erfi[b*x])/E^(b^2*x^2),x]

[Out]

(2*x)/(b^5*Sqrt[Pi]) + (2*x^3)/(3*b^3*Sqrt[Pi]) + x^5/(5*b*Sqrt[Pi]) - Erfi[b*x]/(b^6*E^(b^2*x^2)) - (x^2*Erfi
[b*x])/(b^4*E^(b^2*x^2)) - (x^4*Erfi[b*x])/(2*b^2*E^(b^2*x^2))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

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

Rule 6519

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 6522

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^5 \text {erfi}(b x) \, dx &=-\frac {e^{-b^2 x^2} x^4 \text {erfi}(b x)}{2 b^2}+\frac {2 \int e^{-b^2 x^2} x^3 \text {erfi}(b x) \, dx}{b^2}+\frac {\int x^4 \, dx}{b \sqrt {\pi }}\\ &=\frac {x^5}{5 b \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^2 \text {erfi}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erfi}(b x)}{2 b^2}+\frac {2 \int e^{-b^2 x^2} x \text {erfi}(b x) \, dx}{b^4}+\frac {2 \int x^2 \, dx}{b^3 \sqrt {\pi }}\\ &=\frac {2 x^3}{3 b^3 \sqrt {\pi }}+\frac {x^5}{5 b \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{b^6}-\frac {e^{-b^2 x^2} x^2 \text {erfi}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erfi}(b x)}{2 b^2}+\frac {2 \int 1 \, dx}{b^5 \sqrt {\pi }}\\ &=\frac {2 x}{b^5 \sqrt {\pi }}+\frac {2 x^3}{3 b^3 \sqrt {\pi }}+\frac {x^5}{5 b \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{b^6}-\frac {e^{-b^2 x^2} x^2 \text {erfi}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erfi}(b x)}{2 b^2}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 68, normalized size = 0.64 \begin {gather*} \frac {\frac {60 b x+20 b^3 x^3+6 b^5 x^5}{\sqrt {\pi }}-15 e^{-b^2 x^2} \left (2+2 b^2 x^2+b^4 x^4\right ) \text {Erfi}(b x)}{30 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^5*Erfi[b*x])/E^(b^2*x^2),x]

[Out]

((60*b*x + 20*b^3*x^3 + 6*b^5*x^5)/Sqrt[Pi] - (15*(2 + 2*b^2*x^2 + b^4*x^4)*Erfi[b*x])/E^(b^2*x^2))/(30*b^6)

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Maple [A]
time = 0.40, size = 103, normalized size = 0.96

method	result	size
default	\(\frac {\left (6 x^{5} {\mathrm e}^{b^{2} x^{2}} b^{5}-15 \erfi \left (b x \right ) x^{4} b^{4} \sqrt {\pi }+20 b^{3} x^{3} {\mathrm e}^{b^{2} x^{2}}-30 \erfi \left (b x \right ) \sqrt {\pi }\, b^{2} x^{2}+60 \,{\mathrm e}^{b^{2} x^{2}} b x -30 \erfi \left (b x \right ) \sqrt {\pi }\right ) {\mathrm e}^{-b^{2} x^{2}}}{30 b^{6} \sqrt {\pi }}\)	\(103\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*erfi(b*x)/exp(b^2*x^2),x,method=_RETURNVERBOSE)

[Out]

1/30*(6*x^5*exp(b^2*x^2)*b^5-15*erfi(b*x)*x^4*b^4*Pi^(1/2)+20*b^3*x^3*exp(b^2*x^2)-30*erfi(b*x)*Pi^(1/2)*b^2*x
^2+60*exp(b^2*x^2)*b*x-30*erfi(b*x)*Pi^(1/2))/b^6/Pi^(1/2)/exp(b^2*x^2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^5*erfi(b*x)*e^(-b^2*x^2), x)

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Fricas [A]
time = 0.39, size = 79, normalized size = 0.74 \begin {gather*} \frac {{\left (2 \, \sqrt {\pi } {\left (3 \, b^{5} x^{5} + 10 \, b^{3} x^{3} + 30 \, b x\right )} e^{\left (b^{2} x^{2}\right )} - 15 \, {\left (2 \, \pi + \pi b^{4} x^{4} + 2 \, \pi b^{2} x^{2}\right )} \operatorname {erfi}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{30 \, \pi b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(2*sqrt(pi)*(3*b^5*x^5 + 10*b^3*x^3 + 30*b*x)*e^(b^2*x^2) - 15*(2*pi + pi*b^4*x^4 + 2*pi*b^2*x^2)*erfi(b*
x))*e^(-b^2*x^2)/(pi*b^6)

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Sympy [A]
time = 138.69, size = 99, normalized size = 0.93 \begin {gather*} \begin {cases} \frac {x^{5}}{5 \sqrt {\pi } b} - \frac {x^{4} e^{- b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{2 b^{2}} + \frac {2 x^{3}}{3 \sqrt {\pi } b^{3}} - \frac {x^{2} e^{- b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{b^{4}} + \frac {2 x}{\sqrt {\pi } b^{5}} - \frac {e^{- b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{b^{6}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*erfi(b*x)/exp(b**2*x**2),x)

[Out]

Piecewise((x**5/(5*sqrt(pi)*b) - x**4*exp(-b**2*x**2)*erfi(b*x)/(2*b**2) + 2*x**3/(3*sqrt(pi)*b**3) - x**2*exp
(-b**2*x**2)*erfi(b*x)/b**4 + 2*x/(sqrt(pi)*b**5) - exp(-b**2*x**2)*erfi(b*x)/b**6, Ne(b, 0)), (0, True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^5*erfi(b*x)*e^(-b^2*x^2), x)

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Mupad [B]
time = 0.25, size = 82, normalized size = 0.77 \begin {gather*} \frac {3\,b^4\,x^5+10\,b^2\,x^3+30\,x}{15\,b^5\,\sqrt {\pi }}-\mathrm {erfi}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{-b^2\,x^2}}{b^6}+\frac {x^4\,{\mathrm {e}}^{-b^2\,x^2}}{2\,b^2}+\frac {x^2\,{\mathrm {e}}^{-b^2\,x^2}}{b^4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*exp(-b^2*x^2)*erfi(b*x),x)

[Out]

(30*x + 10*b^2*x^3 + 3*b^4*x^5)/(15*b^5*pi^(1/2)) - erfi(b*x)*(exp(-b^2*x^2)/b^6 + (x^4*exp(-b^2*x^2))/(2*b^2)
+ (x^2*exp(-b^2*x^2))/b^4)

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