3.64 \(\int e^{c+b^2 x^2} x^5 \text {erf}(b x) \, dx\)

Optimal. Leaf size=118 \[ -\frac {2 e^c x}{\sqrt {\pi } b^5}+\frac {2 e^c x^3}{3 \sqrt {\pi } b^3}+\frac {x^4 e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}+\frac {e^{b^2 x^2+c} \text {erf}(b x)}{b^6}-\frac {x^2 e^{b^2 x^2+c} \text {erf}(b x)}{b^4}-\frac {e^c x^5}{5 \sqrt {\pi } b} \]

[Out]

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

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Rubi [A]  time = 0.14, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6385, 6382, 8, 12, 30} \[ \frac {x^4 e^{b^2 x^2+c} \text {Erf}(b x)}{2 b^2}-\frac {x^2 e^{b^2 x^2+c} \text {Erf}(b x)}{b^4}+\frac {e^{b^2 x^2+c} \text {Erf}(b x)}{b^6}+\frac {2 e^c x^3}{3 \sqrt {\pi } b^3}-\frac {2 e^c x}{\sqrt {\pi } b^5}-\frac {e^c x^5}{5 \sqrt {\pi } b} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + b^2*x^2)*x^5*Erf[b*x],x]

[Out]

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

Rule 8

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

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

Rule 6382

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Simp[(E^(c + d*x^2)*Erf[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 6385

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m - 1)*E^(c + d*x^2)*Erf
[a + b*x])/(2*d), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erf[a + b*x], x], x] - Dist[b/(d*Sqrt
[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^{c+b^2 x^2} x^5 \text {erf}(b x) \, dx &=\frac {e^{c+b^2 x^2} x^4 \text {erf}(b x)}{2 b^2}-\frac {2 \int e^{c+b^2 x^2} x^3 \text {erf}(b x) \, dx}{b^2}-\frac {\int e^c x^4 \, dx}{b \sqrt {\pi }}\\ &=-\frac {e^{c+b^2 x^2} x^2 \text {erf}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erf}(b x)}{2 b^2}+\frac {2 \int e^{c+b^2 x^2} x \text {erf}(b x) \, dx}{b^4}+\frac {2 \int e^c x^2 \, dx}{b^3 \sqrt {\pi }}-\frac {e^c \int x^4 \, dx}{b \sqrt {\pi }}\\ &=-\frac {e^c x^5}{5 b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erf}(b x)}{b^6}-\frac {e^{c+b^2 x^2} x^2 \text {erf}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erf}(b x)}{2 b^2}-\frac {2 \int e^c \, dx}{b^5 \sqrt {\pi }}+\frac {\left (2 e^c\right ) \int x^2 \, dx}{b^3 \sqrt {\pi }}\\ &=-\frac {2 e^c x}{b^5 \sqrt {\pi }}+\frac {2 e^c x^3}{3 b^3 \sqrt {\pi }}-\frac {e^c x^5}{5 b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erf}(b x)}{b^6}-\frac {e^{c+b^2 x^2} x^2 \text {erf}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erf}(b x)}{2 b^2}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 73, normalized size = 0.62 \[ \frac {e^c \left (-6 b^5 x^5+20 b^3 x^3+15 \sqrt {\pi } e^{b^2 x^2} \left (b^4 x^4-2 b^2 x^2+2\right ) \text {erf}(b x)-60 b x\right )}{30 \sqrt {\pi } b^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.52, size = 74, normalized size = 0.63 \[ \frac {15 \, {\left (2 \, \pi + \pi b^{4} x^{4} - 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} - 2 \, \sqrt {\pi } {\left (3 \, b^{5} x^{5} - 10 \, b^{3} x^{3} + 30 \, b x\right )} e^{c}}{30 \, \pi b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.23, size = 118, normalized size = 1.00 \[ \frac {1}{2} \, {\left (\frac {c^{2} e^{\left (b^{2} x^{2} + c\right )}}{b^{6}} - \frac {{\left (2 \, b^{2} x^{2} - {\left (b^{2} x^{2} + c\right )}^{2} + 2 \, {\left (b^{2} x^{2} + c\right )} c - 2\right )} e^{\left (b^{2} x^{2} + c\right )}}{b^{6}}\right )} \operatorname {erf}\left (b x\right ) - \frac {3 \, \sqrt {\pi } b^{4} x^{5} e^{c} - 10 \, \sqrt {\pi } b^{2} x^{3} e^{c} + 30 \, \sqrt {\pi } x e^{c}}{15 \, \pi b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.07, size = 88, normalized size = 0.75 \[ \frac {\frac {\erf \left (b x \right ) {\mathrm e}^{c} \left (\frac {{\mathrm e}^{b^{2} x^{2}} b^{4} x^{4}}{2}-b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}+{\mathrm e}^{b^{2} x^{2}}\right )}{b^{5}}-\frac {{\mathrm e}^{c} \left (\frac {1}{5} b^{5} x^{5}-\frac {2}{3} b^{3} x^{3}+2 b x \right )}{\sqrt {\pi }\, b^{5}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.45, size = 82, normalized size = 0.69 \[ -\frac {6 \, b^{5} x^{5} e^{c} - 20 \, b^{3} x^{3} e^{c} - 15 \, {\left (\sqrt {\pi } b^{4} x^{4} e^{c} - 2 \, \sqrt {\pi } b^{2} x^{2} e^{c} + 2 \, \sqrt {\pi } e^{c}\right )} \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} + 60 \, b x e^{c}}{30 \, \sqrt {\pi } b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(6*b^5*x^5*e^c - 20*b^3*x^3*e^c - 15*(sqrt(pi)*b^4*x^4*e^c - 2*sqrt(pi)*b^2*x^2*e^c + 2*sqrt(pi)*e^c)*er
f(b*x)*e^(b^2*x^2) + 60*b*x*e^c)/(sqrt(pi)*b^6)

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mupad [B]  time = 0.30, size = 91, normalized size = 0.77 \[ \mathrm {erf}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{b^2\,x^2+c}}{b^6}+\frac {x^4\,{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^2}-\frac {x^2\,{\mathrm {e}}^{b^2\,x^2+c}}{b^4}\right )-\frac {3\,{\mathrm {e}}^c\,b^4\,x^5-10\,{\mathrm {e}}^c\,b^2\,x^3+30\,{\mathrm {e}}^c\,x}{15\,b^5\,\sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*exp(c + b^2*x^2)*erf(b*x),x)

[Out]

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

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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(exp(b**2*x**2+c)*x**5*erf(b*x),x)

[Out]

Timed out

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