\(\int x^5 \text {erf}(b x) \, dx\) [1]

Optimal result

Integrand size = 8, antiderivative size = 96 \[ \int x^5 \text {erf}(b x) \, dx=\frac {5 e^{-b^2 x^2} x}{8 b^5 \sqrt {\pi }}+\frac {5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}-\frac {5 \text {erf}(b x)}{16 b^6}+\frac {1}{6} x^6 \text {erf}(b x) \]

[Out]

-5/16*erf(b*x)/b^6+1/6*x^6*erf(b*x)+5/8*x/b^5/exp(b^2*x^2)/Pi^(1/2)+5/12*x^3/b^3/exp(b^2*x^2)/Pi^(1/2)+1/6*x^5
/b/exp(b^2*x^2)/Pi^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6496, 2243, 2236} \[ \int x^5 \text {erf}(b x) \, dx=-\frac {5 \text {erf}(b x)}{16 b^6}+\frac {x^5 e^{-b^2 x^2}}{6 \sqrt {\pi } b}+\frac {5 x e^{-b^2 x^2}}{8 \sqrt {\pi } b^5}+\frac {5 x^3 e^{-b^2 x^2}}{12 \sqrt {\pi } b^3}+\frac {1}{6} x^6 \text {erf}(b x) \]

[In]

Int[x^5*Erf[b*x],x]

[Out]

(5*x)/(8*b^5*E^(b^2*x^2)*Sqrt[Pi]) + (5*x^3)/(12*b^3*E^(b^2*x^2)*Sqrt[Pi]) + x^5/(6*b*E^(b^2*x^2)*Sqrt[Pi]) -
(5*Erf[b*x])/(16*b^6) + (x^6*Erf[b*x])/6

Rule 2236

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

Rule 2243

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])

Rule 6496

Int[Erf[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Erf[a + b*x]/(d*(
m + 1))), x] - Dist[2*(b/(Sqrt[Pi]*d*(m + 1))), Int[(c + d*x)^(m + 1)/E^(a + b*x)^2, x], x] /; FreeQ[{a, b, c,
d, m}, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^6 \text {erf}(b x)-\frac {b \int e^{-b^2 x^2} x^6 \, dx}{3 \sqrt {\pi }} \\ & = \frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erf}(b x)-\frac {5 \int e^{-b^2 x^2} x^4 \, dx}{6 b \sqrt {\pi }} \\ & = \frac {5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erf}(b x)-\frac {5 \int e^{-b^2 x^2} x^2 \, dx}{4 b^3 \sqrt {\pi }} \\ & = \frac {5 e^{-b^2 x^2} x}{8 b^5 \sqrt {\pi }}+\frac {5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erf}(b x)-\frac {5 \int e^{-b^2 x^2} \, dx}{8 b^5 \sqrt {\pi }} \\ & = \frac {5 e^{-b^2 x^2} x}{8 b^5 \sqrt {\pi }}+\frac {5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}-\frac {5 \text {erf}(b x)}{16 b^6}+\frac {1}{6} x^6 \text {erf}(b x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.75 \[ \int x^5 \text {erf}(b x) \, dx=\frac {e^{-b^2 x^2} \left (30 b x+20 b^3 x^3+8 b^5 x^5+e^{b^2 x^2} \sqrt {\pi } \left (-15+8 b^6 x^6\right ) \text {erf}(b x)\right )}{48 b^6 \sqrt {\pi }} \]

[In]

Integrate[x^5*Erf[b*x],x]

[Out]

(30*b*x + 20*b^3*x^3 + 8*b^5*x^5 + E^(b^2*x^2)*Sqrt[Pi]*(-15 + 8*b^6*x^6)*Erf[b*x])/(48*b^6*E^(b^2*x^2)*Sqrt[P
i])

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.62

method	result	size
meijerg	\(\frac {\frac {x b \left (28 b^{4} x^{4}+70 b^{2} x^{2}+105\right ) {\mathrm e}^{-b^{2} x^{2}}}{84}-\frac {\left (-56 b^{6} x^{6}+105\right ) \operatorname {erf}\left (b x \right ) \sqrt {\pi }}{168}}{2 b^{6} \sqrt {\pi }}\)	\(60\)
parallelrisch	\(\frac {8 \,\operatorname {erf}\left (b x \right ) x^{6} b^{6} \sqrt {\pi }+8 \,{\mathrm e}^{-b^{2} x^{2}} x^{5} b^{5}+20 x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}+30 \,{\mathrm e}^{-b^{2} x^{2}} b x -15 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{48 b^{6} \sqrt {\pi }}\)	\(81\)
derivativedivides	\(\frac {\frac {\operatorname {erf}\left (b x \right ) b^{6} x^{6}}{6}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{5} b^{5}}{2}-\frac {5 x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}}{4}-\frac {15 \,{\mathrm e}^{-b^{2} x^{2}} b x}{8}+\frac {15 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{16}}{3 \sqrt {\pi }}}{b^{6}}\)	\(83\)
default	\(\frac {\frac {\operatorname {erf}\left (b x \right ) b^{6} x^{6}}{6}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{5} b^{5}}{2}-\frac {5 x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}}{4}-\frac {15 \,{\mathrm e}^{-b^{2} x^{2}} b x}{8}+\frac {15 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{16}}{3 \sqrt {\pi }}}{b^{6}}\)	\(83\)
parts	\(\frac {x^{6} \operatorname {erf}\left (b x \right )}{6}-\frac {b \left (-\frac {x^{5} {\mathrm e}^{-b^{2} x^{2}}}{2 b^{2}}+\frac {-\frac {5 x^{3} {\mathrm e}^{-b^{2} x^{2}}}{4 b^{2}}+\frac {5 \left (-\frac {3 x \,{\mathrm e}^{-b^{2} x^{2}}}{4 b^{2}}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (b x \right )}{8 b^{3}}\right )}{2 b^{2}}}{b^{2}}\right )}{3 \sqrt {\pi }}\)	\(91\)

[In]

int(x^5*erf(b*x),x,method=_RETURNVERBOSE)

[Out]

1/2/b^6/Pi^(1/2)*(1/84*x*b*(28*b^4*x^4+70*b^2*x^2+105)*exp(-b^2*x^2)-1/168*(-56*b^6*x^6+105)*erf(b*x)*Pi^(1/2)
)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.66 \[ \int x^5 \text {erf}(b x) \, dx=\frac {2 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} + 10 \, b^{3} x^{3} + 15 \, b x\right )} e^{\left (-b^{2} x^{2}\right )} - {\left (15 \, \pi - 8 \, \pi b^{6} x^{6}\right )} \operatorname {erf}\left (b x\right )}{48 \, \pi b^{6}} \]

[In]

integrate(x^5*erf(b*x),x, algorithm="fricas")

[Out]

1/48*(2*sqrt(pi)*(4*b^5*x^5 + 10*b^3*x^3 + 15*b*x)*e^(-b^2*x^2) - (15*pi - 8*pi*b^6*x^6)*erf(b*x))/(pi*b^6)

Sympy [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.92 \[ \int x^5 \text {erf}(b x) \, dx=\begin {cases} \frac {x^{6} \operatorname {erf}{\left (b x \right )}}{6} + \frac {x^{5} e^{- b^{2} x^{2}}}{6 \sqrt {\pi } b} + \frac {5 x^{3} e^{- b^{2} x^{2}}}{12 \sqrt {\pi } b^{3}} + \frac {5 x e^{- b^{2} x^{2}}}{8 \sqrt {\pi } b^{5}} - \frac {5 \operatorname {erf}{\left (b x \right )}}{16 b^{6}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(x**5*erf(b*x),x)

[Out]

Piecewise((x**6*erf(b*x)/6 + x**5*exp(-b**2*x**2)/(6*sqrt(pi)*b) + 5*x**3*exp(-b**2*x**2)/(12*sqrt(pi)*b**3) +
5*x*exp(-b**2*x**2)/(8*sqrt(pi)*b**5) - 5*erf(b*x)/(16*b**6), Ne(b, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.66 \[ \int x^5 \text {erf}(b x) \, dx=\frac {1}{6} \, x^{6} \operatorname {erf}\left (b x\right ) + \frac {b {\left (\frac {2 \, {\left (4 \, b^{4} x^{5} + 10 \, b^{2} x^{3} + 15 \, x\right )} e^{\left (-b^{2} x^{2}\right )}}{b^{6}} - \frac {15 \, \sqrt {\pi } \operatorname {erf}\left (b x\right )}{b^{7}}\right )}}{48 \, \sqrt {\pi }} \]

[In]

integrate(x^5*erf(b*x),x, algorithm="maxima")

[Out]

1/6*x^6*erf(b*x) + 1/48*b*(2*(4*b^4*x^5 + 10*b^2*x^3 + 15*x)*e^(-b^2*x^2)/b^6 - 15*sqrt(pi)*erf(b*x)/b^7)/sqrt
(pi)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.67 \[ \int x^5 \text {erf}(b x) \, dx=\frac {1}{6} \, x^{6} \operatorname {erf}\left (b x\right ) + \frac {b {\left (\frac {2 \, {\left (4 \, b^{4} x^{5} + 10 \, b^{2} x^{3} + 15 \, x\right )} e^{\left (-b^{2} x^{2}\right )}}{b^{6}} + \frac {15 \, \sqrt {\pi } \operatorname {erf}\left (-b x\right )}{b^{7}}\right )}}{48 \, \sqrt {\pi }} \]

[In]

integrate(x^5*erf(b*x),x, algorithm="giac")

[Out]

1/6*x^6*erf(b*x) + 1/48*b*(2*(4*b^4*x^5 + 10*b^2*x^3 + 15*x)*e^(-b^2*x^2)/b^6 + 15*sqrt(pi)*erf(-b*x)/b^7)/sqr
t(pi)

Mupad [B] (verification not implemented)

Time = 5.19 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.12 \[ \int x^5 \text {erf}(b x) \, dx=\frac {x^6\,\mathrm {erf}\left (b\,x\right )}{6}-\frac {5\,b\,x^7}{16\,{\left (b^2\,x^2\right )}^{7/2}}+\frac {x^5\,{\mathrm {e}}^{-b^2\,x^2}}{6\,b\,\sqrt {\pi }}+\frac {5\,x^3\,{\mathrm {e}}^{-b^2\,x^2}}{12\,b^3\,\sqrt {\pi }}+\frac {5\,x\,{\mathrm {e}}^{-b^2\,x^2}}{8\,b^5\,\sqrt {\pi }}+\frac {5\,b\,x^7\,\mathrm {erfc}\left (\sqrt {b^2\,x^2}\right )}{16\,{\left (b^2\,x^2\right )}^{7/2}} \]

[In]

int(x^5*erf(b*x),x)

[Out]

(x^6*erf(b*x))/6 - (5*b*x^7)/(16*(b^2*x^2)^(7/2)) + (x^5*exp(-b^2*x^2))/(6*b*pi^(1/2)) + (5*x^3*exp(-b^2*x^2))
/(12*b^3*pi^(1/2)) + (5*x*exp(-b^2*x^2))/(8*b^5*pi^(1/2)) + (5*b*x^7*erfc((b^2*x^2)^(1/2)))/(16*(b^2*x^2)^(7/2
))