\(\int e^{c+d x^2} x^5 \text {erf}(b x) \, dx\) [53]

Optimal result

Integrand size = 17, antiderivative size = 285 \[ \int e^{c+d x^2} x^5 \text {erf}(b x) \, dx=-\frac {b e^{c-\left (b^2-d\right ) x^2} x}{\left (b^2-d\right ) d^2 \sqrt {\pi }}+\frac {3 b e^{c-\left (b^2-d\right ) x^2} x}{4 \left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {b e^{c-\left (b^2-d\right ) x^2} x^3}{2 \left (b^2-d\right ) d \sqrt {\pi }}+\frac {e^{c+d x^2} \text {erf}(b x)}{d^3}-\frac {e^{c+d x^2} x^2 \text {erf}(b x)}{d^2}+\frac {e^{c+d x^2} x^4 \text {erf}(b x)}{2 d}-\frac {b e^c \text {erf}\left (\sqrt {b^2-d} x\right )}{\sqrt {b^2-d} d^3}+\frac {b e^c \text {erf}\left (\sqrt {b^2-d} x\right )}{2 \left (b^2-d\right )^{3/2} d^2}-\frac {3 b e^c \text {erf}\left (\sqrt {b^2-d} x\right )}{8 \left (b^2-d\right )^{5/2} d} \]

[Out]

exp(d*x^2+c)*erf(b*x)/d^3-exp(d*x^2+c)*x^2*erf(b*x)/d^2+1/2*exp(d*x^2+c)*x^4*erf(b*x)/d+1/2*b*exp(c)*erf(x*(b^
2-d)^(1/2))/(b^2-d)^(3/2)/d^2-3/8*b*exp(c)*erf(x*(b^2-d)^(1/2))/(b^2-d)^(5/2)/d-b*exp(c)*erf(x*(b^2-d)^(1/2))/
d^3/(b^2-d)^(1/2)-b*exp(c-(b^2-d)*x^2)*x/(b^2-d)/d^2/Pi^(1/2)+3/4*b*exp(c-(b^2-d)*x^2)*x/(b^2-d)^2/d/Pi^(1/2)+
1/2*b*exp(c-(b^2-d)*x^2)*x^3/(b^2-d)/d/Pi^(1/2)

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6520, 6517, 2236, 2243} \[ \int e^{c+d x^2} x^5 \text {erf}(b x) \, dx=-\frac {b e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{d^3 \sqrt {b^2-d}}+\frac {b e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{2 d^2 \left (b^2-d\right )^{3/2}}-\frac {b x e^{c-x^2 \left (b^2-d\right )}}{\sqrt {\pi } d^2 \left (b^2-d\right )}-\frac {3 b e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{8 d \left (b^2-d\right )^{5/2}}+\frac {3 b x e^{c-x^2 \left (b^2-d\right )}}{4 \sqrt {\pi } d \left (b^2-d\right )^2}+\frac {b x^3 e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt {\pi } d \left (b^2-d\right )}+\frac {\text {erf}(b x) e^{c+d x^2}}{d^3}-\frac {x^2 \text {erf}(b x) e^{c+d x^2}}{d^2}+\frac {x^4 \text {erf}(b x) e^{c+d x^2}}{2 d} \]

[In]

Int[E^(c + d*x^2)*x^5*Erf[b*x],x]

[Out]

-((b*E^(c - (b^2 - d)*x^2)*x)/((b^2 - d)*d^2*Sqrt[Pi])) + (3*b*E^(c - (b^2 - d)*x^2)*x)/(4*(b^2 - d)^2*d*Sqrt[
Pi]) + (b*E^(c - (b^2 - d)*x^2)*x^3)/(2*(b^2 - d)*d*Sqrt[Pi]) + (E^(c + d*x^2)*Erf[b*x])/d^3 - (E^(c + d*x^2)*
x^2*Erf[b*x])/d^2 + (E^(c + d*x^2)*x^4*Erf[b*x])/(2*d) - (b*E^c*Erf[Sqrt[b^2 - d]*x])/(Sqrt[b^2 - d]*d^3) + (b
*E^c*Erf[Sqrt[b^2 - d]*x])/(2*(b^2 - d)^(3/2)*d^2) - (3*b*E^c*Erf[Sqrt[b^2 - d]*x])/(8*(b^2 - d)^(5/2)*d)

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 6517

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 6520

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*} \text {integral}& = \frac {e^{c+d x^2} x^4 \text {erf}(b x)}{2 d}-\frac {2 \int e^{c+d x^2} x^3 \text {erf}(b x) \, dx}{d}-\frac {b \int e^{c-\left (b^2-d\right ) x^2} x^4 \, dx}{d \sqrt {\pi }} \\ & = \frac {b e^{c-\left (b^2-d\right ) x^2} x^3}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {e^{c+d x^2} x^2 \text {erf}(b x)}{d^2}+\frac {e^{c+d x^2} x^4 \text {erf}(b x)}{2 d}+\frac {2 \int e^{c+d x^2} x \text {erf}(b x) \, dx}{d^2}+\frac {(2 b) \int e^{c-\left (b^2-d\right ) x^2} x^2 \, dx}{d^2 \sqrt {\pi }}-\frac {(3 b) \int e^{c+\left (-b^2+d\right ) x^2} x^2 \, dx}{2 \left (b^2-d\right ) d \sqrt {\pi }} \\ & = -\frac {b e^{c-\left (b^2-d\right ) x^2} x}{\left (b^2-d\right ) d^2 \sqrt {\pi }}+\frac {3 b e^{c-\left (b^2-d\right ) x^2} x}{4 \left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {b e^{c-\left (b^2-d\right ) x^2} x^3}{2 \left (b^2-d\right ) d \sqrt {\pi }}+\frac {e^{c+d x^2} \text {erf}(b x)}{d^3}-\frac {e^{c+d x^2} x^2 \text {erf}(b x)}{d^2}+\frac {e^{c+d x^2} x^4 \text {erf}(b x)}{2 d}-\frac {(2 b) \int e^{c-\left (b^2-d\right ) x^2} \, dx}{d^3 \sqrt {\pi }}+\frac {b \int e^{c+\left (-b^2+d\right ) x^2} \, dx}{\left (b^2-d\right ) d^2 \sqrt {\pi }}-\frac {(3 b) \int e^{c+\left (-b^2+d\right ) x^2} \, dx}{4 \left (b^2-d\right )^2 d \sqrt {\pi }} \\ & = -\frac {b e^{c-\left (b^2-d\right ) x^2} x}{\left (b^2-d\right ) d^2 \sqrt {\pi }}+\frac {3 b e^{c-\left (b^2-d\right ) x^2} x}{4 \left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {b e^{c-\left (b^2-d\right ) x^2} x^3}{2 \left (b^2-d\right ) d \sqrt {\pi }}+\frac {e^{c+d x^2} \text {erf}(b x)}{d^3}-\frac {e^{c+d x^2} x^2 \text {erf}(b x)}{d^2}+\frac {e^{c+d x^2} x^4 \text {erf}(b x)}{2 d}-\frac {b e^c \text {erf}\left (\sqrt {b^2-d} x\right )}{\sqrt {b^2-d} d^3}+\frac {b e^c \text {erf}\left (\sqrt {b^2-d} x\right )}{2 \left (b^2-d\right )^{3/2} d^2}-\frac {3 b e^c \text {erf}\left (\sqrt {b^2-d} x\right )}{8 \left (b^2-d\right )^{5/2} d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.48 \[ \int e^{c+d x^2} x^5 \text {erf}(b x) \, dx=\frac {e^c \left (\frac {2 b d e^{\left (-b^2+d\right ) x^2} x \left (d \left (7-2 d x^2\right )+2 b^2 \left (-2+d x^2\right )\right )}{\left (b^2-d\right )^2 \sqrt {\pi }}+4 e^{d x^2} \left (2-2 d x^2+d^2 x^4\right ) \text {erf}(b x)+\frac {b \left (-8 b^4+20 b^2 d-15 d^2\right ) \text {erfi}\left (\sqrt {-b^2+d} x\right )}{\left (-b^2+d\right )^{5/2}}\right )}{8 d^3} \]

[In]

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

[Out]

(E^c*((2*b*d*E^((-b^2 + d)*x^2)*x*(d*(7 - 2*d*x^2) + 2*b^2*(-2 + d*x^2)))/((b^2 - d)^2*Sqrt[Pi]) + 4*E^(d*x^2)
*(2 - 2*d*x^2 + d^2*x^4)*Erf[b*x] + (b*(-8*b^4 + 20*b^2*d - 15*d^2)*Erfi[Sqrt[-b^2 + d]*x])/(-b^2 + d)^(5/2)))
/(8*d^3)

Maple [A] (verified)

Time = 5.24 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.09

method	result	size
default	\(\frac {\frac {\operatorname {erf}\left (b x \right ) {\mathrm e}^{c} \left (\frac {{\mathrm e}^{d \,x^{2}} b^{6} x^{4}}{2 d}-\frac {2 b^{2} \left (\frac {b^{4} x^{2} {\mathrm e}^{d \,x^{2}}}{2 d}-\frac {b^{4} {\mathrm e}^{d \,x^{2}}}{2 d^{2}}\right )}{d}\right )}{b^{5}}-\frac {{\mathrm e}^{c} \left (\frac {b^{2} \left (\frac {b^{3} x^{3} {\mathrm e}^{\left (-1+\frac {d}{b^{2}}\right ) b^{2} x^{2}}}{-2+\frac {2 d}{b^{2}}}-\frac {3 \left (\frac {b x \,{\mathrm e}^{\left (-1+\frac {d}{b^{2}}\right ) b^{2} x^{2}}}{-2+\frac {2 d}{b^{2}}}-\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {1-\frac {d}{b^{2}}}\, b x \right )}{4 \left (-1+\frac {d}{b^{2}}\right ) \sqrt {1-\frac {d}{b^{2}}}}\right )}{2 \left (-1+\frac {d}{b^{2}}\right )}\right )}{d}+\frac {b^{6} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {1-\frac {d}{b^{2}}}\, b x \right )}{d^{3} \sqrt {1-\frac {d}{b^{2}}}}-\frac {2 b^{4} \left (\frac {b x \,{\mathrm e}^{\left (-1+\frac {d}{b^{2}}\right ) b^{2} x^{2}}}{-2+\frac {2 d}{b^{2}}}-\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {1-\frac {d}{b^{2}}}\, b x \right )}{4 \left (-1+\frac {d}{b^{2}}\right ) \sqrt {1-\frac {d}{b^{2}}}}\right )}{d^{2}}\right )}{\sqrt {\pi }\, b^{5}}}{b}\)	\(312\)

[In]

int(exp(d*x^2+c)*x^5*erf(b*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.91 \[ \int e^{c+d x^2} x^5 \text {erf}(b x) \, dx=-\frac {\pi {\left (8 \, b^{5} - 20 \, b^{3} d + 15 \, b d^{2}\right )} \sqrt {b^{2} - d} \operatorname {erf}\left (\sqrt {b^{2} - d} x\right ) e^{c} - 4 \, {\left (\pi {\left (b^{6} d^{2} - 3 \, b^{4} d^{3} + 3 \, b^{2} d^{4} - d^{5}\right )} x^{4} - 2 \, \pi {\left (b^{6} d - 3 \, b^{4} d^{2} + 3 \, b^{2} d^{3} - d^{4}\right )} x^{2} + 2 \, \pi {\left (b^{6} - 3 \, b^{4} d + 3 \, b^{2} d^{2} - d^{3}\right )}\right )} \operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )} - 2 \, \sqrt {\pi } {\left (2 \, {\left (b^{5} d^{2} - 2 \, b^{3} d^{3} + b d^{4}\right )} x^{3} - {\left (4 \, b^{5} d - 11 \, b^{3} d^{2} + 7 \, b d^{3}\right )} x\right )} e^{\left (-b^{2} x^{2} + d x^{2} + c\right )}}{8 \, \pi {\left (b^{6} d^{3} - 3 \, b^{4} d^{4} + 3 \, b^{2} d^{5} - d^{6}\right )}} \]

[In]

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

[Out]

-1/8*(pi*(8*b^5 - 20*b^3*d + 15*b*d^2)*sqrt(b^2 - d)*erf(sqrt(b^2 - d)*x)*e^c - 4*(pi*(b^6*d^2 - 3*b^4*d^3 + 3
*b^2*d^4 - d^5)*x^4 - 2*pi*(b^6*d - 3*b^4*d^2 + 3*b^2*d^3 - d^4)*x^2 + 2*pi*(b^6 - 3*b^4*d + 3*b^2*d^2 - d^3))
*erf(b*x)*e^(d*x^2 + c) - 2*sqrt(pi)*(2*(b^5*d^2 - 2*b^3*d^3 + b*d^4)*x^3 - (4*b^5*d - 11*b^3*d^2 + 7*b*d^3)*x
)*e^(-b^2*x^2 + d*x^2 + c))/(pi*(b^6*d^3 - 3*b^4*d^4 + 3*b^2*d^5 - d^6))

Sympy [F]

\[ \int e^{c+d x^2} x^5 \text {erf}(b x) \, dx=e^{c} \int x^{5} e^{d x^{2}} \operatorname {erf}{\left (b x \right )}\, dx \]

[In]

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

[Out]

exp(c)*Integral(x**5*exp(d*x**2)*erf(b*x), x)

Maxima [F]

\[ \int e^{c+d x^2} x^5 \text {erf}(b x) \, dx=\int { x^{5} \operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]

[In]

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

[Out]

1/2*(d^2*x^4*e^c - 2*d*x^2*e^c + 2*e^c)*erf(b*x)*e^(d*x^2)/d^3 - integrate((b*d^2*x^4*e^c - 2*b*d*x^2*e^c + 2*
b*e^c)*e^(-b^2*x^2 + d*x^2), x)/(sqrt(pi)*d^3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.95 \[ \int e^{c+d x^2} x^5 \text {erf}(b x) \, dx=\frac {1}{2} \, {\left (\frac {c^{2} e^{\left (d x^{2} + c\right )}}{d^{3}} - \frac {{\left (2 \, d x^{2} - {\left (d x^{2} + c\right )}^{2} + 2 \, {\left (d x^{2} + c\right )} c - 2\right )} e^{\left (d x^{2} + c\right )}}{d^{3}}\right )} \operatorname {erf}\left (b x\right ) + \frac {\sqrt {\pi } b d^{2} {\left (\frac {2 \, {\left (2 \, b^{2} x^{3} - 2 \, d x^{3} + 3 \, x\right )} e^{\left (-b^{2} x^{2} + d x^{2} + c\right )}}{b^{4} - 2 \, b^{2} d + d^{2}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {b^{2} - d} x\right ) e^{c}}{{\left (b^{4} - 2 \, b^{2} d + d^{2}\right )} \sqrt {b^{2} - d}}\right )} - 4 \, \sqrt {\pi } b d {\left (\frac {2 \, x e^{\left (-b^{2} x^{2} + d x^{2} + c\right )}}{b^{2} - d} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {b^{2} - d} x\right ) e^{c}}{{\left (b^{2} - d\right )}^{\frac {3}{2}}}\right )} + \frac {8 \, \pi b \operatorname {erf}\left (-\sqrt {b^{2} - d} x\right ) e^{c}}{\sqrt {b^{2} - d}}}{8 \, \pi d^{3}} \]

[In]

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

[Out]

1/2*(c^2*e^(d*x^2 + c)/d^3 - (2*d*x^2 - (d*x^2 + c)^2 + 2*(d*x^2 + c)*c - 2)*e^(d*x^2 + c)/d^3)*erf(b*x) + 1/8
*(sqrt(pi)*b*d^2*(2*(2*b^2*x^3 - 2*d*x^3 + 3*x)*e^(-b^2*x^2 + d*x^2 + c)/(b^4 - 2*b^2*d + d^2) + 3*sqrt(pi)*er
f(-sqrt(b^2 - d)*x)*e^c/((b^4 - 2*b^2*d + d^2)*sqrt(b^2 - d))) - 4*sqrt(pi)*b*d*(2*x*e^(-b^2*x^2 + d*x^2 + c)/
(b^2 - d) + sqrt(pi)*erf(-sqrt(b^2 - d)*x)*e^c/(b^2 - d)^(3/2)) + 8*pi*b*erf(-sqrt(b^2 - d)*x)*e^c/sqrt(b^2 -
d))/(pi*d^3)

Mupad [B] (verification not implemented)

Time = 0.88 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.86 \[ \int e^{c+d x^2} x^5 \text {erf}(b x) \, dx=\mathrm {erf}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{d\,x^2+c}}{d^3}-\frac {x^2\,{\mathrm {e}}^{d\,x^2+c}}{d^2}+\frac {x^4\,{\mathrm {e}}^{d\,x^2+c}}{2\,d}\right )-\frac {b\,{\mathrm {e}}^c\,\mathrm {erf}\left (x\,\sqrt {b^2-d}\right )}{d^3\,\sqrt {b^2-d}}-\frac {b\,{\mathrm {e}}^c\,\mathrm {erfi}\left (x\,\sqrt {d-b^2}\right )}{2\,d^2\,{\left (d-b^2\right )}^{3/2}}+\frac {b\,x\,{\mathrm {e}}^{-b^2\,x^2+d\,x^2+c}}{d^2\,\sqrt {\pi }\,\left (d-b^2\right )}+\frac {b\,x^5\,{\mathrm {e}}^c\,\left ({\mathrm {e}}^{d\,x^2-b^2\,x^2}\,\left (\frac {3\,\sqrt {-x^2\,\left (d-b^2\right )}}{2}+{\left (-x^2\,\left (d-b^2\right )\right )}^{3/2}\right )-\frac {3\,\sqrt {\pi }}{4}+\frac {3\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-x^2\,\left (d-b^2\right )}\right )}{4}\right )}{2\,d\,\sqrt {\pi }\,{\left (-x^2\,\left (d-b^2\right )\right )}^{5/2}} \]

[In]

int(x^5*exp(c + d*x^2)*erf(b*x),x)

[Out]

erf(b*x)*(exp(c + d*x^2)/d^3 - (x^2*exp(c + d*x^2))/d^2 + (x^4*exp(c + d*x^2))/(2*d)) - (b*exp(c)*erf(x*(b^2 -
d)^(1/2)))/(d^3*(b^2 - d)^(1/2)) - (b*exp(c)*erfi(x*(d - b^2)^(1/2)))/(2*d^2*(d - b^2)^(3/2)) + (b*x*exp(c +
d*x^2 - b^2*x^2))/(d^2*pi^(1/2)*(d - b^2)) + (b*x^5*exp(c)*(exp(d*x^2 - b^2*x^2)*((3*(-x^2*(d - b^2))^(1/2))/2
+ (-x^2*(d - b^2))^(3/2)) - (3*pi^(1/2))/4 + (3*pi^(1/2)*erfc((-x^2*(d - b^2))^(1/2)))/4))/(2*d*pi^(1/2)*(-x^
2*(d - b^2))^(5/2))