\(\int e^{c+b^2 x^2} x^5 \text {erfi}(b x) \, dx\) [283]

Optimal result

Integrand size = 19, antiderivative size = 144 \[ \int e^{c+b^2 x^2} x^5 \text {erfi}(b x) \, dx=\frac {11 e^{c+2 b^2 x^2} x}{16 b^5 \sqrt {\pi }}-\frac {e^{c+2 b^2 x^2} x^3}{4 b^3 \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{b^6}-\frac {e^{c+b^2 x^2} x^2 \text {erfi}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erfi}(b x)}{2 b^2}-\frac {43 e^c \text {erfi}\left (\sqrt {2} b x\right )}{32 \sqrt {2} b^6} \]

[Out]

exp(b^2*x^2+c)*erfi(b*x)/b^6-exp(b^2*x^2+c)*x^2*erfi(b*x)/b^4+1/2*exp(b^2*x^2+c)*x^4*erfi(b*x)/b^2-43/64*exp(c
)*erfi(b*x*2^(1/2))/b^6*2^(1/2)+11/16*exp(2*b^2*x^2+c)*x/b^5/Pi^(1/2)-1/4*exp(2*b^2*x^2+c)*x^3/b^3/Pi^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6522, 6519, 2235, 2243} \[ \int e^{c+b^2 x^2} x^5 \text {erfi}(b x) \, dx=-\frac {43 e^c \text {erfi}\left (\sqrt {2} b x\right )}{32 \sqrt {2} b^6}+\frac {x^4 e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}+\frac {e^{b^2 x^2+c} \text {erfi}(b x)}{b^6}+\frac {11 x e^{2 b^2 x^2+c}}{16 \sqrt {\pi } b^5}-\frac {x^2 e^{b^2 x^2+c} \text {erfi}(b x)}{b^4}-\frac {x^3 e^{2 b^2 x^2+c}}{4 \sqrt {\pi } b^3} \]

[In]

Int[E^(c + b^2*x^2)*x^5*Erfi[b*x],x]

[Out]

(11*E^(c + 2*b^2*x^2)*x)/(16*b^5*Sqrt[Pi]) - (E^(c + 2*b^2*x^2)*x^3)/(4*b^3*Sqrt[Pi]) + (E^(c + b^2*x^2)*Erfi[
b*x])/b^6 - (E^(c + b^2*x^2)*x^2*Erfi[b*x])/b^4 + (E^(c + b^2*x^2)*x^4*Erfi[b*x])/(2*b^2) - (43*E^c*Erfi[Sqrt[
2]*b*x])/(32*Sqrt[2]*b^6)

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[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 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*} \text {integral}& = \frac {e^{c+b^2 x^2} x^4 \text {erfi}(b x)}{2 b^2}-\frac {2 \int e^{c+b^2 x^2} x^3 \text {erfi}(b x) \, dx}{b^2}-\frac {\int e^{c+2 b^2 x^2} x^4 \, dx}{b \sqrt {\pi }} \\ & = -\frac {e^{c+2 b^2 x^2} x^3}{4 b^3 \sqrt {\pi }}-\frac {e^{c+b^2 x^2} x^2 \text {erfi}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erfi}(b x)}{2 b^2}+\frac {2 \int e^{c+b^2 x^2} x \text {erfi}(b x) \, dx}{b^4}+\frac {3 \int e^{c+2 b^2 x^2} x^2 \, dx}{4 b^3 \sqrt {\pi }}+\frac {2 \int e^{c+2 b^2 x^2} x^2 \, dx}{b^3 \sqrt {\pi }} \\ & = \frac {11 e^{c+2 b^2 x^2} x}{16 b^5 \sqrt {\pi }}-\frac {e^{c+2 b^2 x^2} x^3}{4 b^3 \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{b^6}-\frac {e^{c+b^2 x^2} x^2 \text {erfi}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erfi}(b x)}{2 b^2}-\frac {3 \int e^{c+2 b^2 x^2} \, dx}{16 b^5 \sqrt {\pi }}-\frac {\int e^{c+2 b^2 x^2} \, dx}{2 b^5 \sqrt {\pi }}-\frac {2 \int e^{c+2 b^2 x^2} \, dx}{b^5 \sqrt {\pi }} \\ & = \frac {11 e^{c+2 b^2 x^2} x}{16 b^5 \sqrt {\pi }}-\frac {e^{c+2 b^2 x^2} x^3}{4 b^3 \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{b^6}-\frac {e^{c+b^2 x^2} x^2 \text {erfi}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erfi}(b x)}{2 b^2}-\frac {43 e^c \text {erfi}\left (\sqrt {2} b x\right )}{32 \sqrt {2} b^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.66 \[ \int e^{c+b^2 x^2} x^5 \text {erfi}(b x) \, dx=\frac {e^c \left (-4 b e^{2 b^2 x^2} x \left (-11+4 b^2 x^2\right )+32 e^{b^2 x^2} \sqrt {\pi } \left (2-2 b^2 x^2+b^4 x^4\right ) \text {erfi}(b x)-43 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} b x\right )\right )}{64 b^6 \sqrt {\pi }} \]

[In]

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

[Out]

(E^c*(-4*b*E^(2*b^2*x^2)*x*(-11 + 4*b^2*x^2) + 32*E^(b^2*x^2)*Sqrt[Pi]*(2 - 2*b^2*x^2 + b^4*x^4)*Erfi[b*x] - 4
3*Sqrt[2*Pi]*Erfi[Sqrt[2]*b*x]))/(64*b^6*Sqrt[Pi])

Maple [F]

\[\int {\mathrm e}^{b^{2} x^{2}+c} x^{5} \operatorname {erfi}\left (b x \right )d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.74 \[ \int e^{c+b^2 x^2} x^5 \text {erfi}(b x) \, dx=\frac {43 \, \sqrt {2} \pi \sqrt {-b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {-b^{2}} x\right ) e^{c} + 32 \, {\left (\pi b^{5} x^{4} - 2 \, \pi b^{3} x^{2} + 2 \, \pi b\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} - 4 \, \sqrt {\pi } {\left (4 \, b^{4} x^{3} - 11 \, b^{2} x\right )} e^{\left (2 \, b^{2} x^{2} + c\right )}}{64 \, \pi b^{7}} \]

[In]

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

[Out]

1/64*(43*sqrt(2)*pi*sqrt(-b^2)*erf(sqrt(2)*sqrt(-b^2)*x)*e^c + 32*(pi*b^5*x^4 - 2*pi*b^3*x^2 + 2*pi*b)*erfi(b*
x)*e^(b^2*x^2 + c) - 4*sqrt(pi)*(4*b^4*x^3 - 11*b^2*x)*e^(2*b^2*x^2 + c))/(pi*b^7)

Sympy [F]

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

[In]

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

[Out]

exp(c)*Integral(x**5*exp(b**2*x**2)*erfi(b*x), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.27 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.43 \[ \int e^{c+b^2 x^2} x^5 \text {erfi}(b x) \, dx=\mathrm {erfi}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{b^2\,x^2+c}}{b^6}+\frac {x^4\,{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^2}-\frac {x^2\,{\mathrm {e}}^{b^2\,x^2+c}}{b^4}\right )-\frac {3\,x^5\,{\mathrm {e}}^c}{8\,b\,{\left (-2\,b^2\,x^2\right )}^{5/2}}+\frac {11\,x\,{\mathrm {e}}^{2\,b^2\,x^2+c}}{16\,b^5\,\sqrt {\pi }}-\frac {x^3\,{\mathrm {e}}^{2\,b^2\,x^2+c}}{4\,b^3\,\sqrt {\pi }}-\frac {\sqrt {2}\,{\mathrm {e}}^c\,\mathrm {erfi}\left (\sqrt {2}\,x\,\sqrt {b^2}\right )}{8\,b^3\,{\left (b^2\right )}^{3/2}}+\frac {3\,x^5\,{\mathrm {e}}^c\,\mathrm {erfc}\left (\sqrt {-2\,b^2\,x^2}\right )}{8\,b\,{\left (-2\,b^2\,x^2\right )}^{5/2}}-\frac {\sqrt {2}\,\mathrm {erf}\left (\sqrt {2}\,x\,\sqrt {-b^2}\right )\,{\mathrm {e}}^c}{2\,b\,{\left (-b^2\right )}^{5/2}} \]

[In]

int(x^5*exp(c + b^2*x^2)*erfi(b*x),x)

[Out]

erfi(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) - (3*x^5*exp(c)
)/(8*b*(-2*b^2*x^2)^(5/2)) + (11*x*exp(c + 2*b^2*x^2))/(16*b^5*pi^(1/2)) - (x^3*exp(c + 2*b^2*x^2))/(4*b^3*pi^
(1/2)) - (2^(1/2)*exp(c)*erfi(2^(1/2)*x*(b^2)^(1/2)))/(8*b^3*(b^2)^(3/2)) + (3*x^5*exp(c)*erfc((-2*b^2*x^2)^(1
/2)))/(8*b*(-2*b^2*x^2)^(5/2)) - (2^(1/2)*erf(2^(1/2)*x*(-b^2)^(1/2))*exp(c))/(2*b*(-b^2)^(5/2))