\(\int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx\) [172]

Optimal result

Integrand size = 19, antiderivative size = 134 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\frac {b e^c}{6 \sqrt {\pi } x^3}+\frac {b^3 e^c}{2 \sqrt {\pi } x}-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{4 x^4}-\frac {b^2 e^{c+b^2 x^2} \text {erfc}(b x)}{4 x^2}+\frac {1}{4} b^4 e^c \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\frac {b^5 e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }} \]

[Out]

1/4*b^4*exp(c)*Ei(b^2*x^2)-1/4*exp(b^2*x^2+c)*erfc(b*x)/x^4-1/4*b^2*exp(b^2*x^2+c)*erfc(b*x)/x^2+1/6*b*exp(c)/
x^3/Pi^(1/2)+1/2*b^3*exp(c)/x/Pi^(1/2)-b^5*exp(c)*x*hypergeom([1/2, 1],[3/2, 3/2],b^2*x^2)/Pi^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6527, 6524, 2241, 6523, 12, 30} \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=-\frac {b^5 e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }}+\frac {b^3 e^c}{2 \sqrt {\pi } x}-\frac {b^2 e^{b^2 x^2+c} \text {erfc}(b x)}{4 x^2}-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{4 x^4}+\frac {1}{4} b^4 e^c \operatorname {ExpIntegralEi}\left (b^2 x^2\right )+\frac {b e^c}{6 \sqrt {\pi } x^3} \]

[In]

Int[(E^(c + b^2*x^2)*Erfc[b*x])/x^5,x]

[Out]

(b*E^c)/(6*Sqrt[Pi]*x^3) + (b^3*E^c)/(2*Sqrt[Pi]*x) - (E^(c + b^2*x^2)*Erfc[b*x])/(4*x^4) - (b^2*E^(c + b^2*x^
2)*Erfc[b*x])/(4*x^2) + (b^4*E^c*ExpIntegralEi[b^2*x^2])/4 - (b^5*E^c*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}
, b^2*x^2])/Sqrt[Pi]

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 2241

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[F^a*(ExpIntegralEi[
b*(c + d*x)^n*Log[F]]/(f*n)), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rule 6523

Int[(E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)])/(x_), x_Symbol] :> Simp[2*b*E^c*(x/Sqrt[Pi])*HypergeometricPFQ[
{1/2, 1}, {3/2, 3/2}, b^2*x^2], x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]

Rule 6524

Int[(E^((c_.) + (d_.)*(x_)^2)*Erfc[(b_.)*(x_)])/(x_), x_Symbol] :> Int[E^(c + d*x^2)/x, x] - Int[E^(c + d*x^2)
*(Erf[b*x]/x), x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]

Rule 6527

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^(m + 1)*E^(c + d*x^2)*(Er
fc[a + b*x]/(m + 1)), x] + (-Dist[2*(d/(m + 1)), Int[x^(m + 2)*E^(c + d*x^2)*Erfc[a + b*x], x], x] + Dist[2*(b
/((m + 1)*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]
&& ILtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{4 x^4}+\frac {1}{2} b^2 \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^3} \, dx-\frac {b \int \frac {e^c}{x^4} \, dx}{2 \sqrt {\pi }} \\ & = -\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{4 x^4}-\frac {b^2 e^{c+b^2 x^2} \text {erfc}(b x)}{4 x^2}+\frac {1}{2} b^4 \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x} \, dx-\frac {b^3 \int \frac {e^c}{x^2} \, dx}{2 \sqrt {\pi }}-\frac {\left (b e^c\right ) \int \frac {1}{x^4} \, dx}{2 \sqrt {\pi }} \\ & = \frac {b e^c}{6 \sqrt {\pi } x^3}-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{4 x^4}-\frac {b^2 e^{c+b^2 x^2} \text {erfc}(b x)}{4 x^2}+\frac {1}{2} b^4 \int \frac {e^{c+b^2 x^2}}{x} \, dx-\frac {1}{2} b^4 \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x} \, dx-\frac {\left (b^3 e^c\right ) \int \frac {1}{x^2} \, dx}{2 \sqrt {\pi }} \\ & = \frac {b e^c}{6 \sqrt {\pi } x^3}+\frac {b^3 e^c}{2 \sqrt {\pi } x}-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{4 x^4}-\frac {b^2 e^{c+b^2 x^2} \text {erfc}(b x)}{4 x^2}+\frac {1}{4} b^4 e^c \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\frac {b^5 e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.62 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=-\frac {e^c \left (3 \sqrt {\pi } \left (e^{b^2 x^2} \left (1+b^2 x^2\right )-b^4 x^4 \operatorname {ExpIntegralEi}\left (b^2 x^2\right )\right )-8 b x \, _2F_2\left (-\frac {3}{2},1;-\frac {1}{2},\frac {3}{2};b^2 x^2\right )\right )}{12 \sqrt {\pi } x^4} \]

[In]

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

[Out]

-1/12*(E^c*(3*Sqrt[Pi]*(E^(b^2*x^2)*(1 + b^2*x^2) - b^4*x^4*ExpIntegralEi[b^2*x^2]) - 8*b*x*HypergeometricPFQ[
{-3/2, 1}, {-1/2, 3/2}, b^2*x^2]))/(Sqrt[Pi]*x^4)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(-(erf(b*x) - 1)*e^(b^2*x^2 + c)/x^5, x)

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\int \frac {{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfc}\left (b\,x\right )}{x^5} \,d x \]

[In]

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

[Out]

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