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

Optimal. Leaf size=231 \[ \frac {1}{2} d^2 \text {Int}\left (\frac {\text {erf}(b x) e^{c+d x^2}}{x},x\right )-\frac {1}{2} b e^c d \sqrt {b^2-d} \text {erf}\left (x \sqrt {b^2-d}\right )+\frac {1}{3} b e^c \left (b^2-d\right )^{3/2} \text {erf}\left (x \sqrt {b^2-d}\right )-\frac {b d e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt {\pi } x}+\frac {b \left (b^2-d\right ) e^{c-x^2 \left (b^2-d\right )}}{3 \sqrt {\pi } x}-\frac {b e^{c-x^2 \left (b^2-d\right )}}{6 \sqrt {\pi } x^3}-\frac {d \text {erf}(b x) e^{c+d x^2}}{4 x^2}-\frac {\text {erf}(b x) e^{c+d x^2}}{4 x^4} \]

[Out]

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

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Rubi [A]  time = 0.34, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {e^{c+d x^2} \text {Erf}(b x)}{x^5} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(E^(c + d*x^2)*Erf[b*x])/x^5,x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {e^{c+d x^2} \text {erf}(b x)}{x^5} \, dx &=-\frac {e^{c+d x^2} \text {erf}(b x)}{4 x^4}+\frac {1}{2} d \int \frac {e^{c+d x^2} \text {erf}(b x)}{x^3} \, dx+\frac {b \int \frac {e^{c-\left (b^2-d\right ) x^2}}{x^4} \, dx}{2 \sqrt {\pi }}\\ &=-\frac {b e^{c-\left (b^2-d\right ) x^2}}{6 \sqrt {\pi } x^3}-\frac {e^{c+d x^2} \text {erf}(b x)}{4 x^4}-\frac {d e^{c+d x^2} \text {erf}(b x)}{4 x^2}+\frac {1}{2} d^2 \int \frac {e^{c+d x^2} \text {erf}(b x)}{x} \, dx-\frac {\left (b \left (b^2-d\right )\right ) \int \frac {e^{c+\left (-b^2+d\right ) x^2}}{x^2} \, dx}{3 \sqrt {\pi }}+\frac {(b d) \int \frac {e^{c-\left (b^2-d\right ) x^2}}{x^2} \, dx}{2 \sqrt {\pi }}\\ &=-\frac {b e^{c-\left (b^2-d\right ) x^2}}{6 \sqrt {\pi } x^3}+\frac {b \left (b^2-d\right ) e^{c-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x}-\frac {b d e^{c-\left (b^2-d\right ) x^2}}{2 \sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erf}(b x)}{4 x^4}-\frac {d e^{c+d x^2} \text {erf}(b x)}{4 x^2}+\frac {1}{2} d^2 \int \frac {e^{c+d x^2} \text {erf}(b x)}{x} \, dx+\frac {\left (2 b \left (b^2-d\right )^2\right ) \int e^{c+\left (-b^2+d\right ) x^2} \, dx}{3 \sqrt {\pi }}-\frac {\left (b \left (b^2-d\right ) d\right ) \int e^{c+\left (-b^2+d\right ) x^2} \, dx}{\sqrt {\pi }}\\ &=-\frac {b e^{c-\left (b^2-d\right ) x^2}}{6 \sqrt {\pi } x^3}+\frac {b \left (b^2-d\right ) e^{c-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x}-\frac {b d e^{c-\left (b^2-d\right ) x^2}}{2 \sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erf}(b x)}{4 x^4}-\frac {d e^{c+d x^2} \text {erf}(b x)}{4 x^2}+\frac {1}{3} b \left (b^2-d\right )^{3/2} e^c \text {erf}\left (\sqrt {b^2-d} x\right )-\frac {1}{2} b \sqrt {b^2-d} d e^c \text {erf}\left (\sqrt {b^2-d} x\right )+\frac {1}{2} d^2 \int \frac {e^{c+d x^2} \text {erf}(b x)}{x} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {e^{c+d x^2} \text {erf}(b x)}{x^5} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{d \,x^{2}+c} \erf \left (b x \right )}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {e}}^{d\,x^2+c}\,\mathrm {erf}\left (b\,x\right )}{x^5} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(c + d*x^2)*erf(b*x))/x^5,x)

[Out]

int((exp(c + d*x^2)*erf(b*x))/x^5, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{c} \int \frac {e^{d x^{2}} \operatorname {erf}{\left (b x \right )}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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