3.288 \(\int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^5} \, dx\)

Optimal. Leaf size=174 \[ \frac {1}{2} b^4 \text {Int}\left (\frac {e^{b^2 x^2+c} \text {erfi}(b x)}{x},x\right )+\frac {2}{3} \sqrt {2} b^4 e^c \text {erfi}\left (\sqrt {2} b x\right )+\frac {b^4 e^c \text {erfi}\left (\sqrt {2} b x\right )}{\sqrt {2}}-\frac {b^2 e^{b^2 x^2+c} \text {erfi}(b x)}{4 x^2}-\frac {e^{b^2 x^2+c} \text {erfi}(b x)}{4 x^4}-\frac {b e^{2 b^2 x^2+c}}{6 \sqrt {\pi } x^3}-\frac {7 b^3 e^{2 b^2 x^2+c}}{6 \sqrt {\pi } x} \]

[Out]

-1/4*exp(b^2*x^2+c)*erfi(b*x)/x^4-1/4*b^2*exp(b^2*x^2+c)*erfi(b*x)/x^2+7/6*b^4*exp(c)*erfi(b*x*2^(1/2))*2^(1/2
)-1/6*b*exp(2*b^2*x^2+c)/x^3/Pi^(1/2)-7/6*b^3*exp(2*b^2*x^2+c)/x/Pi^(1/2)+1/2*b^4*Unintegrable(exp(b^2*x^2+c)*
erfi(b*x)/x,x)

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Rubi [A]  time = 0.22, 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 = {} \[ \int \frac {e^{c+b^2 x^2} \text {Erfi}(b x)}{x^5} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-(b*E^(c + 2*b^2*x^2))/(6*Sqrt[Pi]*x^3) - (7*b^3*E^(c + 2*b^2*x^2))/(6*Sqrt[Pi]*x) - (E^(c + b^2*x^2)*Erfi[b*x
])/(4*x^4) - (b^2*E^(c + b^2*x^2)*Erfi[b*x])/(4*x^2) + (b^4*E^c*Erfi[Sqrt[2]*b*x])/Sqrt[2] + (2*Sqrt[2]*b^4*E^
c*Erfi[Sqrt[2]*b*x])/3 + (b^4*Defer[Int][(E^(c + b^2*x^2)*Erfi[b*x])/x, x])/2

Rubi steps

\begin {align*} \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^5} \, dx &=-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{4 x^4}+\frac {1}{2} b^2 \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^3} \, dx+\frac {b \int \frac {e^{c+2 b^2 x^2}}{x^4} \, dx}{2 \sqrt {\pi }}\\ &=-\frac {b e^{c+2 b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{4 x^4}-\frac {b^2 e^{c+b^2 x^2} \text {erfi}(b x)}{4 x^2}+\frac {1}{2} b^4 \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x} \, dx+\frac {b^3 \int \frac {e^{c+2 b^2 x^2}}{x^2} \, dx}{2 \sqrt {\pi }}+\frac {\left (2 b^3\right ) \int \frac {e^{c+2 b^2 x^2}}{x^2} \, dx}{3 \sqrt {\pi }}\\ &=-\frac {b e^{c+2 b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {7 b^3 e^{c+2 b^2 x^2}}{6 \sqrt {\pi } x}-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{4 x^4}-\frac {b^2 e^{c+b^2 x^2} \text {erfi}(b x)}{4 x^2}+\frac {1}{2} b^4 \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x} \, dx+\frac {\left (2 b^5\right ) \int e^{c+2 b^2 x^2} \, dx}{\sqrt {\pi }}+\frac {\left (8 b^5\right ) \int e^{c+2 b^2 x^2} \, dx}{3 \sqrt {\pi }}\\ &=-\frac {b e^{c+2 b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {7 b^3 e^{c+2 b^2 x^2}}{6 \sqrt {\pi } x}-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{4 x^4}-\frac {b^2 e^{c+b^2 x^2} \text {erfi}(b x)}{4 x^2}+\frac {b^4 e^c \text {erfi}\left (\sqrt {2} b x\right )}{\sqrt {2}}+\frac {2}{3} \sqrt {2} b^4 e^c \text {erfi}\left (\sqrt {2} b x\right )+\frac {1}{2} b^4 \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^5} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{b^{2} x^{2}+c} \erfi \left (b x \right )}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfi}\left (b\,x\right )}{x^5} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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