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

Optimal. Leaf size=173 \[ \frac{1}{2} b^4 \text{Unintegrable}\left (\frac{e^{b^2 x^2+c} \text{Erfi}(b x)}{x},x\right )-\frac{b^2 e^{b^2 x^2+c} \text{Erfi}(b x)}{4 x^2}-\frac{e^{b^2 x^2+c} \text{Erfi}(b x)}{4 x^4}+\frac{2}{3} \sqrt{2} b^4 e^c \text{Erfi}\left (\sqrt{2} b x\right )+\frac{b^4 e^c \text{Erfi}\left (\sqrt{2} b x\right )}{\sqrt{2}}-\frac{7 b^3 e^{2 b^2 x^2+c}}{6 \sqrt{\pi } x}-\frac{b e^{2 b^2 x^2+c}}{6 \sqrt{\pi } x^3} \]

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

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Rubi [A]  time = 0.218594, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [A]  time = 0.192628, size = 0, normalized size = 0. \[ \int \frac{e^{c+b^2 x^2} \text{Erfi}(b x)}{x^5} \, dx \]

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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Maple [A]  time = 0.283, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{{b}^{2}{x}^{2}+c}}{\it erfi} \left ( bx \right ) }{{x}^{5}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{5}}\,{d x} \end{align*}

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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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{5}}, x\right ) \end{align*}

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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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{c} \int \frac{e^{b^{2} x^{2}} \operatorname{erfi}{\left (b x \right )}}{x^{5}}\, dx \end{align*}

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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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{5}}\,{d x} \end{align*}

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)