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

Optimal. Leaf size=211 \[ \frac{1}{2} d^2 \text{Unintegrable}\left (\frac{\text{Erfi}(b x) e^{c+d x^2}}{x},x\right )+\frac{1}{2} b e^c d \sqrt{b^2+d} \text{Erfi}\left (x \sqrt{b^2+d}\right )+\frac{1}{3} b e^c \left (b^2+d\right )^{3/2} \text{Erfi}\left (x \sqrt{b^2+d}\right )-\frac{b d e^{x^2 \left (b^2+d\right )+c}}{2 \sqrt{\pi } x}-\frac{b \left (b^2+d\right ) e^{x^2 \left (b^2+d\right )+c}}{3 \sqrt{\pi } x}-\frac{b e^{x^2 \left (b^2+d\right )+c}}{6 \sqrt{\pi } x^3}-\frac{d \text{Erfi}(b x) e^{c+d x^2}}{4 x^2}-\frac{\text{Erfi}(b x) e^{c+d x^2}}{4 x^4} \]

[Out]

-(b*E^(c + (b^2 + d)*x^2))/(6*Sqrt[Pi]*x^3) - (b*d*E^(c + (b^2 + d)*x^2))/(2*Sqrt[Pi]*x) - (b*(b^2 + d)*E^(c +
(b^2 + d)*x^2))/(3*Sqrt[Pi]*x) - (E^(c + d*x^2)*Erfi[b*x])/(4*x^4) - (d*E^(c + d*x^2)*Erfi[b*x])/(4*x^2) + (b
*d*Sqrt[b^2 + d]*E^c*Erfi[Sqrt[b^2 + d]*x])/2 + (b*(b^2 + d)^(3/2)*E^c*Erfi[Sqrt[b^2 + d]*x])/3 + (d^2*Uninteg
rable[(E^(c + d*x^2)*Erfi[b*x])/x, x])/2

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Rubi [A]  time = 0.320188, 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 = {} \[ \int \frac{e^{c+d x^2} \text{Erfi}(b x)}{x^5} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(E^(c + d*x^2)*Erfi[b*x])/x^5,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.233364, size = 0, normalized size = 0. \[ \int \frac{e^{c+d x^2} \text{Erfi}(b x)}{x^5} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.297, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{d{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(d*x^2+c)*erfi(b*x)/x^5,x)

[Out]

int(exp(d*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 (d x^{2} + c\right )}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfi(b*x)*e^(d*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 (d x^{2} + c\right )}}{x^{5}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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 (d x^{2} + c\right )}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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