3.437 \(\int \frac{e^{a+b x-c x^2}}{x^2} \, dx\)

Optimal. Leaf size=81 \[ b \text{Unintegrable}\left (\frac{e^{a+b x-c x^2}}{x},x\right )+\sqrt{\pi } \sqrt{c} e^{a+\frac{b^2}{4 c}} \text{Erf}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )-\frac{e^{a+b x-c x^2}}{x} \]

[Out]

-(E^(a + b*x - c*x^2)/x) + Sqrt[c]*E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])] + b*Unintegrable[E^
(a + b*x - c*x^2)/x, x]

________________________________________________________________________________________

Rubi [A]  time = 0.0705695, 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 = {} \[ \int \frac{e^{a+b x-c x^2}}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[E^(a + b*x - c*x^2)/x^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.220979, size = 0, normalized size = 0. \[ \int \frac{e^{a+b x-c x^2}}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[E^(a + b*x - c*x^2)/x^2,x]

[Out]

Integrate[E^(a + b*x - c*x^2)/x^2, x]

________________________________________________________________________________________

Maple [A]  time = 0.016, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{-c{x}^{2}+bx+a}}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-c*x^2+b*x+a)/x^2,x)

[Out]

int(exp(-c*x^2+b*x+a)/x^2,x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(-c*x^2 + b*x + a)/x^2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(e^(-c*x^2 + b*x + a)/x^2, x)

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{a} \int \frac{e^{b x} e^{- c x^{2}}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-c*x**2+b*x+a)/x**2,x)

[Out]

exp(a)*Integral(exp(b*x)*exp(-c*x**2)/x**2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(-c*x^2 + b*x + a)/x^2, x)