\(\int \frac {e^{a+b x-c x^2}}{x^2} \, dx\) [437]

Optimal result

Integrand size = 17, antiderivative size = 17 \[ \int \frac {e^{a+b x-c x^2}}{x^2} \, dx=-\frac {e^{a+b x-c x^2}}{x}+\sqrt {c} e^{a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )+b \text {Int}\left (\frac {e^{a+b x-c x^2}}{x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {e^{a+b x-c x^2}}{x^2} \, dx=\int \frac {e^{a+b x-c x^2}}{x^2} \, dx \]

[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*} \text {integral}& = -\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 [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {e^{a+b x-c x^2}}{x^2} \, dx=\int \frac {e^{a+b x-c x^2}}{x^2} \, dx \]

[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 [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94

\[\int \frac {{\mathrm e}^{-c \,x^{2}+b x +a}}{x^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{a+b x-c x^2}}{x^2} \, dx=\int { \frac {e^{\left (-c x^{2} + b x + a\right )}}{x^{2}} \,d x } \]

[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 [N/A]

Not integrable

Time = 2.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {e^{a+b x-c x^2}}{x^2} \, dx=e^{a} \int \frac {e^{b x} e^{- c x^{2}}}{x^{2}}\, dx \]

[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)

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{a+b x-c x^2}}{x^2} \, dx=\int { \frac {e^{\left (-c x^{2} + b x + a\right )}}{x^{2}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{a+b x-c x^2}}{x^2} \, dx=\int { \frac {e^{\left (-c x^{2} + b x + a\right )}}{x^{2}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{a+b x-c x^2}}{x^2} \, dx=\int \frac {{\mathrm {e}}^{-c\,x^2+b\,x+a}}{x^2} \,d x \]

[In]

int(exp(a + b*x - c*x^2)/x^2,x)

[Out]

int(exp(a + b*x - c*x^2)/x^2, x)