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

Optimal. Leaf size=82 \[ -\frac {e^{a+b x-c x^2}}{x}+\sqrt {c} e^{a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )+b \text {Int}\left (\frac {e^{a+b x-c x^2}}{x},x\right ) \]

[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)

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Rubi [A]
time = 0.05, 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 = {} \begin {gather*} \int \frac {e^{a+b x-c x^2}}{x^2} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 0.35, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{a+b x-c x^2}}{x^2} \, dx \end {gather*}

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]

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Maple [A]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{-c \,x^{2}+b x +a}}{x^{2}}\, dx\]

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)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{a} \int \frac {e^{b x} e^{- c x^{2}}}{x^{2}}\, dx \end {gather*}

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)

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {e}}^{-c\,x^2+b\,x+a}}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(a + b*x - c*x^2)/x^2,x)

[Out]

int(exp(a + b*x - c*x^2)/x^2, x)

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