Optimal. Leaf size=127 \[ (a d+b c) \text{Unintegrable}\left (\frac{e^{x (a d+b c)+a c+b d x^2}}{x},x\right )+\sqrt{\pi } \sqrt{b} \sqrt{d} e^{-\frac{(b c-a d)^2}{4 b d}} \text{Erfi}\left (\frac{a d+b c+2 b d x}{2 \sqrt{b} \sqrt{d}}\right )-\frac{e^{x (a d+b c)+a c+b d x^2}}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.264586, 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 = {} \[ \int \frac{e^{(a+b x) (c+d x)}}{x^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int \frac{e^{(a+b x) (c+d x)}}{x^2} \, dx &=\int \frac{e^{a c+(b c+a d) x+b d x^2}}{x^2} \, dx\\ &=-\frac{e^{a c+(b c+a d) x+b d x^2}}{x}+(2 b d) \int e^{a c+(b c+a d) x+b d x^2} \, dx-(-b c-a d) \int \frac{e^{a c+(b c+a d) x+b d x^2}}{x} \, dx\\ &=-\frac{e^{a c+(b c+a d) x+b d x^2}}{x}-(-b c-a d) \int \frac{e^{a c+(b c+a d) x+b d x^2}}{x} \, dx+\left (2 b d e^{-\frac{(b c-a d)^2}{4 b d}}\right ) \int e^{\frac{(b c+a d+2 b d x)^2}{4 b d}} \, dx\\ &=-\frac{e^{a c+(b c+a d) x+b d x^2}}{x}+\sqrt{b} \sqrt{d} e^{-\frac{(b c-a d)^2}{4 b d}} \sqrt{\pi } \text{erfi}\left (\frac{b c+a d+2 b d x}{2 \sqrt{b} \sqrt{d}}\right )-(-b c-a d) \int \frac{e^{a c+(b c+a d) x+b d x^2}}{x} \, dx\\ \end{align*}
Mathematica [A] time = 0.378521, size = 0, normalized size = 0. \[ \int \frac{e^{(a+b x) (c+d x)}}{x^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.017, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left ({\left (b x + a\right )}{\left (d x + c\right )}\right )}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e^{\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )}}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} e^{a c} \int \frac{e^{a d x} e^{b c x} e^{b d x^{2}}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left ({\left (b x + a\right )}{\left (d x + c\right )}\right )}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]