\(\int \frac {e^{(a+b x) (c+d x)}}{x^2} \, dx\) [443]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.17 (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 = {} \[ \int \frac {e^{(a+b x) (c+d x)}}{x^2} \, dx=\int \frac {e^{(a+b x) (c+d x)}}{x^2} \, dx \]

[In]

Int[E^((a + b*x)*(c + d*x))/x^2,x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \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 [N/A]

Not integrable

Time = 0.60 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {e^{(a+b x) (c+d x)}}{x^2} \, dx=\int \frac {e^{(a+b x) (c+d x)}}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

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

\[\int \frac {{\mathrm e}^{\left (b x +a \right ) \left (d x +c \right )}}{x^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \frac {e^{(a+b x) (c+d x)}}{x^2} \, dx=\int { \frac {e^{\left ({\left (b x + a\right )} {\left (d x + c\right )}\right )}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 9.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82 \[ \int \frac {e^{(a+b x) (c+d x)}}{x^2} \, dx=e^{a c} \int \frac {e^{a d x} e^{b c x} e^{b d x^{2}}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{(a+b x) (c+d x)}}{x^2} \, dx=\int { \frac {e^{\left ({\left (b x + a\right )} {\left (d x + c\right )}\right )}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{(a+b x) (c+d x)}}{x^2} \, dx=\int { \frac {e^{\left ({\left (b x + a\right )} {\left (d x + c\right )}\right )}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{(a+b x) (c+d x)}}{x^2} \, dx=\int \frac {{\mathrm {e}}^{\left (a+b\,x\right )\,\left (c+d\,x\right )}}{x^2} \,d x \]

[In]

int(exp((a + b*x)*(c + d*x))/x^2,x)

[Out]

int(exp((a + b*x)*(c + d*x))/x^2, x)