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

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]

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

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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]

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*} \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]

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

[Out]

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

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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]

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

[Out]

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

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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]

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)

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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]

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)

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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]

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)

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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]

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)