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

Optimal. Leaf size=128 \[ (a d+b c) \text{Int}\left (\frac{e^{x (a d+b c)+a c+b d x^2}}{x},x\right )+\sqrt{\pi } \sqrt{b} \sqrt{d} e^{-\frac{(b c-a d)^2}{4 b d}} \text{Erfi}\left (\frac{a d+b c+2 b d x}{2 \sqrt{b} \sqrt{d}}\right )-\frac{e^{x (a d+b c)+a c+b d x^2}}{x} \]

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

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Rubi [A]  time = 0.422996, 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. \[ \text{Int}\left (\frac{e^{(a+b x) (c+d x)}}{x^2},x\right ) \]

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

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Rubi in Sympy [A]  time = 0., size = 0, normalized size = 0. \[ 2 b d e^{a c - \frac{\left (a d + b c\right )^{2}}{4 b d}} \int e^{\frac{\left (a d + b c + 2 b d x\right )^{2}}{4 b d}}\, dx + \left (a d + b c\right ) \int \frac{e^{a c + b d x^{2} + x \left (a d + b c\right )}}{x}\, dx - \frac{e^{a c + b d x^{2} + x \left (a d + b c\right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp((b*x+a)*(d*x+c))/x**2,x)

[Out]

2*b*d*exp(a*c - (a*d + b*c)**2/(4*b*d))*Integral(exp((a*d + b*c + 2*b*d*x)**2/(4
*b*d)), x) + (a*d + b*c)*Integral(exp(a*c + b*d*x**2 + x*(a*d + b*c))/x, x) - ex
p(a*c + b*d*x**2 + x*(a*d + b*c))/x

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Mathematica [A]  time = 0.753913, 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 = 0.026, size = 0, normalized size = 0. \[ \int{\frac{{{\rm e}^{ \left ( dx+c \right ) \left ( bx+a \right ) }}}{{x}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp((d*x+c)*(b*x+a))/x^2,x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e^{\left ({\left (b x + a\right )}{\left (d x + c\right )}\right )}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^((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. \[{\rm integral}\left (\frac{e^{\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^((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. \[ e^{a c} \int \frac{e^{a d x} e^{b c x} e^{b d x^{2}}}{x^{2}}\, dx \]

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/XCAS [A]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e^{\left ({\left (b x + a\right )}{\left (d x + c\right )}\right )}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^((b*x + a)*(d*x + c))/x^2,x, algorithm="giac")

[Out]

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