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

Optimal. Leaf size=68 \[ \frac{\sqrt{\pi } e^{-\frac{(b c-a d)^2}{4 b d}} \text{Erfi}\left (\frac{a d+b c+2 b d x}{2 \sqrt{b} \sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.050062, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\sqrt{\pi } e^{-\frac{(b c-a d)^2}{4 b d}} \text{Erfi}\left (\frac{a d+b c+2 b d x}{2 \sqrt{b} \sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]  Int[E^((a + b*x)*(c + d*x)),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.021842, size = 68, normalized size = 1. \[ \frac{\sqrt{\pi } e^{-\frac{(b c-a d)^2}{4 b d}} \text{Erfi}\left (\frac{a d+b (c+2 d x)}{2 \sqrt{b} \sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 60, normalized size = 0.9 \[ -{\frac{\sqrt{\pi }}{2}{{\rm e}^{ac-{\frac{ \left ( ad+cb \right ) ^{2}}{4\,bd}}}}{\it Erf} \left ( -\sqrt{-bd}x+{\frac{ad+cb}{2}{\frac{1}{\sqrt{-bd}}}} \right ){\frac{1}{\sqrt{-bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp((d*x+c)*(b*x+a)),x)

[Out]

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

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Maxima [A]  time = 0.757819, size = 78, normalized size = 1.15 \[ \frac{\sqrt{\pi } \operatorname{erf}\left (\sqrt{-b d} x - \frac{b c + a d}{2 \, \sqrt{-b d}}\right ) e^{\left (a c - \frac{{\left (b c + a d\right )}^{2}}{4 \, b d}\right )}}{2 \, \sqrt{-b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(pi)*erf(sqrt(-b*d)*x - 1/2*(b*c + a*d)/sqrt(-b*d))*e^(a*c - 1/4*(b*c +
a*d)^2/(b*d))/sqrt(-b*d)

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Fricas [A]  time = 0.288946, size = 92, normalized size = 1.35 \[ \frac{\sqrt{\pi } \operatorname{erf}\left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, b d}\right ) e^{\left (-\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{4 \, b d}\right )}}{2 \, \sqrt{-b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(pi)*erf(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(b*d))*e^(-1/4*(b^2*c^2 -
2*a*b*c*d + a^2*d^2)/(b*d))/sqrt(-b*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ e^{a c} \int e^{a d x} e^{b c x} e^{b d x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp((b*x+a)*(d*x+c)),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.27236, size = 92, normalized size = 1.35 \[ -\frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-b d}{\left (2 \, x + \frac{b c + a d}{b d}\right )}\right ) e^{\left (-\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{4 \, b d}\right )}}{2 \, \sqrt{-b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(pi)*erf(-1/2*sqrt(-b*d)*(2*x + (b*c + a*d)/(b*d)))*e^(-1/4*(b^2*c^2 -
2*a*b*c*d + a^2*d^2)/(b*d))/sqrt(-b*d)