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

Optimal. Leaf size=107 \[ \frac{e^{x (a d+b c)+a c+b d x^2}}{2 b d}-\frac{\sqrt{\pi } (a d+b c) e^{-\frac{(b c-a d)^2}{4 b d}} \text{Erfi}\left (\frac{a d+b c+2 b d x}{2 \sqrt{b} \sqrt{d}}\right )}{4 b^{3/2} d^{3/2}} \]

[Out]

E^(a*c + (b*c + a*d)*x + b*d*x^2)/(2*b*d) - ((b*c + a*d)*Sqrt[Pi]*Erfi[(b*c + a*
d + 2*b*d*x)/(2*Sqrt[b]*Sqrt[d])])/(4*b^(3/2)*d^(3/2)*E^((b*c - a*d)^2/(4*b*d)))

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Rubi [A]  time = 0.186668, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{e^{x (a d+b c)+a c+b d x^2}}{2 b d}-\frac{\sqrt{\pi } (a d+b c) e^{-\frac{(b c-a d)^2}{4 b d}} \text{Erfi}\left (\frac{a d+b c+2 b d x}{2 \sqrt{b} \sqrt{d}}\right )}{4 b^{3/2} d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

E^(a*c + (b*c + a*d)*x + b*d*x^2)/(2*b*d) - ((b*c + a*d)*Sqrt[Pi]*Erfi[(b*c + a*
d + 2*b*d*x)/(2*Sqrt[b]*Sqrt[d])])/(4*b^(3/2)*d^(3/2)*E^((b*c - a*d)^2/(4*b*d)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.12508, size = 116, normalized size = 1.08 \[ \frac{e^{-\frac{(b c-a d)^2}{4 b d}} \left (2 \sqrt{b} \sqrt{d} e^{\frac{(a d+b (c+2 d x))^2}{4 b d}}-\sqrt{\pi } (a d+b c) \text{Erfi}\left (\frac{a d+b (c+2 d x)}{2 \sqrt{b} \sqrt{d}}\right )\right )}{4 b^{3/2} d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[b]*Sqrt[d]*E^((a*d + b*(c + 2*d*x))^2/(4*b*d)) - (b*c + a*d)*Sqrt[Pi]*Er
fi[(a*d + b*(c + 2*d*x))/(2*Sqrt[b]*Sqrt[d])])/(4*b^(3/2)*d^(3/2)*E^((b*c - a*d)
^2/(4*b*d)))

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Maple [A]  time = 0.004, size = 102, normalized size = 1. \[{\frac{{{\rm e}^{ac+ \left ( ad+cb \right ) x+bd{x}^{2}}}}{2\,bd}}+{\frac{ \left ( ad+cb \right ) \sqrt{\pi }}{4\,bd}{{\rm e}^{ac-{\frac{ \left ( ad+cb \right ) ^{2}}{4\,bd}}}}{\it Erf} \left ( -\sqrt{-bd}x+{\frac{ad+cb}{2}{\frac{1}{\sqrt{-bd}}}} \right ){\frac{1}{\sqrt{-bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp((d*x+c)*(b*x+a))*x,x)

[Out]

1/2*exp(a*c+(a*d+b*c)*x+b*d*x^2)/b/d+1/4*(a*d+b*c)/b/d*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 = 0.827631, size = 193, normalized size = 1.8 \[ -\frac{{\left (\frac{\sqrt{\pi }{\left (2 \, b d x + b c + a d\right )}{\left (b c + a d\right )}{\left (\operatorname{erf}\left (\frac{1}{2} \, \sqrt{-\frac{{\left (2 \, b d x + b c + a d\right )}^{2}}{b d}}\right ) - 1\right )}}{\left (b d\right )^{\frac{3}{2}} \sqrt{-\frac{{\left (2 \, b d x + b c + a d\right )}^{2}}{b d}}} - \frac{2 \, b d e^{\left (\frac{{\left (2 \, b d x + b c + a d\right )}^{2}}{4 \, b d}\right )}}{\left (b d\right )^{\frac{3}{2}}}\right )} e^{\left (a c - \frac{{\left (b c + a d\right )}^{2}}{4 \, b d}\right )}}{4 \, \sqrt{b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(sqrt(pi)*(2*b*d*x + b*c + a*d)*(b*c + a*d)*(erf(1/2*sqrt(-(2*b*d*x + b*c +
a*d)^2/(b*d))) - 1)/((b*d)^(3/2)*sqrt(-(2*b*d*x + b*c + a*d)^2/(b*d))) - 2*b*d*
e^(1/4*(2*b*d*x + b*c + a*d)^2/(b*d))/(b*d)^(3/2))*e^(a*c - 1/4*(b*c + a*d)^2/(b
*d))/sqrt(b*d)

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Fricas [A]  time = 0.296438, size = 150, normalized size = 1.4 \[ -\frac{\sqrt{\pi }{\left (b c + a d\right )} \operatorname{erf}\left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, b d}\right ) e^{\left (-\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{4 \, b d}\right )} - 2 \, \sqrt{-b d} e^{\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )}}{4 \, \sqrt{-b d} b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.235179, size = 140, normalized size = 1.31 \[ \frac{\frac{\sqrt{\pi }{\left (b c + a d\right )} \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-b d}{\left (2 \, x + \frac{b c + a d}{b d}\right )}\right ) e^{\left (-\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{4 \, b d}\right )}}{\sqrt{-b d}} + 2 \, e^{\left (b d x^{2} + b c x + a d x + a c\right )}}{4 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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