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 = 0.104187, 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, Rules used = {2244, 2240, 2234, 2204} \[ \frac{e^{x (a d+b c)+a c+b d x^2}}{2 b d}-\frac{\sqrt{\pi } (a d+b c) e^{-\frac{(b c-a d)^2}{4 b d}} \text{Erfi}\left (\frac{a d+b c+2 b d x}{2 \sqrt{b} \sqrt{d}}\right )}{4 b^{3/2} d^{3/2}} \]

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)))

Rule 2244

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])

Rule 2240

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] - Dist[(b*e - 2*c*d)/(2*c), Int[F^(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e}, x] &&
NeQ[b*e - 2*c*d, 0]

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rubi steps

\begin{align*} \int e^{(a+b x) (c+d x)} x \, dx &=\int 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}}{2 b d}-\frac{(b c+a d) \int e^{a c+(b c+a d) x+b d x^2} \, dx}{2 b d}\\ &=\frac{e^{a c+(b c+a d) x+b d x^2}}{2 b d}-\frac{\left ((b c+a 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}{2 b d}\\ &=\frac{e^{a c+(b c+a d) x+b d x^2}}{2 b d}-\frac{(b c+a 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 )}{4 b^{3/2} d^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0880173, 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]*Erfi[(a*d + b*(c + 2*d*x))/(2*Sq
rt[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. \begin{align*}{\frac{{{\rm e}^{ac+ \left ( ad+bc \right ) x+bd{x}^{2}}}}{2\,bd}}+{\frac{ \left ( ad+bc \right ) \sqrt{\pi }}{4\,bd}{{\rm e}^{ac-{\frac{ \left ( ad+bc \right ) ^{2}}{4\,bd}}}}{\it Erf} \left ( -\sqrt{-bd}x+{\frac{ad+bc}{2}{\frac{1}{\sqrt{-bd}}}} \right ){\frac{1}{\sqrt{-bd}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp((b*x+a)*(d*x+c))*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 = 1.16907, size = 193, normalized size = 1.8 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp((b*x+a)*(d*x+c))*x,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 = 1.52755, size = 251, normalized size = 2.35 \begin{align*} \frac{\sqrt{\pi }{\left (b c + a d\right )} \sqrt{-b d} \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 \, b d e^{\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )}}{4 \, b^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{a c} \int x e^{a d x} e^{b c x} e^{b d 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,x)

[Out]

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

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Giac [A]  time = 1.22384, size = 140, normalized size = 1.31 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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