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]

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

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Rubi [A]  time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2235, 2234, 2204} \[ \frac {\sqrt {\pi } e^{-\frac {(b c-a d)^2}{4 b d}} \text {Erfi}\left (\frac {a d+b c+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}} \]

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)))

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]

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 2235

Int[(F_)^(v_), x_Symbol] :> Int[F^ExpandToSum[v, x], x] /; FreeQ[F, x] && QuadraticQ[v, x] &&  !QuadraticMatch
Q[v, x]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 68, normalized size = 1.00 \[ \frac {\sqrt {\pi } e^{-\frac {(b c-a d)^2}{4 b d}} \text {erfi}\left (\frac {a d+b (c+2 d x)}{2 \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}} \]

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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fricas [A]  time = 0.40, size = 74, normalized size = 1.09 \[ -\frac {\sqrt {\pi } \sqrt {-b d} \operatorname {erf}\left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d}}{2 \, b d}\right ) e^{\left (-\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{4 \, b d}\right )}}{2 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(pi)*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))/(b*d)

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giac [A]  time = 0.33, size = 68, normalized size = 1.00 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-b d} {\left (2 \, x + \frac {b c + a d}{b d}\right )}\right ) e^{\left (-\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{4 \, b d}\right )}}{2 \, \sqrt {-b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp((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))/sq
rt(-b*d)

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maple [A]  time = 0.01, size = 60, normalized size = 0.88 \[ -\frac {\sqrt {\pi }\, \erf \left (-\sqrt {-b d}\, x +\frac {a d +b c}{2 \sqrt {-b d}}\right ) {\mathrm e}^{a c -\frac {\left (a d +b c \right )^{2}}{4 b d}}}{2 \sqrt {-b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp((b*x+a)*(d*x+c)),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 = 0.85, size = 58, normalized size = 0.85 \[ \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {-b d} x - \frac {b c + a d}{2 \, \sqrt {-b d}}\right ) e^{\left (a c - \frac {{\left (b c + a d\right )}^{2}}{4 \, b d}\right )}}{2 \, \sqrt {-b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp((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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mupad [B]  time = 0.04, size = 60, normalized size = 0.88 \[ -\frac {\sqrt {\pi }\,{\mathrm {e}}^{\frac {a\,c}{2}-\frac {a^2\,d}{4\,b}-\frac {b\,c^2}{4\,d}}\,\mathrm {erf}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{2\,\sqrt {b\,d}}\right )\,1{}\mathrm {i}}{2\,\sqrt {b\,d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp((a + b*x)*(c + d*x)),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{a c} \int e^{a d x} e^{b c x} e^{b d x^{2}}\, dx \]

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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