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

Optimal. Leaf size=12 \[ e^{a+b x+c x^2} \]

[Out]

E^(a + b*x + c*x^2)

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Rubi [A]  time = 0.0192453, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2236} \[ e^{a+b x+c x^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(a + b*x + c*x^2)*(b + 2*c*x),x]

[Out]

E^(a + b*x + c*x^2)

Rule 2236

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] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rubi steps

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

Mathematica [A]  time = 0.0373032, size = 12, normalized size = 1. \[ e^{a+b x+c x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(a + b*x + c*x^2)*(b + 2*c*x),x]

[Out]

E^(a + b*x + c*x^2)

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Maple [A]  time = 0.036, size = 12, normalized size = 1. \begin{align*}{{\rm e}^{c{x}^{2}+bx+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(c*x^2+b*x+a)

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Maxima [A]  time = 0.974464, size = 15, normalized size = 1.25 \begin{align*} e^{\left (c x^{2} + b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*x^2+b*x+a)*(2*c*x+b),x, algorithm="maxima")

[Out]

e^(c*x^2 + b*x + a)

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Fricas [A]  time = 0.706163, size = 28, normalized size = 2.33 \begin{align*} e^{\left (c x^{2} + b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*x^2+b*x+a)*(2*c*x+b),x, algorithm="fricas")

[Out]

e^(c*x^2 + b*x + a)

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Sympy [A]  time = 0.100673, size = 10, normalized size = 0.83 \begin{align*} e^{a + b x + c x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*x**2+b*x+a)*(2*c*x+b),x)

[Out]

exp(a + b*x + c*x**2)

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Giac [A]  time = 1.25708, size = 15, normalized size = 1.25 \begin{align*} e^{\left (c x^{2} + b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*x^2+b*x+a)*(2*c*x+b),x, algorithm="giac")

[Out]

e^(c*x^2 + b*x + a)