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]

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

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Rubi [A]  time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.04, size = 12, normalized size = 1.00 \[ e^{a+b x+c x^2} \]

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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fricas [A]  time = 0.40, size = 11, normalized size = 0.92 \[ e^{\left (c x^{2} + b x + a\right )} \]

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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giac [A]  time = 0.21, size = 11, normalized size = 0.92 \[ e^{\left (c x^{2} + b x + a\right )} \]

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)

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maple [A]  time = 0.03, size = 12, normalized size = 1.00 \[ {\mathrm e}^{c \,x^{2}+b x +a} \]

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 = 1.27, size = 11, normalized size = 0.92 \[ e^{\left (c x^{2} + b x + a\right )} \]

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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mupad [B]  time = 0.08, size = 13, normalized size = 1.08 \[ {\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,{\mathrm {e}}^{c\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.11, size = 10, normalized size = 0.83 \[ e^{a + b x + c x^{2}} \]

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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