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

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 = 0.0278753, 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 \[ e^{a+b x+c x^2} \]

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)

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Rubi in Sympy [A]  time = 3.41435, size = 10, normalized size = 0.83 \[ e^{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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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Mathematica [A]  time = 0.006791, 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 = 0.003, size = 12, normalized size = 1. \[{{\rm e}^{c{x}^{2}+bx+a}} \]

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 = 0.786515, size = 15, normalized size = 1.25 \[ e^{\left (c x^{2} + b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)*e^(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

e^(c*x^2 + b*x + a)

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Fricas [A]  time = 0.253149, size = 15, normalized size = 1.25 \[ e^{\left (c x^{2} + b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)*e^(c*x^2 + b*x + a),x, algorithm="fricas")

[Out]

e^(c*x^2 + b*x + a)

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Sympy [A]  time = 0.094107, 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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GIAC/XCAS [A]  time = 0.242233, size = 15, normalized size = 1.25 \[ e^{\left (c x^{2} + b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)*e^(c*x^2 + b*x + a),x, algorithm="giac")

[Out]

e^(c*x^2 + b*x + a)