3.347 \(\int e^{3 x} x^2 \, dx\)

Optimal. Leaf size=32 \[ \frac{1}{3} e^{3 x} x^2-\frac{2}{9} e^{3 x} x+\frac{2 e^{3 x}}{27} \]

[Out]

(2*E^(3*x))/27 - (2*E^(3*x)*x)/9 + (E^(3*x)*x^2)/3

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Rubi [A]  time = 0.0322933, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{1}{3} e^{3 x} x^2-\frac{2}{9} e^{3 x} x+\frac{2 e^{3 x}}{27} \]

Antiderivative was successfully verified.

[In]  Int[E^(3*x)*x^2,x]

[Out]

(2*E^(3*x))/27 - (2*E^(3*x)*x)/9 + (E^(3*x)*x^2)/3

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Rubi in Sympy [A]  time = 2.15312, size = 27, normalized size = 0.84 \[ \frac{x^{2} e^{3 x}}{3} - \frac{2 x e^{3 x}}{9} + \frac{2 e^{3 x}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(3*x)*x**2,x)

[Out]

x**2*exp(3*x)/3 - 2*x*exp(3*x)/9 + 2*exp(3*x)/27

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Mathematica [A]  time = 0.00356269, size = 19, normalized size = 0.59 \[ \frac{1}{27} e^{3 x} \left (9 x^2-6 x+2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^(3*x)*x^2,x]

[Out]

(E^(3*x)*(2 - 6*x + 9*x^2))/27

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Maple [A]  time = 0.005, size = 17, normalized size = 0.5 \[{\frac{ \left ( 9\,{x}^{2}-6\,x+2 \right ){{\rm e}^{3\,x}}}{27}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(3*x)*x^2,x)

[Out]

1/27*(9*x^2-6*x+2)*exp(3*x)

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Maxima [A]  time = 1.35018, size = 22, normalized size = 0.69 \[ \frac{1}{27} \,{\left (9 \, x^{2} - 6 \, x + 2\right )} e^{\left (3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*e^(3*x),x, algorithm="maxima")

[Out]

1/27*(9*x^2 - 6*x + 2)*e^(3*x)

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Fricas [A]  time = 0.198966, size = 22, normalized size = 0.69 \[ \frac{1}{27} \,{\left (9 \, x^{2} - 6 \, x + 2\right )} e^{\left (3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*e^(3*x),x, algorithm="fricas")

[Out]

1/27*(9*x^2 - 6*x + 2)*e^(3*x)

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Sympy [A]  time = 0.085207, size = 15, normalized size = 0.47 \[ \frac{\left (9 x^{2} - 6 x + 2\right ) e^{3 x}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(3*x)*x**2,x)

[Out]

(9*x**2 - 6*x + 2)*exp(3*x)/27

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GIAC/XCAS [A]  time = 0.208055, size = 22, normalized size = 0.69 \[ \frac{1}{27} \,{\left (9 \, x^{2} - 6 \, x + 2\right )} e^{\left (3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*e^(3*x),x, algorithm="giac")

[Out]

1/27*(9*x^2 - 6*x + 2)*e^(3*x)