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

Optimal. Leaf size=22 \[ \frac{1}{2} e^{x^2} x^2-\frac{e^{x^2}}{2} \]

[Out]

-E^x^2/2 + (E^x^2*x^2)/2

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Rubi [A]  time = 0.0321641, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{1}{2} e^{x^2} x^2-\frac{e^{x^2}}{2} \]

Antiderivative was successfully verified.

[In]  Int[E^x^2*x^3,x]

[Out]

-E^x^2/2 + (E^x^2*x^2)/2

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Rubi in Sympy [A]  time = 3.78177, size = 15, normalized size = 0.68 \[ \frac{x^{2} e^{x^{2}}}{2} - \frac{e^{x^{2}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x**2)*x**3,x)

[Out]

x**2*exp(x**2)/2 - exp(x**2)/2

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Mathematica [A]  time = 0.00335246, size = 14, normalized size = 0.64 \[ \frac{1}{2} e^{x^2} \left (x^2-1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^x^2*x^3,x]

[Out]

(E^x^2*(-1 + x^2))/2

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Maple [A]  time = 0.004, size = 12, normalized size = 0.6 \[{\frac{ \left ({x}^{2}-1 \right ){{\rm e}^{{x}^{2}}}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x^2)*x^3,x)

[Out]

1/2*(x^2-1)*exp(x^2)

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Maxima [A]  time = 0.785365, size = 15, normalized size = 0.68 \[ \frac{1}{2} \,{\left (x^{2} - 1\right )} e^{\left (x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(x^2 - 1)*e^(x^2)

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Fricas [A]  time = 0.234235, size = 15, normalized size = 0.68 \[ \frac{1}{2} \,{\left (x^{2} - 1\right )} e^{\left (x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(x^2 - 1)*e^(x^2)

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Sympy [A]  time = 0.059248, size = 10, normalized size = 0.45 \[ \frac{\left (x^{2} - 1\right ) e^{x^{2}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x**2)*x**3,x)

[Out]

(x**2 - 1)*exp(x**2)/2

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GIAC/XCAS [A]  time = 0.223341, size = 15, normalized size = 0.68 \[ \frac{1}{2} \,{\left (x^{2} - 1\right )} e^{\left (x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(x^2 - 1)*e^(x^2)