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

Optimal. Leaf size=28 \[ \frac{2}{3} e^{x^{3/2}} x^{3/2}-\frac{2 e^{x^{3/2}}}{3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0630159, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{2}{3} e^{x^{3/2}} x^{3/2}-\frac{2 e^{x^{3/2}}}{3} \]

Antiderivative was successfully verified.

[In]  Int[E^x^(3/2)*x^2,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.50796, size = 24, normalized size = 0.86 \[ \frac{2 x^{\frac{3}{2}} e^{x^{\frac{3}{2}}}}{3} - \frac{2 e^{x^{\frac{3}{2}}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(3/2)*exp(x**(3/2))/3 - 2*exp(x**(3/2))/3

_______________________________________________________________________________________

Mathematica [A]  time = 0.0049831, size = 18, normalized size = 0.64 \[ \frac{2}{3} e^{x^{3/2}} \left (x^{3/2}-1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^x^(3/2)*x^2,x]

[Out]

(2*E^x^(3/2)*(-1 + x^(3/2)))/3

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 17, normalized size = 0.6 \[ -{\frac{2}{3}{{\rm e}^{{x}^{{\frac{3}{2}}}}}}+{\frac{2}{3}{{\rm e}^{{x}^{{\frac{3}{2}}}}}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x^(3/2))*x^2,x)

[Out]

-2/3*exp(x^(3/2))+2/3*exp(x^(3/2))*x^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.775684, size = 15, normalized size = 0.54 \[ \frac{2}{3} \,{\left (x^{\frac{3}{2}} - 1\right )} e^{\left (x^{\frac{3}{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(x^(3/2) - 1)*e^(x^(3/2))

_______________________________________________________________________________________

Fricas [A]  time = 0.245369, size = 15, normalized size = 0.54 \[ \frac{2}{3} \,{\left (x^{\frac{3}{2}} - 1\right )} e^{\left (x^{\frac{3}{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(x^(3/2) - 1)*e^(x^(3/2))

_______________________________________________________________________________________

Sympy [A]  time = 4.78464, size = 24, normalized size = 0.86 \[ \frac{2 x^{\frac{3}{2}} e^{x^{\frac{3}{2}}}}{3} - \frac{2 e^{x^{\frac{3}{2}}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x**(3/2))*x**2,x)

[Out]

2*x**(3/2)*exp(x**(3/2))/3 - 2*exp(x**(3/2))/3

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.238286, size = 18, normalized size = 0.64 \[ \frac{2}{3} \,{\left (x^{\frac{3}{2}} - 1\right )} e^{\left (x^{\frac{3}{2}}\right )} + \frac{2}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(x^(3/2) - 1)*e^(x^(3/2)) + 2/3