3.82.65 \(\int \frac {1}{2} (2 e^{4/3} x+3 x^2) \, dx\)

Optimal. Leaf size=14 \[ \frac {1}{2} x^2 \left (e^{4/3}+x\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.43, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {12} \begin {gather*} \frac {x^3}{2}+\frac {1}{2} e^{4/3} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*E^(4/3)*x + 3*x^2)/2,x]

[Out]

(E^(4/3)*x^2)/2 + x^3/2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \left (2 e^{4/3} x+3 x^2\right ) \, dx\\ &=\frac {1}{2} e^{4/3} x^2+\frac {x^3}{2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 20, normalized size = 1.43 \begin {gather*} \frac {1}{2} e^{4/3} x^2+\frac {x^3}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*E^(4/3)*x + 3*x^2)/2,x]

[Out]

(E^(4/3)*x^2)/2 + x^3/2

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fricas [A]  time = 0.51, size = 13, normalized size = 0.93 \begin {gather*} \frac {1}{2} \, x^{3} + \frac {1}{2} \, x^{2} e^{\frac {4}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*exp(4/3)+3/2*x^2,x, algorithm="fricas")

[Out]

1/2*x^3 + 1/2*x^2*e^(4/3)

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giac [A]  time = 0.22, size = 13, normalized size = 0.93 \begin {gather*} \frac {1}{2} \, x^{3} + \frac {1}{2} \, x^{2} e^{\frac {4}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*exp(4/3)+3/2*x^2,x, algorithm="giac")

[Out]

1/2*x^3 + 1/2*x^2*e^(4/3)

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maple [A]  time = 0.02, size = 10, normalized size = 0.71

method	result	size
gosper	\(\frac {\left (x +{\mathrm e}^{\frac {4}{3}}\right ) x^{2}}{2}\)	\(10\)
default	\(\frac {{\mathrm e}^{\frac {4}{3}} x^{2}}{2}+\frac {x^{3}}{2}\)	\(14\)
norman	\(\frac {{\mathrm e}^{\frac {4}{3}} x^{2}}{2}+\frac {x^{3}}{2}\)	\(14\)
risch	\(\frac {{\mathrm e}^{\frac {4}{3}} x^{2}}{2}+\frac {x^{3}}{2}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*exp(4/3)+3/2*x^2,x,method=_RETURNVERBOSE)

[Out]

1/2*(x+exp(4/3))*x^2

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maxima [A]  time = 0.37, size = 13, normalized size = 0.93 \begin {gather*} \frac {1}{2} \, x^{3} + \frac {1}{2} \, x^{2} e^{\frac {4}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*exp(4/3)+3/2*x^2,x, algorithm="maxima")

[Out]

1/2*x^3 + 1/2*x^2*e^(4/3)

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mupad [B]  time = 5.28, size = 9, normalized size = 0.64 \begin {gather*} \frac {x^2\,\left (x+{\mathrm {e}}^{4/3}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*exp(4/3) + (3*x^2)/2,x)

[Out]

(x^2*(x + exp(4/3)))/2

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sympy [A]  time = 0.05, size = 14, normalized size = 1.00 \begin {gather*} \frac {x^{3}}{2} + \frac {x^{2} e^{\frac {4}{3}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*exp(4/3)+3/2*x**2,x)

[Out]

x**3/2 + x**2*exp(4/3)/2

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