\(\int \frac {1}{2} (2 e^{4/3} x+3 x^2) \, dx\) [8165]

Optimal result

Integrand size = 18, antiderivative size = 14 \[ \int \frac {1}{2} \left (2 e^{4/3} x+3 x^2\right ) \, dx=\frac {1}{2} x^2 \left (e^{4/3}+x\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {12} \[ \int \frac {1}{2} \left (2 e^{4/3} x+3 x^2\right ) \, dx=\frac {x^3}{2}+\frac {1}{2} e^{4/3} x^2 \]

[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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int \frac {1}{2} \left (2 e^{4/3} x+3 x^2\right ) \, dx=\frac {1}{2} e^{4/3} x^2+\frac {x^3}{2} \]

[In]

Integrate[(2*E^(4/3)*x + 3*x^2)/2,x]

[Out]

(E^(4/3)*x^2)/2 + x^3/2

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {1}{2} \left (2 e^{4/3} x+3 x^2\right ) \, dx=\frac {1}{2} \, x^{3} + \frac {1}{2} \, x^{2} e^{\frac {4}{3}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2} \left (2 e^{4/3} x+3 x^2\right ) \, dx=\frac {x^{3}}{2} + \frac {x^{2} e^{\frac {4}{3}}}{2} \]

[In]

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

[Out]

x**3/2 + x**2*exp(4/3)/2

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {1}{2} \left (2 e^{4/3} x+3 x^2\right ) \, dx=\frac {1}{2} \, x^{3} + \frac {1}{2} \, x^{2} e^{\frac {4}{3}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {1}{2} \left (2 e^{4/3} x+3 x^2\right ) \, dx=\frac {1}{2} \, x^{3} + \frac {1}{2} \, x^{2} e^{\frac {4}{3}} \]

[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)

Mupad [B] (verification not implemented)

Time = 12.71 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.64 \[ \int \frac {1}{2} \left (2 e^{4/3} x+3 x^2\right ) \, dx=\frac {x^2\,\left (x+{\mathrm {e}}^{4/3}\right )}{2} \]

[In]

int(x*exp(4/3) + (3*x^2)/2,x)

[Out]

(x^2*(x + exp(4/3)))/2