\(\int (\frac {x^2}{2}+\frac {x^3}{3}) \, dx\) [1897]

Optimal result

Integrand size = 15, antiderivative size = 15 \[ \int \left (\frac {x^2}{2}+\frac {x^3}{3}\right ) \, dx=\frac {x^3}{6}+\frac {x^4}{12} \]

[Out]

1/6*x^3+1/12*x^4

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (\frac {x^2}{2}+\frac {x^3}{3}\right ) \, dx=\frac {x^4}{12}+\frac {x^3}{6} \]

[In]

Int[x^2/2 + x^3/3,x]

[Out]

x^3/6 + x^4/12

Rubi steps \begin{align*} \text {integral}& = \frac {x^3}{6}+\frac {x^4}{12} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \left (\frac {x^2}{2}+\frac {x^3}{3}\right ) \, dx=\frac {x^3}{6}+\frac {x^4}{12} \]

[In]

Integrate[x^2/2 + x^3/3,x]

[Out]

x^3/6 + x^4/12

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60

method	result	size
gosper	\(\frac {x^{3} \left (2+x \right )}{12}\)	\(9\)
default	\(\frac {1}{6} x^{3}+\frac {1}{12} x^{4}\)	\(12\)
norman	\(\frac {1}{6} x^{3}+\frac {1}{12} x^{4}\)	\(12\)
risch	\(\frac {1}{6} x^{3}+\frac {1}{12} x^{4}\)	\(12\)
parallelrisch	\(\frac {1}{6} x^{3}+\frac {1}{12} x^{4}\)	\(12\)
parts	\(\frac {1}{6} x^{3}+\frac {1}{12} x^{4}\)	\(12\)

[In]

int(1/2*x^2+1/3*x^3,x,method=_RETURNVERBOSE)

[Out]

1/12*x^3*(2+x)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \left (\frac {x^2}{2}+\frac {x^3}{3}\right ) \, dx=\frac {1}{12} \, x^{4} + \frac {1}{6} \, x^{3} \]

[In]

integrate(1/2*x^2+1/3*x^3,x, algorithm="fricas")

[Out]

1/12*x^4 + 1/6*x^3

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.53 \[ \int \left (\frac {x^2}{2}+\frac {x^3}{3}\right ) \, dx=\frac {x^{4}}{12} + \frac {x^{3}}{6} \]

[In]

integrate(1/2*x**2+1/3*x**3,x)

[Out]

x**4/12 + x**3/6

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \left (\frac {x^2}{2}+\frac {x^3}{3}\right ) \, dx=\frac {1}{12} \, x^{4} + \frac {1}{6} \, x^{3} \]

[In]

integrate(1/2*x^2+1/3*x^3,x, algorithm="maxima")

[Out]

1/12*x^4 + 1/6*x^3

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \left (\frac {x^2}{2}+\frac {x^3}{3}\right ) \, dx=\frac {1}{12} \, x^{4} + \frac {1}{6} \, x^{3} \]

[In]

integrate(1/2*x^2+1/3*x^3,x, algorithm="giac")

[Out]

1/12*x^4 + 1/6*x^3

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.53 \[ \int \left (\frac {x^2}{2}+\frac {x^3}{3}\right ) \, dx=\frac {x^3\,\left (x+2\right )}{12} \]

[In]

int(x^2/2 + x^3/3,x)

[Out]

(x^3*(x + 2))/12