\(\int (\frac {2}{\sqrt {x}}+\sqrt {x}-\frac {x}{2}) \, dx\) [1916]

Optimal result

Integrand size = 18, antiderivative size = 24 \[ \int \left (\frac {2}{\sqrt {x}}+\sqrt {x}-\frac {x}{2}\right ) \, dx=4 \sqrt {x}+\frac {2 x^{3/2}}{3}-\frac {x^2}{4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 24, 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 = {} \[ \int \left (\frac {2}{\sqrt {x}}+\sqrt {x}-\frac {x}{2}\right ) \, dx=\frac {2 x^{3/2}}{3}-\frac {x^2}{4}+4 \sqrt {x} \]

[In]

Int[2/Sqrt[x] + Sqrt[x] - x/2,x]

[Out]

4*Sqrt[x] + (2*x^(3/2))/3 - x^2/4

Rubi steps \begin{align*} \text {integral}& = 4 \sqrt {x}+\frac {2 x^{3/2}}{3}-\frac {x^2}{4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \left (\frac {2}{\sqrt {x}}+\sqrt {x}-\frac {x}{2}\right ) \, dx=4 \sqrt {x}+\frac {2 x^{3/2}}{3}-\frac {x^2}{4} \]

[In]

Integrate[2/Sqrt[x] + Sqrt[x] - x/2,x]

[Out]

4*Sqrt[x] + (2*x^(3/2))/3 - x^2/4

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71

method	result	size
derivativedivides	\(\frac {2 x^{\frac {3}{2}}}{3}-\frac {x^{2}}{4}+4 \sqrt {x}\)	\(17\)
default	\(\frac {2 x^{\frac {3}{2}}}{3}-\frac {x^{2}}{4}+4 \sqrt {x}\)	\(17\)
risch	\(\frac {2 x^{\frac {3}{2}}}{3}-\frac {x^{2}}{4}+4 \sqrt {x}\)	\(17\)
trager	\(-\frac {\left (-1+x \right ) \left (1+x \right )}{4}+\frac {\left (8+\frac {4 x}{3}\right ) \sqrt {x}}{2}\)	\(20\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.58 \[ \int \left (\frac {2}{\sqrt {x}}+\sqrt {x}-\frac {x}{2}\right ) \, dx=-\frac {1}{4} \, x^{2} + \frac {2}{3} \, {\left (x + 6\right )} \sqrt {x} \]

[In]

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

[Out]

-1/4*x^2 + 2/3*(x + 6)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \left (\frac {2}{\sqrt {x}}+\sqrt {x}-\frac {x}{2}\right ) \, dx=\frac {2 x^{\frac {3}{2}}}{3} + 4 \sqrt {x} - \frac {x^{2}}{4} \]

[In]

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

[Out]

2*x**(3/2)/3 + 4*sqrt(x) - x**2/4

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \left (\frac {2}{\sqrt {x}}+\sqrt {x}-\frac {x}{2}\right ) \, dx=-\frac {1}{4} \, x^{2} + \frac {2}{3} \, x^{\frac {3}{2}} + 4 \, \sqrt {x} \]

[In]

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

[Out]

-1/4*x^2 + 2/3*x^(3/2) + 4*sqrt(x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \left (\frac {2}{\sqrt {x}}+\sqrt {x}-\frac {x}{2}\right ) \, dx=-\frac {1}{4} \, x^{2} + \frac {2}{3} \, x^{\frac {3}{2}} + 4 \, \sqrt {x} \]

[In]

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

[Out]

-1/4*x^2 + 2/3*x^(3/2) + 4*sqrt(x)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \left (\frac {2}{\sqrt {x}}+\sqrt {x}-\frac {x}{2}\right ) \, dx=\frac {\sqrt {x}\,\left (8\,x-3\,x^{3/2}+48\right )}{12} \]

[In]

int(2/x^(1/2) - x/2 + x^(1/2),x)

[Out]

(x^(1/2)*(8*x - 3*x^(3/2) + 48))/12