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

Optimal. Leaf size=24 \[ 4 \sqrt {x}+\frac {2 x^{3/2}}{3}-\frac {x^2}{4} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \frac {2 x^{3/2}}{3}-\frac {x^2}{4}+4 \sqrt {x} \end {gather*}

Antiderivative was successfully verified.

[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*} \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}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} 4 \sqrt {x}+\frac {2 x^{3/2}}{3}-\frac {x^2}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4*Sqrt[x] + (2*x^(3/2))/3 - x^2/4

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Mathics [A]
time = 1.64, size = 16, normalized size = 0.67 \begin {gather*} 4 \sqrt {x}+\frac {2 x^{\frac {3}{2}}}{3}-\frac {x^2}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[2/Sqrt[x] + Sqrt[x] - x/2,x]')

[Out]

4 Sqrt[x] + 2 x ^ (3 / 2) / 3 - x ^ 2 / 4

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Maple [A]
time = 0.02, size = 17, normalized size = 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	\(-\frac {\left (1+x \right ) \left (-1+x \right )}{4}+\frac {\left (8+\frac {4 x}{3}\right ) \sqrt {x}}{2}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]
time = 0.26, size = 16, normalized size = 0.67 \begin {gather*} -\frac {1}{4} \, x^{2} + \frac {2}{3} \, x^{\frac {3}{2}} + 4 \, \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]
time = 0.30, size = 14, normalized size = 0.58 \begin {gather*} -\frac {1}{4} \, x^{2} + \frac {2}{3} \, {\left (x + 6\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]
time = 0.03, size = 19, normalized size = 0.79 \begin {gather*} \frac {2 x^{\frac {3}{2}}}{3} + 4 \sqrt {x} - \frac {x^{2}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]
time = 0.00, size = 28, normalized size = 1.17 \begin {gather*} -\frac {x^{2}}{2\cdot 2}+2\cdot 2 \sqrt {x}+\frac {2}{3} \sqrt {x} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x^2 + 2/3*x^(3/2) + 4*sqrt(x)

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Mupad [B]
time = 0.03, size = 15, normalized size = 0.62 \begin {gather*} \frac {\sqrt {x}\,\left (8\,x-3\,x^{3/2}+48\right )}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2/x^(1/2) - x/2 + x^(1/2),x)

[Out]

(x^(1/2)*(8*x - 3*x^(3/2) + 48))/12

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