3.17.2 \(\int (\frac {2}{\sqrt {x}}+\sqrt {x}-\frac {x}{2}) \, dx\)

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

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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 {x^2}{4}+\frac {2 x^{3/2}}{3}+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*} \frac {2 x^{3/2}}{3}-\frac {x^2}{4}+4 \sqrt {x} \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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IntegrateAlgebraic [A]  time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} \frac {1}{12} \left (8 x^{3/2}-3 x^2+48 \sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(48*Sqrt[x] + 8*x^(3/2) - 3*x^2)/12

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fricas [A]  time = 1.19, 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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giac [A]  time = 1.09, 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="giac")

[Out]

-1/4*x^2 + 2/3*x^(3/2) + 4*sqrt(x)

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maple [A]  time = 0.00, size = 17, normalized size = 0.71 \begin {gather*} -\frac {x^{2}}{4}+\frac {2 x^{\frac {3}{2}}}{3}+4 \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.05, 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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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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sympy [A]  time = 0.06, 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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