3.1916 \(\int (\frac{2}{\sqrt{x}}+\sqrt{x}-\frac{x}{2}) \, dx\)

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

[Out]

4*Sqrt[x] + (2*x^(3/2))/3 - x^2/4

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Rubi [A]  time = 0.0024671, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ -\frac{x^2}{4}+\frac{2 x^{3/2}}{3}+4 \sqrt{x} \]

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*}

Mathematica [A]  time = 0.0043234, size = 24, normalized size = 1. \[ -\frac{x^2}{4}+\frac{2 x^{3/2}}{3}+4 \sqrt{x} \]

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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Maple [A]  time = 0.001, size = 17, normalized size = 0.7 \begin{align*}{\frac{2}{3}{x}^{{\frac{3}{2}}}}-{\frac{{x}^{2}}{4}}+4\,\sqrt{x} \end{align*}

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 = 0.972828, size = 22, normalized size = 0.92 \begin{align*} -\frac{1}{4} \, x^{2} + \frac{2}{3} \, x^{\frac{3}{2}} + 4 \, \sqrt{x} \end{align*}

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 = 2.34832, size = 43, normalized size = 1.79 \begin{align*} -\frac{1}{4} \, x^{2} + \frac{2}{3} \,{\left (x + 6\right )} \sqrt{x} \end{align*}

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 = 0.056591, size = 19, normalized size = 0.79 \begin{align*} \frac{2 x^{\frac{3}{2}}}{3} + 4 \sqrt{x} - \frac{x^{2}}{4} \end{align*}

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 = 1.05495, size = 22, normalized size = 0.92 \begin{align*} -\frac{1}{4} \, x^{2} + \frac{2}{3} \, x^{\frac{3}{2}} + 4 \, \sqrt{x} \end{align*}

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)