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

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

[Out]

4*Sqrt[x] + (2*x^(3/2))/3 - x^2/4

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Rubi [A]  time = 0.00846067, 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. \[ \frac{2 x^{3/2}}{3}-\frac{x^2}{4}+4 \sqrt{x} \]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 x^{\frac{3}{2}}}{3} + 4 \sqrt{x} - \frac{\int x\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(-1/2*x+2/x**(1/2)+x**(1/2),x)

[Out]

2*x**(3/2)/3 + 4*sqrt(x) - Integral(x, x)/2

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Mathematica [A]  time = 0.00719066, size = 24, normalized size = 1. \[ \frac{2 x^{3/2}}{3}-\frac{x^2}{4}+4 \sqrt{x} \]

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 = 0.002, size = 17, normalized size = 0.7 \[{\frac{2}{3}{x}^{{\frac{3}{2}}}}-{\frac{{x}^{2}}{4}}+4\,\sqrt{x} \]

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 = 1.34178, size = 22, normalized size = 0.92 \[ -\frac{1}{4} \, x^{2} + \frac{2}{3} \, x^{\frac{3}{2}} + 4 \, \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2*x + sqrt(x) + 2/sqrt(x),x, algorithm="maxima")

[Out]

-1/4*x^2 + 2/3*x^(3/2) + 4*sqrt(x)

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Fricas [A]  time = 0.214198, size = 19, normalized size = 0.79 \[ -\frac{1}{4} \, x^{2} + \frac{2}{3} \,{\left (x + 6\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2*x + sqrt(x) + 2/sqrt(x),x, algorithm="fricas")

[Out]

-1/4*x^2 + 2/3*(x + 6)*sqrt(x)

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Sympy [A]  time = 0.037598, size = 19, normalized size = 0.79 \[ \frac{2 x^{\frac{3}{2}}}{3} + 4 \sqrt{x} - \frac{x^{2}}{4} \]

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/XCAS [A]  time = 0.227877, size = 22, normalized size = 0.92 \[ -\frac{1}{4} \, x^{2} + \frac{2}{3} \, x^{\frac{3}{2}} + 4 \, \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2*x + sqrt(x) + 2/sqrt(x),x, algorithm="giac")

[Out]

-1/4*x^2 + 2/3*x^(3/2) + 4*sqrt(x)