3.200 \(\int \frac{\sqrt{x} \sqrt{1+x}+\sqrt{x} \sqrt{2+x}+\sqrt{1+x} \sqrt{2+x}}{2 \sqrt{x} \sqrt{1+x} \sqrt{2+x}} \, dx\)

Optimal. Leaf size=20 \[ \sqrt{x}+\sqrt{x+1}+\sqrt{x+2} \]

[Out]

Sqrt[x] + Sqrt[1 + x] + Sqrt[2 + x]

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Rubi [A]  time = 1.70373, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 65, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.031 \[ \sqrt{x}+\sqrt{x+1}+\sqrt{x+2} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[x]*Sqrt[1 + x] + Sqrt[x]*Sqrt[2 + x] + Sqrt[1 + x]*Sqrt[2 + x])/(2*Sqrt[x]*Sqrt[1 + x]*Sqrt[2 + x]),x]

[Out]

Sqrt[x] + Sqrt[1 + x] + Sqrt[2 + x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \sqrt{x + 1} + \sqrt{x + 2} + 2 \int ^{\sqrt{x}} \frac{1}{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x + 1) + sqrt(x + 2) + 2*Integral(1/2, (x, sqrt(x)))

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Mathematica [A]  time = 0.0252207, size = 30, normalized size = 1.5 \[ \frac{1}{2} \left (2 \sqrt{x}+2 \sqrt{x+1}+2 \sqrt{x+2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[x]*Sqrt[1 + x] + Sqrt[x]*Sqrt[2 + x] + Sqrt[1 + x]*Sqrt[2 + x])/(2*Sqrt[x]*Sqrt[1 + x]*Sqrt[2 + x]),x]

[Out]

(2*Sqrt[x] + 2*Sqrt[1 + x] + 2*Sqrt[2 + x])/2

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Maple [A]  time = 0.002, size = 15, normalized size = 0.8 \[ \sqrt{x}+\sqrt{1+x}+\sqrt{2+x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/2*(x^(1/2)*(1+x)^(1/2)+x^(1/2)*(2+x)^(1/2)+(1+x)^(1/2)*(2+x)^(1/2))/x^(1/2)/(1+x)^(1/2)/(2+x)^(1/2),x)

[Out]

x^(1/2)+(1+x)^(1/2)+(2+x)^(1/2)

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Maxima [A]  time = 1.53255, size = 19, normalized size = 0.95 \[ \sqrt{x + 2} + \sqrt{x + 1} + \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x + 2) + sqrt(x + 1) + sqrt(x)

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Fricas [A]  time = 0.201664, size = 19, normalized size = 0.95 \[ \sqrt{x + 2} + \sqrt{x + 1} + \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x + 2) + sqrt(x + 1) + sqrt(x)

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Sympy [A]  time = 1.60816, size = 17, normalized size = 0.85 \[ \sqrt{x} + \sqrt{x + 1} + \sqrt{x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x) + sqrt(x + 1) + sqrt(x + 2)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x + 2} \sqrt{x + 1} + \sqrt{x + 2} \sqrt{x} + \sqrt{x + 1} \sqrt{x}}{2 \, \sqrt{x + 2} \sqrt{x + 1} \sqrt{x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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