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]

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

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Rubi [A]  time = 0.91, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {12, 6688} \[ \sqrt {x}+\sqrt {x+1}+\sqrt {x+2} \]

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]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {align*} \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 &=\frac {1}{2} \int \frac {\sqrt {x} \sqrt {1+x}+\sqrt {x} \sqrt {2+x}+\sqrt {1+x} \sqrt {2+x}}{\sqrt {x} \sqrt {1+x} \sqrt {2+x}} \, dx\\ &=\frac {1}{2} \int \left (\frac {1}{\sqrt {x}}+\frac {1}{\sqrt {1+x}}+\frac {1}{\sqrt {2+x}}\right ) \, dx\\ &=\sqrt {x}+\sqrt {1+x}+\sqrt {2+x}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 30, normalized size = 1.50 \[ \frac {1}{2} \left (2 \sqrt {x}+2 \sqrt {x+1}+2 \sqrt {x+2}\right ) \]

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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fricas [A]  time = 0.40, size = 14, normalized size = 0.70 \[ \sqrt {x + 2} + \sqrt {x + 1} + \sqrt {x} \]

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, algorithm="fricas")

[Out]

sqrt(x + 2) + sqrt(x + 1) + sqrt(x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x + 2} \sqrt {x + 1} + \sqrt {x + 2} \sqrt {x} + \sqrt {x + 1} \sqrt {x}}{2 \, \sqrt {x + 2} \sqrt {x + 1} \sqrt {x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(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, 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)*s
qrt(x)), x)

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maple [A]  time = 0.00, size = 15, normalized size = 0.75 \[ \sqrt {x}+\sqrt {x +1}+\sqrt {x +2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.42, size = 14, normalized size = 0.70 \[ \sqrt {x + 2} + \sqrt {x + 1} + \sqrt {x} \]

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, algorithm="maxima")

[Out]

sqrt(x + 2) + sqrt(x + 1) + sqrt(x)

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mupad [B]  time = 0.34, size = 14, normalized size = 0.70 \[ \sqrt {x+1}+\sqrt {x+2}+\sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.17, 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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