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 = 0.912736, 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, 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*}

Mathematica [A]  time = 0.0176992, 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 \begin{align*} \sqrt{x}+\sqrt{1+x}+\sqrt{2+x} \end{align*}

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 = 0.946515, size = 19, normalized size = 0.95 \begin{align*} \sqrt{x + 2} + \sqrt{x + 1} + \sqrt{x} \end{align*}

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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Fricas [A]  time = 1.51155, size = 50, normalized size = 2.5 \begin{align*} \sqrt{x + 2} + \sqrt{x + 1} + \sqrt{x} \end{align*}

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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Sympy [A]  time = 1.21279, size = 17, normalized size = 0.85 \begin{align*} \sqrt{x} + \sqrt{x + 1} + \sqrt{x + 2} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)