3.2.100 \(\int \frac {\sqrt {x} \sqrt {1+x}+\sqrt {x} \sqrt {2+x}+\sqrt {1+x} \sqrt {2+x}}{2 \sqrt {x} \sqrt {1+x} \sqrt {2+x}} \, dx\) [200]

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

[Out]

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

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Rubi [A]
time = 0.60, 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, 6820} \begin {gather*} \sqrt {x}+\sqrt {x+1}+\sqrt {x+2} \end {gather*}

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 6820

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 \begin {gather*} \frac {1}{2} \left (2 \sqrt {x}+2 \sqrt {1+x}+2 \sqrt {2+x}\right ) \end {gather*}

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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Mathics [A]
time = 1.82, size = 14, normalized size = 0.70 \begin {gather*} \sqrt {x}+\sqrt {1+x}+\sqrt {2+x} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('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]

Sqrt[x] + Sqrt[1 + x] + Sqrt[2 + x]

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Maple [A]
time = 0.05, size = 15, normalized size = 0.75

method	result	size
default	\(\sqrt {x}+\sqrt {1+x}+\sqrt {2+x}\)	\(15\)

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,me
thod=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 14, normalized size = 0.70 \begin {gather*} \sqrt {x + 2} + \sqrt {x + 1} + \sqrt {x} \end {gather*}

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 = 0.30, size = 14, normalized size = 0.70 \begin {gather*} \sqrt {x + 2} + \sqrt {x + 1} + \sqrt {x} \end {gather*}

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 = 0.26, size = 17, normalized size = 0.85 \begin {gather*} \sqrt {x} + \sqrt {x + 1} + \sqrt {x + 2} \end {gather*}

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] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}

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]

Could not integrate

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Mupad [B]
time = 0.34, size = 14, normalized size = 0.70 \begin {gather*} \sqrt {x+1}+\sqrt {x+2}+\sqrt {x} \end {gather*}

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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