Optimal. Leaf size=20 \[ \sqrt{x}+\sqrt{x+1}+\sqrt{x+2} \]
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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.
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Rule 12
Rule 6688
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.
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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.
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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.
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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.
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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.
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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.
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