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