Optimal. Leaf size=77 \[ \frac {2 \sqrt {\sqrt {x}+\sqrt {2 x+2 \sqrt {x}+1}+1} \left (6 x^{3/2}+\sqrt {x}-\left (2-\sqrt {x}\right ) \sqrt {2 x+2 \sqrt {x}+1}+2\right )}{15 \sqrt {x}} \]
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Rubi [A] time = 0.07, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2114} \[ \frac {2 \sqrt {\sqrt {x}+\sqrt {2 x+2 \sqrt {x}+1}+1} \left (6 x^{3/2}+\sqrt {x}-\left (2-\sqrt {x}\right ) \sqrt {2 x+2 \sqrt {x}+1}+2\right )}{15 \sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 2114
Rubi steps
\begin {align*} \int \sqrt {1+\sqrt {x}+\sqrt {1+2 \sqrt {x}+2 x}} \, dx &=2 \operatorname {Subst}\left (\int x \sqrt {1+x+\sqrt {1+2 x+2 x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \sqrt {1+\sqrt {x}+\sqrt {1+2 \sqrt {x}+2 x}} \left (2+\sqrt {x}+6 x^{3/2}-\left (2-\sqrt {x}\right ) \sqrt {1+2 \sqrt {x}+2 x}\right )}{15 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 74, normalized size = 0.96 \[ \frac {2 \sqrt {\sqrt {x}+\sqrt {2 x+2 \sqrt {x}+1}+1} \left (6 x^{3/2}+\sqrt {x}+\left (\sqrt {x}-2\right ) \sqrt {2 x+2 \sqrt {x}+1}+2\right )}{15 \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 56, normalized size = 0.73 \[ \frac {2 \, {\left (6 \, x^{2} + \sqrt {2 \, x + 2 \, \sqrt {x} + 1} {\left (x - 2 \, \sqrt {x}\right )} + x + 2 \, \sqrt {x}\right )} \sqrt {\sqrt {2 \, x + 2 \, \sqrt {x} + 1} + \sqrt {x} + 1}}{15 \, x} \]
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 \[ \int \sqrt {\sqrt {2 \, x + 2 \, \sqrt {x} + 1} + \sqrt {x} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \sqrt {\sqrt {x}+1+\sqrt {2 x +2 \sqrt {x}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sqrt {2 \, x + 2 \, \sqrt {x} + 1} + \sqrt {x} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {\sqrt {2\,x+2\,\sqrt {x}+1}+\sqrt {x}+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sqrt {x} + \sqrt {2 \sqrt {x} + 2 x + 1} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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