Optimal. Leaf size=118 \[ \frac {2 \sqrt {2} \sqrt {\sqrt {2}+\sqrt {x}+\sqrt {2} \sqrt {1+\sqrt {2} \sqrt {x}+x}} \left (4+\sqrt {2} \sqrt {x}+3 \sqrt {2} x^{3/2}-\sqrt {2} \left (2 \sqrt {2}-\sqrt {x}\right ) \sqrt {1+\sqrt {2} \sqrt {x}+x}\right )}{15 \sqrt {x}} \]
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Rubi [A]
time = 0.13, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2140, 2139}
\begin {gather*} \frac {2 \sqrt {2} \sqrt {\sqrt {x}+\sqrt {2} \sqrt {x+\sqrt {2} \sqrt {x}+1}+\sqrt {2}} \left (3 \sqrt {2} x^{3/2}+\sqrt {2} \sqrt {x}-\sqrt {2} \left (2 \sqrt {2}-\sqrt {x}\right ) \sqrt {x+\sqrt {2} \sqrt {x}+1}+4\right )}{15 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2139
Rule 2140
Rubi steps
\begin {align*} \int \sqrt {\sqrt {2}+\sqrt {x}+\sqrt {2+2 \sqrt {2} \sqrt {x}+2 x}} \, dx &=2 \text {Subst}\left (\int x \sqrt {x+\sqrt {2} \left (1+\sqrt {1+\sqrt {2} x+x^2}\right )} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int x \sqrt {\sqrt {2}+x+\sqrt {2} \sqrt {1+\sqrt {2} x+x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \sqrt {2} \sqrt {\sqrt {2}+\sqrt {x}+\sqrt {2} \sqrt {1+\sqrt {2} \sqrt {x}+x}} \left (4+\sqrt {2} \sqrt {x}+3 \sqrt {2} x^{3/2}-\sqrt {2} \left (2 \sqrt {2}-\sqrt {x}\right ) \sqrt {1+\sqrt {2} \sqrt {x}+x}\right )}{15 \sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 10.05, size = 112, normalized size = 0.95 \begin {gather*} \frac {2 \sqrt {2} \left (4+\sqrt {2} \sqrt {x}+3 \sqrt {2} x^{3/2}+\sqrt {2} \left (-2 \sqrt {2}+\sqrt {x}\right ) \sqrt {1+\sqrt {2} \sqrt {x}+x}\right ) \sqrt {\sqrt {x}+\sqrt {2} \left (1+\sqrt {1+\sqrt {2} \sqrt {x}+x}\right )}}{15 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \sqrt {\sqrt {2}+\sqrt {x}+\sqrt {2+2 x +2 \sqrt {2}\, \sqrt {x}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.83, size = 73, normalized size = 0.62 \begin {gather*} \frac {2 \, {\left (6 \, x^{2} + {\left (\sqrt {2} x - 4 \, \sqrt {x}\right )} \sqrt {2 \, \sqrt {2} \sqrt {x} + 2 \, x + 2} + 4 \, \sqrt {2} \sqrt {x} + 2 \, x\right )} \sqrt {\sqrt {2} + \sqrt {2 \, \sqrt {2} \sqrt {x} + 2 \, x + 2} + \sqrt {x}}}{15 \, x} \end {gather*}
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 \begin {gather*} \int \sqrt {\sqrt {x} + \sqrt {2 \sqrt {2} \sqrt {x} + 2 x + 2} + \sqrt {2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {\sqrt {2\,x+2\,\sqrt {2}\,\sqrt {x}+2}+\sqrt {2}+\sqrt {x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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