Optimal. Leaf size=118 \[ \frac {2 \sqrt {2} \sqrt {\sqrt {x}+\sqrt {2} \sqrt {x+\sqrt {2} \sqrt {x}+1}+\sqrt {2}} \left (3 \sqrt {2} x^{3/2}+\sqrt {2} \sqrt {x}-\sqrt {2} \left (2 \sqrt {2}-\sqrt {x}\right ) \sqrt {x+\sqrt {2} \sqrt {x}+1}+4\right )}{15 \sqrt {x}} \]
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Rubi [A] time = 0.19, 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 = {2115, 2114} \[ \frac {2 \sqrt {2} \sqrt {\sqrt {x}+\sqrt {2} \sqrt {x+\sqrt {2} \sqrt {x}+1}+\sqrt {2}} \left (3 \sqrt {2} x^{3/2}+\sqrt {2} \sqrt {x}-\sqrt {2} \left (2 \sqrt {2}-\sqrt {x}\right ) \sqrt {x+\sqrt {2} \sqrt {x}+1}+4\right )}{15 \sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 2114
Rule 2115
Rubi steps
\begin {align*} \int \sqrt {\sqrt {2}+\sqrt {x}+\sqrt {2+2 \sqrt {2} \sqrt {x}+2 x}} \, dx &=2 \operatorname {Subst}\left (\int x \sqrt {x+\sqrt {2} \left (1+\sqrt {1+\sqrt {2} x+x^2}\right )} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {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 = 0.09, size = 112, normalized size = 0.95 \[ \frac {2 \sqrt {2} \left (3 \sqrt {2} x^{3/2}+\sqrt {2} \sqrt {x}+\sqrt {2} \left (\sqrt {x}-2 \sqrt {2}\right ) \sqrt {x+\sqrt {2} \sqrt {x}+1}+4\right ) \sqrt {\sqrt {2} \left (\sqrt {x+\sqrt {2} \sqrt {x}+1}+1\right )+\sqrt {x}}}{15 \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 73, normalized size = 0.62 \[ \frac {2 \, {\left (6 \, x^{2} + {\left (\sqrt {2} x - 4 \, \sqrt {x}\right )} \sqrt {2 \, \sqrt {2} \sqrt {x} + 2 \, x + 2} + 4 \, \sqrt {2} \sqrt {x} + 2 \, x\right )} \sqrt {\sqrt {2} + \sqrt {2 \, \sqrt {2} \sqrt {x} + 2 \, x + 2} + \sqrt {x}}}{15 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \sqrt {\sqrt {x}+\sqrt {2}+\sqrt {2 x +2 \sqrt {2}\, \sqrt {x}+2}}\, dx \]
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 \[ \int \sqrt {\sqrt {2} + \sqrt {2 \, \sqrt {2} \sqrt {x} + 2 \, x + 2} + \sqrt {x}}\,{d x} \]
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 \[ \int \sqrt {\sqrt {2\,x+2\,\sqrt {2}\,\sqrt {x}+2}+\sqrt {2}+\sqrt {x}} \,d x \]
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 \[ \int \sqrt {\sqrt {x} + \sqrt {2 \sqrt {2} \sqrt {x} + 2 x + 2} + \sqrt {2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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