Optimal. Leaf size=74 \[ \frac {2}{3} \sqrt {x+\sqrt {x}} x+\frac {1}{6} \sqrt {x+\sqrt {x}} \sqrt {x}-\frac {\sqrt {x+\sqrt {x}}}{4}+\frac {1}{4} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {x+\sqrt {x}}}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {2004, 2018, 670, 640, 620, 206} \[ \frac {2}{3} \sqrt {x+\sqrt {x}} x+\frac {1}{6} \sqrt {x+\sqrt {x}} \sqrt {x}-\frac {\sqrt {x+\sqrt {x}}}{4}+\frac {1}{4} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {x+\sqrt {x}}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 640
Rule 670
Rule 2004
Rule 2018
Rubi steps
\begin {align*} \int \sqrt {\sqrt {x}+x} \, dx &=\frac {2}{3} x \sqrt {\sqrt {x}+x}+\frac {1}{6} \int \frac {\sqrt {x}}{\sqrt {\sqrt {x}+x}} \, dx\\ &=\frac {2}{3} x \sqrt {\sqrt {x}+x}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {x+x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {1}{6} \sqrt {x} \sqrt {\sqrt {x}+x}+\frac {2}{3} x \sqrt {\sqrt {x}+x}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {x}{\sqrt {x+x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{4} \sqrt {\sqrt {x}+x}+\frac {1}{6} \sqrt {x} \sqrt {\sqrt {x}+x}+\frac {2}{3} x \sqrt {\sqrt {x}+x}+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {x+x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{4} \sqrt {\sqrt {x}+x}+\frac {1}{6} \sqrt {x} \sqrt {\sqrt {x}+x}+\frac {2}{3} x \sqrt {\sqrt {x}+x}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {\sqrt {x}+x}}\right )\\ &=-\frac {1}{4} \sqrt {\sqrt {x}+x}+\frac {1}{6} \sqrt {x} \sqrt {\sqrt {x}+x}+\frac {2}{3} x \sqrt {\sqrt {x}+x}+\frac {1}{4} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {\sqrt {x}+x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 51, normalized size = 0.69 \[ \frac {1}{12} \sqrt {x+\sqrt {x}} \left (8 x+2 \sqrt {x}+\frac {3 \sinh ^{-1}\left (\sqrt [4]{x}\right )}{\sqrt {\sqrt {x}+1} \sqrt [4]{x}}-3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 49, normalized size = 0.66 \[ \frac {1}{12} \, {\left (8 \, x + 2 \, \sqrt {x} - 3\right )} \sqrt {x + \sqrt {x}} + \frac {1}{16} \, \log \left (4 \, \sqrt {x + \sqrt {x}} {\left (2 \, \sqrt {x} + 1\right )} + 8 \, x + 8 \, \sqrt {x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 43, normalized size = 0.58 \[ \frac {1}{12} \, {\left (2 \, \sqrt {x} {\left (4 \, \sqrt {x} + 1\right )} - 3\right )} \sqrt {x + \sqrt {x}} - \frac {1}{8} \, \log \left (-2 \, \sqrt {x + \sqrt {x}} + 2 \, \sqrt {x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 42, normalized size = 0.57 \[ \frac {\ln \left (\sqrt {x}+\frac {1}{2}+\sqrt {x +\sqrt {x}}\right )}{8}+\frac {2 \left (x +\sqrt {x}\right )^{\frac {3}{2}}}{3}-\frac {\left (2 \sqrt {x}+1\right ) \sqrt {x +\sqrt {x}}}{4} \]
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 \[ \int \sqrt {x + \sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.16, size = 27, normalized size = 0.36 \[ \frac {4\,x\,\sqrt {x+\sqrt {x}}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {5}{2};\ \frac {7}{2};\ -\sqrt {x}\right )}{5\,\sqrt {\sqrt {x}+1}} \]
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 \[ \int \sqrt {\sqrt {x} + x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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