Optimal. Leaf size=59 \[ \frac {2}{3} \sqrt {x+\sqrt [4]{x}} x+\frac {1}{3} \sqrt {x+\sqrt [4]{x}} \sqrt [4]{x}-\frac {1}{3} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {x+\sqrt [4]{x}}}\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {2004, 2018, 2024, 2029, 206} \[ \frac {2}{3} \sqrt {x+\sqrt [4]{x}} x+\frac {1}{3} \sqrt {x+\sqrt [4]{x}} \sqrt [4]{x}-\frac {1}{3} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {x+\sqrt [4]{x}}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 2004
Rule 2018
Rule 2024
Rule 2029
Rubi steps
\begin {align*} \int \sqrt {\sqrt [4]{x}+x} \, dx &=\frac {2}{3} x \sqrt {\sqrt [4]{x}+x}+\frac {1}{4} \int \frac {\sqrt [4]{x}}{\sqrt {\sqrt [4]{x}+x}} \, dx\\ &=\frac {2}{3} x \sqrt {\sqrt [4]{x}+x}+\operatorname {Subst}\left (\int \frac {x^4}{\sqrt {x+x^4}} \, dx,x,\sqrt [4]{x}\right )\\ &=\frac {1}{3} \sqrt [4]{x} \sqrt {\sqrt [4]{x}+x}+\frac {2}{3} x \sqrt {\sqrt [4]{x}+x}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\sqrt {x+x^4}} \, dx,x,\sqrt [4]{x}\right )\\ &=\frac {1}{3} \sqrt [4]{x} \sqrt {\sqrt [4]{x}+x}+\frac {2}{3} x \sqrt {\sqrt [4]{x}+x}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {\sqrt [4]{x}+x}}\right )\\ &=\frac {1}{3} \sqrt [4]{x} \sqrt {\sqrt [4]{x}+x}+\frac {2}{3} x \sqrt {\sqrt [4]{x}+x}-\frac {1}{3} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {\sqrt [4]{x}+x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 57, normalized size = 0.97 \[ \frac {3 x^{5/4}-\sqrt {x^{3/4}+1} \sqrt [8]{x} \sinh ^{-1}\left (x^{3/8}\right )+2 x^2+\sqrt {x}}{3 \sqrt {x+\sqrt [4]{x}}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 4.57, size = 45, normalized size = 0.76 \[ \frac {1}{3} \, \sqrt {x + x^{\frac {1}{4}}} x^{\frac {1}{4}} {\left (2 \, x^{\frac {3}{4}} + 1\right )} - \frac {1}{6} \, \log \left (\sqrt {\frac {1}{x^{\frac {3}{4}}} + 1} + 1\right ) + \frac {1}{6} \, \log \left ({\left | \sqrt {\frac {1}{x^{\frac {3}{4}}} + 1} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 342, normalized size = 5.80 \[ \frac {2 \sqrt {x +x^{\frac {1}{4}}}\, x}{3}+\frac {\sqrt {x +x^{\frac {1}{4}}}\, x^{\frac {1}{4}}}{3}+\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x^{\frac {1}{4}}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x^{\frac {1}{4}}+1\right )}}\, \left (x^{\frac {1}{4}}+1\right )^{2} \sqrt {-\frac {x^{\frac {1}{4}}-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x^{\frac {1}{4}}+1\right )}}\, \sqrt {-\frac {x^{\frac {1}{4}}-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x^{\frac {1}{4}}+1\right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x^{\frac {1}{4}}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x^{\frac {1}{4}}+1\right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x^{\frac {1}{4}}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x^{\frac {1}{4}}+1\right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\left (x^{\frac {1}{4}}+1\right ) \left (x^{\frac {1}{4}}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x^{\frac {1}{4}}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{\frac {1}{4}}}} \]
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 \[ \int \sqrt {x + x^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.53, size = 27, normalized size = 0.46 \[ \frac {8\,x\,\sqrt {x+x^{1/4}}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {3}{2};\ \frac {5}{2};\ -x^{3/4}\right )}{9\,\sqrt {x^{3/4}+1}} \]
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 \[ \int \sqrt {\sqrt [4]{x} + x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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