Optimal. Leaf size=59 \[ \frac {1}{3} \sqrt [4]{x} \sqrt {\sqrt [4]{x}+x}+\frac {2}{3} x \sqrt {\sqrt [4]{x}+x}-\frac {1}{3} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {\sqrt [4]{x}+x}}\right ) \]
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Rubi [A]
time = 0.04, 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 = {2029, 2043,
2049, 2054, 212} \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2029
Rule 2043
Rule 2049
Rule 2054
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}+\text {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} \text {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} \text {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 = 0.34, size = 47, normalized size = 0.80 \begin {gather*} \frac {1}{3} \sqrt {\sqrt [4]{x}+x} \left (\sqrt [4]{x}+2 x\right )-\frac {1}{3} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {\sqrt [4]{x}+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.53, size = 342, normalized size = 5.80
method | result | size |
meijerg | \(-\frac {2 \left (-\frac {\sqrt {\pi }\, x^{\frac {3}{8}} \left (6 x^{\frac {3}{4}}+3\right ) \sqrt {x^{\frac {3}{4}}+1}}{6}+\frac {\sqrt {\pi }\, \arcsinh \left (x^{\frac {3}{8}}\right )}{2}\right )}{3 \sqrt {\pi }}\) | \(38\) |
derivativedivides | \(\frac {2 x \sqrt {x^{\frac {1}{4}}+x}}{3}+\frac {x^{\frac {1}{4}} \sqrt {x^{\frac {1}{4}}+x}}{3}+\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x^{\frac {1}{4}}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x^{\frac {1}{4}}\right )}}\, \left (1+x^{\frac {1}{4}}\right )^{2} \sqrt {-\frac {x^{\frac {1}{4}}-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x^{\frac {1}{4}}\right )}}\, \sqrt {-\frac {x^{\frac {1}{4}}-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x^{\frac {1}{4}}\right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x^{\frac {1}{4}}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x^{\frac {1}{4}}\right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x^{\frac {1}{4}}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x^{\frac {1}{4}}\right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x^{\frac {1}{4}} \left (1+x^{\frac {1}{4}}\right ) \left (x^{\frac {1}{4}}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x^{\frac {1}{4}}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\) | \(342\) |
default | \(\frac {2 x \sqrt {x^{\frac {1}{4}}+x}}{3}+\frac {x^{\frac {1}{4}} \sqrt {x^{\frac {1}{4}}+x}}{3}+\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x^{\frac {1}{4}}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x^{\frac {1}{4}}\right )}}\, \left (1+x^{\frac {1}{4}}\right )^{2} \sqrt {-\frac {x^{\frac {1}{4}}-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x^{\frac {1}{4}}\right )}}\, \sqrt {-\frac {x^{\frac {1}{4}}-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x^{\frac {1}{4}}\right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x^{\frac {1}{4}}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x^{\frac {1}{4}}\right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x^{\frac {1}{4}}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x^{\frac {1}{4}}\right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x^{\frac {1}{4}} \left (1+x^{\frac {1}{4}}\right ) \left (x^{\frac {1}{4}}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x^{\frac {1}{4}}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\) | \(342\) |
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 [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 [4]{x} + x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.97, size = 45, normalized size = 0.76 \begin {gather*} \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 ) \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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