Optimal. Leaf size=41 \[ \frac {6 x^{7/6}}{7}-\frac {6 x^{5/6}}{5}+2 \sqrt {x}-6 \sqrt [6]{x}+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {341, 50, 63, 203} \[ \frac {6 x^{7/6}}{7}-\frac {6 x^{5/6}}{5}+2 \sqrt {x}-6 \sqrt [6]{x}+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 341
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{1+\sqrt [3]{x}} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^{7/2}}{1+x} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {6 x^{7/6}}{7}-3 \operatorname {Subst}\left (\int \frac {x^{5/2}}{1+x} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {6 x^{5/6}}{5}+\frac {6 x^{7/6}}{7}+3 \operatorname {Subst}\left (\int \frac {x^{3/2}}{1+x} \, dx,x,\sqrt [3]{x}\right )\\ &=2 \sqrt {x}-\frac {6 x^{5/6}}{5}+\frac {6 x^{7/6}}{7}-3 \operatorname {Subst}\left (\int \frac {\sqrt {x}}{1+x} \, dx,x,\sqrt [3]{x}\right )\\ &=-6 \sqrt [6]{x}+2 \sqrt {x}-\frac {6 x^{5/6}}{5}+\frac {6 x^{7/6}}{7}+3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} (1+x)} \, dx,x,\sqrt [3]{x}\right )\\ &=-6 \sqrt [6]{x}+2 \sqrt {x}-\frac {6 x^{5/6}}{5}+\frac {6 x^{7/6}}{7}+6 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=-6 \sqrt [6]{x}+2 \sqrt {x}-\frac {6 x^{5/6}}{5}+\frac {6 x^{7/6}}{7}+6 \tan ^{-1}\left (\sqrt [6]{x}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 41, normalized size = 1.00 \[ \frac {6 x^{7/6}}{7}-\frac {6 x^{5/6}}{5}+2 \sqrt {x}-6 \sqrt [6]{x}+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 25, normalized size = 0.61 \[ \frac {6}{7} \, {\left (x - 7\right )} x^{\frac {1}{6}} - \frac {6}{5} \, x^{\frac {5}{6}} + 2 \, \sqrt {x} + 6 \, \arctan \left (x^{\frac {1}{6}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 27, normalized size = 0.66 \[ \frac {6}{7} \, x^{\frac {7}{6}} - \frac {6}{5} \, x^{\frac {5}{6}} + 2 \, \sqrt {x} - 6 \, x^{\frac {1}{6}} + 6 \, \arctan \left (x^{\frac {1}{6}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 28, normalized size = 0.68 \[ \frac {6 x^{\frac {7}{6}}}{7}+6 \arctan \left (x^{\frac {1}{6}}\right )-\frac {6 x^{\frac {5}{6}}}{5}+2 \sqrt {x}-6 x^{\frac {1}{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.63, size = 27, normalized size = 0.66 \[ \frac {6}{7} \, x^{\frac {7}{6}} - \frac {6}{5} \, x^{\frac {5}{6}} + 2 \, \sqrt {x} - 6 \, x^{\frac {1}{6}} + 6 \, \arctan \left (x^{\frac {1}{6}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 27, normalized size = 0.66 \[ 6\,\mathrm {atan}\left (x^{1/6}\right )+2\,\sqrt {x}-6\,x^{1/6}-\frac {6\,x^{5/6}}{5}+\frac {6\,x^{7/6}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.36, size = 37, normalized size = 0.90 \[ \frac {6 x^{\frac {7}{6}}}{7} - \frac {6 x^{\frac {5}{6}}}{5} - 6 \sqrt [6]{x} + 2 \sqrt {x} + 6 \operatorname {atan}{\left (\sqrt [6]{x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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