Integrand size = 21, antiderivative size = 26 \[ \int \frac {1+\sqrt {x}}{x^{5/6}+x^{7/6}} \, dx=3 \sqrt [3]{x}+6 \arctan \left (\sqrt [6]{x}\right )-3 \log \left (1+\sqrt [3]{x}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1607, 1833, 1824, 649, 209, 266} \[ \int \frac {1+\sqrt {x}}{x^{5/6}+x^{7/6}} \, dx=6 \arctan \left (\sqrt [6]{x}\right )+3 \sqrt [3]{x}-3 \log \left (\sqrt [3]{x}+1\right ) \]
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Rule 209
Rule 266
Rule 649
Rule 1607
Rule 1824
Rule 1833
Rubi steps \begin{align*} \text {integral}& = \int \frac {1+\sqrt {x}}{\left (1+\sqrt [3]{x}\right ) x^{5/6}} \, dx \\ & = 6 \text {Subst}\left (\int \frac {1+x^3}{1+x^2} \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \text {Subst}\left (\int \left (x+\frac {1-x}{1+x^2}\right ) \, dx,x,\sqrt [6]{x}\right ) \\ & = 3 \sqrt [3]{x}+6 \text {Subst}\left (\int \frac {1-x}{1+x^2} \, dx,x,\sqrt [6]{x}\right ) \\ & = 3 \sqrt [3]{x}+6 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [6]{x}\right )-6 \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,\sqrt [6]{x}\right ) \\ & = 3 \sqrt [3]{x}+6 \tan ^{-1}\left (\sqrt [6]{x}\right )-3 \log \left (1+\sqrt [3]{x}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {1+\sqrt {x}}{x^{5/6}+x^{7/6}} \, dx=3 \sqrt [3]{x}+6 \arctan \left (\sqrt [6]{x}\right )-3 \log \left (1+\sqrt [3]{x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81
\[3 x^{\frac {1}{3}}+6 \arctan \left (x^{\frac {1}{6}}\right )-3 \ln \left (1+x^{\frac {1}{3}}\right )\]
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Time = 0.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {1+\sqrt {x}}{x^{5/6}+x^{7/6}} \, dx=3 \, x^{\frac {1}{3}} + 6 \, \arctan \left (x^{\frac {1}{6}}\right ) - 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]
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Time = 0.95 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {1+\sqrt {x}}{x^{5/6}+x^{7/6}} \, dx=3 \sqrt [3]{x} - 3 \log {\left (\sqrt [3]{x} + 1 \right )} + 6 \operatorname {atan}{\left (\sqrt [6]{x} \right )} \]
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Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {1+\sqrt {x}}{x^{5/6}+x^{7/6}} \, dx=3 \, x^{\frac {1}{3}} + 6 \, \arctan \left (x^{\frac {1}{6}}\right ) - 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]
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Time = 0.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {1+\sqrt {x}}{x^{5/6}+x^{7/6}} \, dx=3 \, x^{\frac {1}{3}} + 6 \, \arctan \left (x^{\frac {1}{6}}\right ) - 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]
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Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {1+\sqrt {x}}{x^{5/6}+x^{7/6}} \, dx=6\,\mathrm {atan}\left (x^{1/6}\right )-3\,\ln \left (36\,x^{1/3}+36\right )+3\,x^{1/3} \]
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