Integrand size = 11, antiderivative size = 42 \[ \int \frac {1}{\sqrt {1+\sqrt [3]{x}}} \, dx=6 \sqrt {1+\sqrt [3]{x}}-4 \left (1+\sqrt [3]{x}\right )^{3/2}+\frac {6}{5} \left (1+\sqrt [3]{x}\right )^{5/2} \]
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Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {196, 45} \[ \int \frac {1}{\sqrt {1+\sqrt [3]{x}}} \, dx=\frac {6}{5} \left (\sqrt [3]{x}+1\right )^{5/2}-4 \left (\sqrt [3]{x}+1\right )^{3/2}+6 \sqrt {\sqrt [3]{x}+1} \]
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Rule 45
Rule 196
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x}} \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {Subst}\left (\int \left (\frac {1}{\sqrt {1+x}}-2 \sqrt {1+x}+(1+x)^{3/2}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = 6 \sqrt {1+\sqrt [3]{x}}-4 \left (1+\sqrt [3]{x}\right )^{3/2}+\frac {6}{5} \left (1+\sqrt [3]{x}\right )^{5/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\sqrt {1+\sqrt [3]{x}}} \, dx=\frac {2}{5} \sqrt {1+\sqrt [3]{x}} \left (8-4 \sqrt [3]{x}+3 x^{2/3}\right ) \]
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Time = 3.61 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.69
method | result | size |
derivativedivides | \(-4 \left (1+x^{\frac {1}{3}}\right )^{\frac {3}{2}}+\frac {6 \left (1+x^{\frac {1}{3}}\right )^{\frac {5}{2}}}{5}+6 \sqrt {1+x^{\frac {1}{3}}}\) | \(29\) |
default | \(-4 \left (1+x^{\frac {1}{3}}\right )^{\frac {3}{2}}+\frac {6 \left (1+x^{\frac {1}{3}}\right )^{\frac {5}{2}}}{5}+6 \sqrt {1+x^{\frac {1}{3}}}\) | \(29\) |
meijerg | \(\frac {-\frac {16 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (6 x^{\frac {2}{3}}-8 x^{\frac {1}{3}}+16\right ) \sqrt {1+x^{\frac {1}{3}}}}{5}}{\sqrt {\pi }}\) | \(36\) |
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Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.50 \[ \int \frac {1}{\sqrt {1+\sqrt [3]{x}}} \, dx=\frac {2}{5} \, {\left (3 \, x^{\frac {2}{3}} - 4 \, x^{\frac {1}{3}} + 8\right )} \sqrt {x^{\frac {1}{3}} + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (36) = 72\).
Time = 0.76 (sec) , antiderivative size = 359, normalized size of antiderivative = 8.55 \[ \int \frac {1}{\sqrt {1+\sqrt [3]{x}}} \, dx=\frac {6 x^{\frac {14}{3}} \sqrt {\sqrt [3]{x} + 1}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} + \frac {10 x^{\frac {13}{3}} \sqrt {\sqrt [3]{x} + 1}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} + \frac {30 x^{\frac {11}{3}} \sqrt {\sqrt [3]{x} + 1}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} - \frac {48 x^{\frac {11}{3}}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} + \frac {40 x^{\frac {10}{3}} \sqrt {\sqrt [3]{x} + 1}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} - \frac {48 x^{\frac {10}{3}}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} + \frac {10 x^{4} \sqrt {\sqrt [3]{x} + 1}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} - \frac {16 x^{4}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} + \frac {16 x^{3} \sqrt {\sqrt [3]{x} + 1}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} - \frac {16 x^{3}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} \]
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Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\sqrt {1+\sqrt [3]{x}}} \, dx=\frac {6}{5} \, {\left (x^{\frac {1}{3}} + 1\right )}^{\frac {5}{2}} - 4 \, {\left (x^{\frac {1}{3}} + 1\right )}^{\frac {3}{2}} + 6 \, \sqrt {x^{\frac {1}{3}} + 1} \]
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Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\sqrt {1+\sqrt [3]{x}}} \, dx=\frac {6}{5} \, {\left (x^{\frac {1}{3}} + 1\right )}^{\frac {5}{2}} - 4 \, {\left (x^{\frac {1}{3}} + 1\right )}^{\frac {3}{2}} + 6 \, \sqrt {x^{\frac {1}{3}} + 1} \]
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Time = 5.91 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\sqrt {1+\sqrt [3]{x}}} \, dx=x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},3;\ 4;\ -x^{1/3}\right ) \]
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