Optimal. Leaf size=42 \[ \frac {6}{5} \left (\sqrt [3]{x}+1\right )^{5/2}-4 \left (\sqrt [3]{x}+1\right )^{3/2}+6 \sqrt {\sqrt [3]{x}+1} \]
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Rubi [A] time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac {6}{5} \left (\sqrt [3]{x}+1\right )^{5/2}-4 \left (\sqrt [3]{x}+1\right )^{3/2}+6 \sqrt {\sqrt [3]{x}+1} \]
Antiderivative was successfully verified.
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Rule 43
Rule 190
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+\sqrt [3]{x}}} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {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*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 0.74 \[ \frac {2}{5} \sqrt {\sqrt [3]{x}+1} \left (3 x^{2/3}-4 \sqrt [3]{x}+8\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 21, normalized size = 0.50 \[ \frac {2}{5} \, {\left (3 \, x^{\frac {2}{3}} - 4 \, x^{\frac {1}{3}} + 8\right )} \sqrt {x^{\frac {1}{3}} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 28, normalized size = 0.67 \[ \frac {6}{5} \, {\left (x^{\frac {1}{3}} + 1\right )}^{\frac {5}{2}} - 4 \, {\left (x^{\frac {1}{3}} + 1\right )}^{\frac {3}{2}} + 6 \, \sqrt {x^{\frac {1}{3}} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 29, normalized size = 0.69 \[ -4 \left (x^{\frac {1}{3}}+1\right )^{\frac {3}{2}}+\frac {6 \left (x^{\frac {1}{3}}+1\right )^{\frac {5}{2}}}{5}+6 \sqrt {x^{\frac {1}{3}}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 28, normalized size = 0.67 \[ \frac {6}{5} \, {\left (x^{\frac {1}{3}} + 1\right )}^{\frac {5}{2}} - 4 \, {\left (x^{\frac {1}{3}} + 1\right )}^{\frac {3}{2}} + 6 \, \sqrt {x^{\frac {1}{3}} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 12, normalized size = 0.29 \[ x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},3;\ 4;\ -x^{1/3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.36, size = 359, normalized size = 8.55 \[ \frac {6 x^{\frac {14}{3}} \sqrt {\sqrt [3]{x} + 1}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} + \frac {10 x^{\frac {13}{3}} \sqrt {\sqrt [3]{x} + 1}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} + \frac {30 x^{\frac {11}{3}} \sqrt {\sqrt [3]{x} + 1}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} - \frac {48 x^{\frac {11}{3}}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} + \frac {40 x^{\frac {10}{3}} \sqrt {\sqrt [3]{x} + 1}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} - \frac {48 x^{\frac {10}{3}}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} + \frac {10 x^{4} \sqrt {\sqrt [3]{x} + 1}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} - \frac {16 x^{4}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} + \frac {16 x^{3} \sqrt {\sqrt [3]{x} + 1}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} - \frac {16 x^{3}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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