Integrand size = 26, antiderivative size = 88 \[ \int \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \, dx=\frac {3 b \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} x^{2/3}}{2 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {a \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} x}{a+\frac {b}{\sqrt [3]{x}}} \]
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Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1355, 1369, 14} \[ \int \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \, dx=\frac {3 b x^{2/3} \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{2 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {a x \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{a+\frac {b}{\sqrt [3]{x}}} \]
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Rule 14
Rule 1355
Rule 1369
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \sqrt {a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}} x^2 \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {\left (3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}\right ) \text {Subst}\left (\int \left (a b+\frac {b^2}{x}\right ) x^2 \, dx,x,\sqrt [3]{x}\right )}{a b+\frac {b^2}{\sqrt [3]{x}}} \\ & = \frac {\left (3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}\right ) \text {Subst}\left (\int \left (b^2 x+a b x^2\right ) \, dx,x,\sqrt [3]{x}\right )}{a b+\frac {b^2}{\sqrt [3]{x}}} \\ & = \frac {3 b^2 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} x^{2/3}}{2 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )}+\frac {a \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} x}{a+\frac {b}{\sqrt [3]{x}}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.56 \[ \int \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \, dx=\frac {\left (3 b+2 a \sqrt [3]{x}\right ) \sqrt {\frac {\left (b+a \sqrt [3]{x}\right )^2}{x^{2/3}}} x}{2 \left (b+a \sqrt [3]{x}\right )} \]
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Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.53
method | result | size |
derivativedivides | \(\frac {\sqrt {\frac {x^{\frac {2}{3}} a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2}}{x^{\frac {2}{3}}}}\, x \left (2 a \,x^{\frac {1}{3}}+3 b \right )}{2 b +2 a \,x^{\frac {1}{3}}}\) | \(47\) |
default | \(\frac {\sqrt {\frac {x^{\frac {2}{3}} a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2}}{x^{\frac {2}{3}}}}\, x^{\frac {1}{3}} \left (3 b \,x^{\frac {2}{3}}+2 a x \right )}{2 b +2 a \,x^{\frac {1}{3}}}\) | \(50\) |
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Timed out. \[ \int \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \, dx=\text {Timed out} \]
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\[ \int \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \, dx=\int \sqrt {a^{2} + \frac {2 a b}{\sqrt [3]{x}} + \frac {b^{2}}{x^{\frac {2}{3}}}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.11 \[ \int \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \, dx=a x + \frac {3}{2} \, b x^{\frac {2}{3}} \]
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Time = 0.31 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.39 \[ \int \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \, dx=a x \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) + \frac {3}{2} \, b x^{\frac {2}{3}} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) \]
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Time = 8.48 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.44 \[ \int \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \, dx=\frac {x\,\left (a+\frac {3\,b}{2\,x^{1/3}}\right )\,\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2\,a\,b}{x^{1/3}}}}{a+\frac {b}{x^{1/3}}} \]
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