Integrand size = 26, antiderivative size = 190 \[ \int \frac {1}{\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}} \, dx=\frac {3 b^2 \left (a+\frac {b}{\sqrt [3]{x}}\right ) \sqrt [3]{x}}{a^3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}-\frac {3 b \left (a+\frac {b}{\sqrt [3]{x}}\right ) x^{2/3}}{2 a^2 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}+\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right ) x}{a \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}-\frac {3 b^3 \left (a+\frac {b}{\sqrt [3]{x}}\right ) \log \left (b+a \sqrt [3]{x}\right )}{a^4 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}} \]
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Time = 0.08 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1355, 1369, 269, 45} \[ \int \frac {1}{\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}} \, dx=-\frac {3 b x^{2/3} \left (a+\frac {b}{\sqrt [3]{x}}\right )}{2 a^2 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}+\frac {x \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}-\frac {3 b^3 \left (a+\frac {b}{\sqrt [3]{x}}\right ) \log \left (a \sqrt [3]{x}+b\right )}{a^4 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}+\frac {3 b^2 \sqrt [3]{x} \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^3 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}} \]
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Rule 45
Rule 269
Rule 1355
Rule 1369
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {x^2}{\sqrt {a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}}} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {\left (3 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )\right ) \text {Subst}\left (\int \frac {x^2}{a b+\frac {b^2}{x}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}} \\ & = \frac {\left (3 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )\right ) \text {Subst}\left (\int \frac {x^3}{b^2+a b x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}} \\ & = \frac {\left (3 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )\right ) \text {Subst}\left (\int \left (\frac {b}{a^3}-\frac {x}{a^2}+\frac {x^2}{a b}-\frac {b^2}{a^3 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}} \\ & = \frac {3 \left (a b^2+\frac {b^3}{\sqrt [3]{x}}\right ) \sqrt [3]{x}}{a^3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}-\frac {3 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right ) x^{2/3}}{2 a^2 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}+\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right ) x}{a \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}-\frac {3 \left (a b^3+\frac {b^4}{\sqrt [3]{x}}\right ) \log \left (b+a \sqrt [3]{x}\right )}{a^4 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.45 \[ \int \frac {1}{\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}} \, dx=\frac {\left (b+a \sqrt [3]{x}\right ) \left (6 a b^2 \sqrt [3]{x}-3 a^2 b x^{2/3}+2 a^3 x-6 b^3 \log \left (b+a \sqrt [3]{x}\right )\right )}{2 a^4 \sqrt {\frac {\left (b+a \sqrt [3]{x}\right )^2}{x^{2/3}}} \sqrt [3]{x}} \]
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Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.41
method | result | size |
derivativedivides | \(-\frac {\left (b +a \,x^{\frac {1}{3}}\right ) \left (-2 a^{3} x +3 a^{2} b \,x^{\frac {2}{3}}+6 b^{3} \ln \left (b +a \,x^{\frac {1}{3}}\right )-6 b^{2} a \,x^{\frac {1}{3}}\right )}{2 \sqrt {\frac {x^{\frac {2}{3}} a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2}}{x^{\frac {2}{3}}}}\, x^{\frac {1}{3}} a^{4}}\) | \(78\) |
default | \(-\frac {\left (b +a \,x^{\frac {1}{3}}\right ) \left (-2 a^{3} x +3 a^{2} b \,x^{\frac {2}{3}}+6 b^{3} \ln \left (b +a \,x^{\frac {1}{3}}\right )-6 b^{2} a \,x^{\frac {1}{3}}\right )}{2 \sqrt {\frac {x^{\frac {2}{3}} a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2}}{x^{\frac {2}{3}}}}\, x^{\frac {1}{3}} a^{4}}\) | \(78\) |
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Timed out. \[ \int \frac {1}{\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}} \, dx=\int \frac {1}{\sqrt {a^{2} + \frac {2 a b}{\sqrt [3]{x}} + \frac {b^{2}}{x^{\frac {2}{3}}}}}\, dx \]
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none
Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.23 \[ \int \frac {1}{\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}} \, dx=-\frac {3 \, b^{3} \log \left (a x^{\frac {1}{3}} + b\right )}{a^{4}} + \frac {2 \, a^{2} x - 3 \, a b x^{\frac {2}{3}} + 6 \, b^{2} x^{\frac {1}{3}}}{2 \, a^{3}} \]
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Time = 0.34 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.41 \[ \int \frac {1}{\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}} \, dx=-\frac {3 \, b^{3} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{4} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right )} + \frac {2 \, a^{2} x - 3 \, a b x^{\frac {2}{3}} + 6 \, b^{2} x^{\frac {1}{3}}}{2 \, a^{3} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {1}{\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}} \, dx=\int \frac {1}{\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2\,a\,b}{x^{1/3}}}} \,d x \]
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