Integrand size = 16, antiderivative size = 33 \[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x}-b x^2}}\right )}{3 \sqrt {b}} \]
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Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2004, 2033, 209} \[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x}-b x^2}}\right )}{3 \sqrt {b}} \]
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Rule 209
Rule 2004
Rule 2033
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {\frac {a}{x}-b x^2}} \, dx \\ & = \frac {2}{3} \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {x}{\sqrt {\frac {a}{x}-b x^2}}\right ) \\ & = \frac {2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x}-b x^2}}\right )}{3 \sqrt {b}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.18 \[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=-\frac {2 i \sqrt {a-b x^3} \log \left (i \sqrt {b} x^{3/2}+\sqrt {a-b x^3}\right )}{3 \sqrt {b} \sqrt {x} \sqrt {\frac {a-b x^3}{x}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(25)=50\).
Time = 2.44 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.82
method | result | size |
default | \(-\frac {2 \left (-b \,x^{3}+a \right ) \arctan \left (\frac {\sqrt {x \left (-b \,x^{3}+a \right )}}{x^{2} \sqrt {b}}\right )}{3 \sqrt {\frac {-b \,x^{3}+a}{x}}\, \sqrt {x \left (-b \,x^{3}+a \right )}\, \sqrt {b}}\) | \(60\) |
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none
Time = 0.57 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.36 \[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=\left [-\frac {\sqrt {-b} \log \left (-8 \, b^{2} x^{6} + 8 \, a b x^{3} - a^{2} + 4 \, {\left (2 \, b x^{5} - a x^{2}\right )} \sqrt {-b} \sqrt {-\frac {b x^{3} - a}{x}}\right )}{6 \, b}, -\frac {\arctan \left (\frac {2 \, \sqrt {b} x^{2} \sqrt {-\frac {b x^{3} - a}{x}}}{2 \, b x^{3} - a}\right )}{3 \, \sqrt {b}}\right ] \]
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\[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=\int \frac {1}{\sqrt {\frac {a - b x^{3}}{x}}}\, dx \]
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\[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=\int { \frac {1}{\sqrt {-\frac {b x^{3} - a}{x}}} \,d x } \]
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Exception generated. \[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=\int \frac {1}{\sqrt {\frac {a-b\,x^3}{x}}} \,d x \]
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