Integrand size = 26, antiderivative size = 29 \[ \int \frac {\sqrt {b-\frac {a}{x}}}{x^2 \sqrt {a-b x}} \, dx=-\frac {2 \sqrt {b-\frac {a}{x}}}{3 x \sqrt {a-b x}} \]
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Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {529, 23, 30} \[ \int \frac {\sqrt {b-\frac {a}{x}}}{x^2 \sqrt {a-b x}} \, dx=-\frac {2 \sqrt {b-\frac {a}{x}}}{3 x \sqrt {a-b x}} \]
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Rule 23
Rule 30
Rule 529
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {b-\frac {a}{x}} \sqrt {x}\right ) \int \frac {\sqrt {-a+b x}}{x^{5/2} \sqrt {a-b x}} \, dx}{\sqrt {-a+b x}} \\ & = \frac {\left (\sqrt {b-\frac {a}{x}} \sqrt {x}\right ) \int \frac {1}{x^{5/2}} \, dx}{\sqrt {a-b x}} \\ & = -\frac {2 \sqrt {b-\frac {a}{x}}}{3 x \sqrt {a-b x}} \\ \end{align*}
Time = 3.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {b-\frac {a}{x}}}{x^2 \sqrt {a-b x}} \, dx=\frac {2 \left (b-\frac {a}{x}\right )^{3/2}}{3 (a-b x)^{3/2}} \]
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Time = 1.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93
method | result | size |
gosper | \(-\frac {2 \sqrt {-\frac {-b x +a}{x}}}{3 x \sqrt {-b x +a}}\) | \(27\) |
default | \(-\frac {2 \sqrt {-\frac {-b x +a}{x}}}{3 x \sqrt {-b x +a}}\) | \(27\) |
risch | \(\frac {2 \sqrt {-\frac {-b x +a}{x}}\, \sqrt {-\left (-b x +a \right ) x}\, \sqrt {\frac {x \left (-b x +a \right )}{b x -a}}\, \left (b x -a \right )}{3 \left (-b x +a \right )^{\frac {3}{2}} \sqrt {\left (b x -a \right ) x}\, \sqrt {-x}\, x}\) | \(80\) |
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none
Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {b-\frac {a}{x}}}{x^2 \sqrt {a-b x}} \, dx=\frac {2 \, \sqrt {-b x + a} \sqrt {\frac {b x - a}{x}}}{3 \, {\left (b x^{2} - a x\right )}} \]
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\[ \int \frac {\sqrt {b-\frac {a}{x}}}{x^2 \sqrt {a-b x}} \, dx=\int \frac {\sqrt {- \frac {a}{x} + b}}{x^{2} \sqrt {a - b x}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.17 \[ \int \frac {\sqrt {b-\frac {a}{x}}}{x^2 \sqrt {a-b x}} \, dx=\frac {2 i}{3 \, x^{\frac {3}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (23) = 46\).
Time = 0.38 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.07 \[ \int \frac {\sqrt {b-\frac {a}{x}}}{x^2 \sqrt {a-b x}} \, dx=\frac {2 \, {\left (\frac {b^{5}}{{\left ({\left (b x - a\right )} b + a b\right )} \sqrt {-{\left (b x - a\right )} b - a b}} - \frac {b^{4}}{\sqrt {-a b} a}\right )} {\left | b \right |} \mathrm {sgn}\left (x\right )}{3 \, b^{3}} \]
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Time = 16.74 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {b-\frac {a}{x}}}{x^2 \sqrt {a-b x}} \, dx=-\frac {2\,\sqrt {b-\frac {a}{x}}}{3\,x\,\sqrt {a-b\,x}} \]
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