Integrand size = 26, antiderivative size = 36 \[ \int \frac {\sqrt {b-\frac {a}{x}} x^m}{\sqrt {a-b x}} \, dx=\frac {2 \sqrt {b-\frac {a}{x}} x^{1+m}}{(1+2 m) \sqrt {a-b x}} \]
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Time = 0.03 (sec) , antiderivative size = 36, 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} \[ \int \frac {\sqrt {b-\frac {a}{x}} x^m}{\sqrt {a-b x}} \, dx=\frac {2 x^{m+1} \sqrt {b-\frac {a}{x}}}{(2 m+1) \sqrt {a-b x}} \]
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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 {x^{-\frac {1}{2}+m} \sqrt {-a+b x}}{\sqrt {a-b x}} \, dx}{\sqrt {-a+b x}} \\ & = \frac {\left (\sqrt {b-\frac {a}{x}} \sqrt {x}\right ) \int x^{-\frac {1}{2}+m} \, dx}{\sqrt {a-b x}} \\ & = \frac {2 \sqrt {b-\frac {a}{x}} x^{1+m}}{(1+2 m) \sqrt {a-b x}} \\ \end{align*}
Time = 1.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {b-\frac {a}{x}} x^m}{\sqrt {a-b x}} \, dx=\frac {\sqrt {b-\frac {a}{x}} x^{1+m}}{\left (\frac {1}{2}+m\right ) \sqrt {a-b x}} \]
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Time = 0.69 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00
method | result | size |
gosper | \(\frac {2 x^{1+m} \sqrt {-\frac {-b x +a}{x}}}{\left (1+2 m \right ) \sqrt {-b x +a}}\) | \(36\) |
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none
Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt {b-\frac {a}{x}} x^m}{\sqrt {a-b x}} \, dx=\frac {2 \, \sqrt {-b x + a} x x^{m} \sqrt {\frac {b x - a}{x}}}{2 \, a m - {\left (2 \, b m + b\right )} x + a} \]
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\[ \int \frac {\sqrt {b-\frac {a}{x}} x^m}{\sqrt {a-b x}} \, dx=\int \frac {x^{m} \sqrt {- \frac {a}{x} + b}}{\sqrt {a - b x}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt {b-\frac {a}{x}} x^m}{\sqrt {a-b x}} \, dx=\frac {2 \, \sqrt {x} x^{m}}{2 i \, m + i} \]
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Exception generated. \[ \int \frac {\sqrt {b-\frac {a}{x}} x^m}{\sqrt {a-b x}} \, dx=\text {Exception raised: TypeError} \]
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Time = 17.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {b-\frac {a}{x}} x^m}{\sqrt {a-b x}} \, dx=\frac {2\,x^{m+1}\,\sqrt {b-\frac {a}{x}}}{\left (2\,m+1\right )\,\sqrt {a-b\,x}} \]
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