Integrand size = 19, antiderivative size = 23 \[ \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {b x+c x^2}}{c \sqrt {x}} \]
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Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {662} \[ \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {b x+c x^2}}{c \sqrt {x}} \]
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Rule 662
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {b x+c x^2}}{c \sqrt {x}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {x (b+c x)}}{c \sqrt {x}} \]
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Time = 2.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78
method | result | size |
default | \(\frac {2 \sqrt {x \left (c x +b \right )}}{\sqrt {x}\, c}\) | \(18\) |
risch | \(\frac {2 \left (c x +b \right ) \sqrt {x}}{\sqrt {x \left (c x +b \right )}\, c}\) | \(23\) |
gosper | \(\frac {2 \left (c x +b \right ) \sqrt {x}}{c \sqrt {c \,x^{2}+b x}}\) | \(25\) |
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none
Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \, \sqrt {c x^{2} + b x}}{c \sqrt {x}} \]
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\[ \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx=\int \frac {\sqrt {x}}{\sqrt {x \left (b + c x\right )}}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \, \sqrt {c x + b}}{c} \]
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none
Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \, \sqrt {c x + b}}{c} - \frac {2 \, \sqrt {b}}{c} \]
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Timed out. \[ \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx=\int \frac {\sqrt {x}}{\sqrt {c\,x^2+b\,x}} \,d x \]
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