Integrand size = 35, antiderivative size = 26 \[ \int \frac {\sqrt {b x+\sqrt {a+b^2 x^2}}}{\sqrt {a+b^2 x^2}} \, dx=\frac {2 \sqrt {b x+\sqrt {a+b^2 x^2}}}{b} \]
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Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {2147, 30} \[ \int \frac {\sqrt {b x+\sqrt {a+b^2 x^2}}}{\sqrt {a+b^2 x^2}} \, dx=\frac {2 \sqrt {\sqrt {a+b^2 x^2}+b x}}{b} \]
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Rule 30
Rule 2147
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\sqrt {x}} \, dx,x,b x+\sqrt {a+b^2 x^2}\right )}{b} \\ & = \frac {2 \sqrt {b x+\sqrt {a+b^2 x^2}}}{b} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {b x+\sqrt {a+b^2 x^2}}}{\sqrt {a+b^2 x^2}} \, dx=\frac {2 \sqrt {b x+\sqrt {a+b^2 x^2}}}{b} \]
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Time = 0.34 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {2 \sqrt {b x +\sqrt {x^{2} b^{2}+a}}}{b}\) | \(23\) |
default | \(\frac {2 \sqrt {b x +\sqrt {x^{2} b^{2}+a}}}{b}\) | \(23\) |
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none
Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {b x+\sqrt {a+b^2 x^2}}}{\sqrt {a+b^2 x^2}} \, dx=\frac {2 \, \sqrt {b x + \sqrt {b^{2} x^{2} + a}}}{b} \]
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Time = 0.45 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {b x+\sqrt {a+b^2 x^2}}}{\sqrt {a+b^2 x^2}} \, dx=\begin {cases} \frac {2 \sqrt {b x + \sqrt {a + b^{2} x^{2}}}}{b} & \text {for}\: b \neq 0 \\\frac {x}{\sqrt [4]{a}} & \text {otherwise} \end {cases} \]
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\[ \int \frac {\sqrt {b x+\sqrt {a+b^2 x^2}}}{\sqrt {a+b^2 x^2}} \, dx=\int { \frac {\sqrt {b x + \sqrt {b^{2} x^{2} + a}}}{\sqrt {b^{2} x^{2} + a}} \,d x } \]
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none
Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {b x+\sqrt {a+b^2 x^2}}}{\sqrt {a+b^2 x^2}} \, dx=\frac {2 \, \sqrt {b x + \sqrt {b^{2} x^{2} + a}}}{b} \]
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Time = 0.55 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {b x+\sqrt {a+b^2 x^2}}}{\sqrt {a+b^2 x^2}} \, dx=\frac {2\,\sqrt {\sqrt {b^2\,x^2+a}+b\,x}}{b} \]
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