Integrand size = 15, antiderivative size = 48 \[ \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx=\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}} \]
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Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {52, 65, 223, 212} \[ \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx=\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}} \]
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Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{2 b} \\ & = \frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{b} \\ & = \frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{b} \\ & = \frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx=\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {2 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )}{b^{3/2}} \]
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Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.35
method | result | size |
default | \(\frac {\sqrt {x}\, \sqrt {b x +a}}{b}-\frac {a \sqrt {x \left (b x +a \right )}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {b x +a}}\) | \(65\) |
risch | \(\frac {\sqrt {x}\, \sqrt {b x +a}}{b}-\frac {a \sqrt {x \left (b x +a \right )}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {b x +a}}\) | \(65\) |
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none
Time = 0.23 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.90 \[ \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx=\left [\frac {a \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, \sqrt {b x + a} b \sqrt {x}}{2 \, b^{2}}, \frac {a \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + \sqrt {b x + a} b \sqrt {x}}{b^{2}}\right ] \]
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Time = 1.32 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx=\frac {\sqrt {a} \sqrt {x} \sqrt {1 + \frac {b x}{a}}}{b} - \frac {a \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (36) = 72\).
Time = 0.37 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.52 \[ \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx=\frac {a \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{2 \, b^{\frac {3}{2}}} - \frac {\sqrt {b x + a} a}{{\left (b^{2} - \frac {{\left (b x + a\right )} b}{x}\right )} \sqrt {x}} \]
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none
Time = 76.91 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx=\frac {{\left (a \sqrt {b} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} + \sqrt {{\left (b x + a\right )} b - a b} \right |}\right ) + \sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a}\right )} {\left | b \right |}}{b^{3}} \]
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Time = 0.64 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx=\frac {\sqrt {x}\,\sqrt {a+b\,x}}{b}-\frac {2\,a\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a+b\,x}-\sqrt {a}}\right )}{b^{3/2}} \]
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