Integrand size = 15, antiderivative size = 44 \[ \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx=\sqrt {x} \sqrt {a+b x}+\frac {a \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{\sqrt {b}} \]
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Time = 0.01 (sec) , antiderivative size = 44, 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} \[ \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx=\frac {a \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{\sqrt {b}}+\sqrt {x} \sqrt {a+b x} \]
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Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \sqrt {x} \sqrt {a+b x}+\frac {1}{2} a \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx \\ & = \sqrt {x} \sqrt {a+b x}+a \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right ) \\ & = \sqrt {x} \sqrt {a+b x}+a \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right ) \\ & = \sqrt {x} \sqrt {a+b x}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{\sqrt {b}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx=\sqrt {x} \sqrt {a+b x}-\frac {a \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{\sqrt {b}} \]
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Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.41
method | result | size |
default | \(\sqrt {x}\, \sqrt {b x +a}+\frac {a \sqrt {x \left (b x +a \right )}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 \sqrt {b x +a}\, \sqrt {x}\, \sqrt {b}}\) | \(62\) |
risch | \(\sqrt {x}\, \sqrt {b x +a}+\frac {a \sqrt {x \left (b x +a \right )}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 \sqrt {b x +a}\, \sqrt {x}\, \sqrt {b}}\) | \(62\) |
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none
Time = 0.23 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.11 \[ \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx=\left [\frac {a \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, \sqrt {b x + a} b \sqrt {x}}{2 \, b}, -\frac {a \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - \sqrt {b x + a} b \sqrt {x}}{b}\right ] \]
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Time = 1.55 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx=\sqrt {a} \sqrt {x} \sqrt {1 + \frac {b x}{a}} + \frac {a \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{\sqrt {b}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (32) = 64\).
Time = 0.31 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.59 \[ \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx=-\frac {a \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{2 \, \sqrt {b}} - \frac {\sqrt {b x + a} a}{{\left (b - \frac {b x + a}{x}\right )} \sqrt {x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (32) = 64\).
Time = 76.15 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.55 \[ \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx=-\frac {{\left (\frac {a \log \left ({\left | -\sqrt {b x + a} \sqrt {b} + \sqrt {{\left (b x + a\right )} b - a b} \right |}\right )}{\sqrt {b}} - \frac {\sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a}}{b}\right )} b}{{\left | b \right |}} \]
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Time = 0.76 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx=\sqrt {x}\,\sqrt {a+b\,x}+\frac {2\,a\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a+b\,x}-\sqrt {a}}\right )}{\sqrt {b}} \]
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