Integrand size = 25, antiderivative size = 41 \[ \int \frac {\sqrt {\frac {a+b x}{c+d x}}}{a+b x} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {a+b x}{c+d x}}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \]
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Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1983, 12, 214} \[ \int \frac {\sqrt {\frac {a+b x}{c+d x}}}{a+b x} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {a+b x}{c+d x}}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \]
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Rule 12
Rule 214
Rule 1983
Rubi steps \begin{align*} \text {integral}& = (2 (b c-a d)) \text {Subst}\left (\int \frac {1}{(b c-a d) \left (b-d x^2\right )} \, dx,x,\sqrt {\frac {a+b x}{c+d x}}\right ) \\ & = 2 \text {Subst}\left (\int \frac {1}{b-d x^2} \, dx,x,\sqrt {\frac {a+b x}{c+d x}}\right ) \\ & = \frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {a+b x}{c+d x}}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.88 \[ \int \frac {\sqrt {\frac {a+b x}{c+d x}}}{a+b x} \, dx=\frac {2 \sqrt {\frac {a+b x}{c+d x}} \sqrt {c+d x} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{\sqrt {b} \sqrt {d} \sqrt {a+b x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs. \(2(31)=62\).
Time = 1.13 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.95
method | result | size |
default | \(\frac {\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \left (d x +c \right ) \sqrt {\frac {b x +a}{d x +c}}}{\sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}\) | \(80\) |
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Time = 0.25 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.56 \[ \int \frac {\sqrt {\frac {a+b x}{c+d x}}}{a+b x} \, dx=\left [\frac {\sqrt {b d} \log \left (2 \, b d x + b c + a d + 2 \, \sqrt {b d} {\left (d x + c\right )} \sqrt {\frac {b x + a}{d x + c}}\right )}{b d}, -\frac {2 \, \sqrt {-b d} \arctan \left (\frac {\sqrt {-b d} {\left (d x + c\right )} \sqrt {\frac {b x + a}{d x + c}}}{b d x + a d}\right )}{b d}\right ] \]
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\[ \int \frac {\sqrt {\frac {a+b x}{c+d x}}}{a+b x} \, dx=\int \frac {\sqrt {\frac {a + b x}{c + d x}}}{a + b x}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.44 \[ \int \frac {\sqrt {\frac {a+b x}{c+d x}}}{a+b x} \, dx=-\frac {\log \left (\frac {d \sqrt {\frac {b x + a}{d x + c}} - \sqrt {b d}}{d \sqrt {\frac {b x + a}{d x + c}} + \sqrt {b d}}\right )}{\sqrt {b d}} \]
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Time = 0.38 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.49 \[ \int \frac {\sqrt {\frac {a+b x}{c+d x}}}{a+b x} \, dx=-\frac {\log \left ({\left | -b c - a d - 2 \, \sqrt {b d} {\left (\sqrt {b d} x - \sqrt {b d x^{2} + b c x + a d x + a c}\right )} \right |}\right ) \mathrm {sgn}\left (d x + c\right )}{\sqrt {b d}} \]
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Time = 0.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {\frac {a+b x}{c+d x}}}{a+b x} \, dx=\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {d}\,\sqrt {\frac {a+b\,x}{c+d\,x}}}{\sqrt {b}}\right )}{\sqrt {b}\,\sqrt {d}} \]
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