Integrand size = 17, antiderivative size = 76 \[ \int \sqrt {\frac {a+b x}{c+d x}} \, dx=\frac {\sqrt {\frac {a+b x}{c+d x}} (c+d x)}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {a+b x}{c+d x}}}{\sqrt {b}}\right )}{\sqrt {b} d^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1979, 294, 214} \[ \int \sqrt {\frac {a+b x}{c+d x}} \, dx=\frac {(c+d x) \sqrt {\frac {a+b x}{c+d x}}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {a+b x}{c+d x}}}{\sqrt {b}}\right )}{\sqrt {b} d^{3/2}} \]
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Rule 214
Rule 294
Rule 1979
Rubi steps \begin{align*} \text {integral}& = (2 (b c-a d)) \text {Subst}\left (\int \frac {x^2}{\left (b-d x^2\right )^2} \, dx,x,\sqrt {\frac {a+b x}{c+d x}}\right ) \\ & = \frac {\sqrt {\frac {a+b x}{c+d x}} (c+d x)}{d}-\frac {(b c-a d) \text {Subst}\left (\int \frac {1}{b-d x^2} \, dx,x,\sqrt {\frac {a+b x}{c+d x}}\right )}{d} \\ & = \frac {\sqrt {\frac {a+b x}{c+d x}} (c+d x)}{d}-\frac {(b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {a+b x}{c+d x}}}{\sqrt {b}}\right )}{\sqrt {b} d^{3/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.28 \[ \int \sqrt {\frac {a+b x}{c+d x}} \, dx=\frac {\sqrt {\frac {a+b x}{c+d x}} \left (\sqrt {d} (c+d x)+\frac {(-b c+a d) \sqrt {c+d x} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{\sqrt {b} \sqrt {a+b x}}\right )}{d^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(151\) vs. \(2(64)=128\).
Time = 0.09 (sec) , antiderivative size = 152, normalized size of antiderivative = 2.00
method | result | size |
default | \(\frac {\sqrt {\frac {b x +a}{d x +c}}\, \left (d x +c \right ) \left (\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a d -\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\right )}{2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, d \sqrt {b d}}\) | \(152\) |
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Time = 0.30 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.37 \[ \int \sqrt {\frac {a+b x}{c+d x}} \, dx=\left [-\frac {{\left (b c - a d\right )} \sqrt {b d} \log \left (2 \, b d x + b c + a d + 2 \, \sqrt {b d} {\left (d x + c\right )} \sqrt {\frac {b x + a}{d x + c}}\right ) - 2 \, {\left (b d^{2} x + b c d\right )} \sqrt {\frac {b x + a}{d x + c}}}{2 \, b d^{2}}, \frac {{\left (b c - a d\right )} \sqrt {-b d} \arctan \left (\frac {\sqrt {-b d} {\left (d x + c\right )} \sqrt {\frac {b x + a}{d x + c}}}{b d x + a d}\right ) + {\left (b d^{2} x + b c d\right )} \sqrt {\frac {b x + a}{d x + c}}}{b d^{2}}\right ] \]
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\[ \int \sqrt {\frac {a+b x}{c+d x}} \, dx=\int \sqrt {\frac {a + b x}{c + d x}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.55 \[ \int \sqrt {\frac {a+b x}{c+d x}} \, dx=\frac {{\left (b c - a d\right )} \sqrt {\frac {b x + a}{d x + c}}}{b d - \frac {{\left (b x + a\right )} d^{2}}{d x + c}} + \frac {{\left (b c - a d\right )} \log \left (\frac {d \sqrt {\frac {b x + a}{d x + c}} - \sqrt {b d}}{d \sqrt {\frac {b x + a}{d x + c}} + \sqrt {b d}}\right )}{2 \, \sqrt {b d} d} \]
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Time = 0.38 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.43 \[ \int \sqrt {\frac {a+b x}{c+d x}} \, dx=\frac {{\left (b c \mathrm {sgn}\left (d x + c\right ) - a d \mathrm {sgn}\left (d x + c\right )\right )} \log \left ({\left | -b c - a d - 2 \, \sqrt {b d} {\left (\sqrt {b d} x - \sqrt {b d x^{2} + b c x + a d x + a c}\right )} \right |}\right )}{2 \, \sqrt {b d} d} + \frac {\sqrt {b d x^{2} + b c x + a d x + a c} \mathrm {sgn}\left (d x + c\right )}{d} \]
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Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.18 \[ \int \sqrt {\frac {a+b x}{c+d x}} \, dx=\frac {\mathrm {atanh}\left (\frac {\sqrt {d}\,\sqrt {\frac {a+b\,x}{c+d\,x}}}{\sqrt {b}}\right )\,\left (a\,d-b\,c\right )}{\sqrt {b}\,d^{3/2}}+\frac {\left (a\,d-b\,c\right )\,\sqrt {\frac {a+b\,x}{c+d\,x}}}{b\,d\,\left (\frac {d\,\left (a+b\,x\right )}{b\,\left (c+d\,x\right )}-1\right )} \]
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