Integrand size = 15, antiderivative size = 61 \[ \int \frac {1}{\sqrt {(a+b x) (c+d x)}} \, dx=\frac {\text {arctanh}\left (\frac {b c+a d+2 b d x}{2 \sqrt {b} \sqrt {d} \sqrt {a c+(b c+a d) x+b d x^2}}\right )}{\sqrt {b} \sqrt {d}} \]
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Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1976, 635, 212} \[ \int \frac {1}{\sqrt {(a+b x) (c+d x)}} \, dx=\frac {\text {arctanh}\left (\frac {a d+b c+2 b d x}{2 \sqrt {b} \sqrt {d} \sqrt {x (a d+b c)+a c+b d x^2}}\right )}{\sqrt {b} \sqrt {d}} \]
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Rule 212
Rule 635
Rule 1976
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {a c+(b c+a d) x+b d x^2}} \, dx \\ & = 2 \text {Subst}\left (\int \frac {1}{4 b d-x^2} \, dx,x,\frac {b c+a d+2 b d x}{\sqrt {a c+(b c+a d) x+b d x^2}}\right ) \\ & = \frac {\tanh ^{-1}\left (\frac {b c+a d+2 b d x}{2 \sqrt {b} \sqrt {d} \sqrt {a c+(b c+a d) x+b d x^2}}\right )}{\sqrt {b} \sqrt {d}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\sqrt {(a+b x) (c+d x)}} \, dx=\frac {2 \sqrt {a+b x} \sqrt {c+d x} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{\sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}} \]
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Time = 1.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {\ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d x}{\sqrt {b d}}+\sqrt {a c +\left (a d +b c \right ) x +b d \,x^{2}}\right )}{\sqrt {b d}}\) | \(49\) |
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none
Time = 0.39 (sec) , antiderivative size = 192, normalized size of antiderivative = 3.15 \[ \int \frac {1}{\sqrt {(a+b x) (c+d x)}} \, dx=\left [\frac {\sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, \sqrt {b d x^{2} + a c + {\left (b c + a d\right )} x} {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )}{2 \, b d}, -\frac {\sqrt {-b d} \arctan \left (\frac {\sqrt {b d x^{2} + a c + {\left (b c + a d\right )} x} {\left (2 \, b d x + b c + a d\right )} \sqrt {-b d}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right )}{b d}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (58) = 116\).
Time = 1.07 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.66 \[ \int \frac {1}{\sqrt {(a+b x) (c+d x)}} \, dx=\begin {cases} \frac {\log {\left (a d + b c + 2 b d x + 2 \sqrt {b d} \sqrt {a c + b d x^{2} + x \left (a d + b c\right )} \right )}}{\sqrt {b d}} & \text {for}\: b d \neq 0 \wedge a c - \frac {\left (a d + b c\right )^{2}}{4 b d} \neq 0 \\\frac {\left (x - \frac {- a d - b c}{2 b d}\right ) \log {\left (x - \frac {- a d - b c}{2 b d} \right )}}{\sqrt {b d \left (x - \frac {- a d - b c}{2 b d}\right )^{2}}} & \text {for}\: b d \neq 0 \\\frac {2 \sqrt {a c + x \left (a d + b c\right )}}{a d + b c} & \text {for}\: a d + b c \neq 0 \\\frac {x}{\sqrt {a c}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {1}{\sqrt {(a+b x) (c+d x)}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (49) = 98\).
Time = 0.31 (sec) , antiderivative size = 123, normalized size of antiderivative = 2.02 \[ \int \frac {1}{\sqrt {(a+b x) (c+d x)}} \, dx=\frac {1}{4} \, \sqrt {b d x^{2} + b c x + a d x + a c} {\left (2 \, x + \frac {b c + a d}{b d}\right )} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | -b c - a d - 2 \, \sqrt {b d} {\left (\sqrt {b d} x - \sqrt {b d x^{2} + b c x + a d x + a c}\right )} \right |}\right )}{8 \, \sqrt {b d} b d} \]
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Timed out. \[ \int \frac {1}{\sqrt {(a+b x) (c+d x)}} \, dx=\int \frac {1}{\sqrt {\left (a+b\,x\right )\,\left (c+d\,x\right )}} \,d x \]
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