Integrand size = 16, antiderivative size = 65 \[ \int \frac {1}{\sqrt {(a+b x) (c-d x)}} \, dx=-\frac {\arctan \left (\frac {b c-a d-2 b d x}{2 \sqrt {b} \sqrt {d} \sqrt {a c+(b c-a d) x-b d x^2}}\right )}{\sqrt {b} \sqrt {d}} \]
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Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1976, 635, 210} \[ \int \frac {1}{\sqrt {(a+b x) (c-d x)}} \, dx=-\frac {\arctan \left (\frac {-a d+b c-2 b d x}{2 \sqrt {b} \sqrt {d} \sqrt {x (b c-a d)+a c-b d x^2}}\right )}{\sqrt {b} \sqrt {d}} \]
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Rule 210
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 {\tan ^{-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 = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\sqrt {(a+b x) (c-d x)}} \, dx=-\frac {2 \sqrt {a+b x} \sqrt {c-d x} \arctan \left (\frac {\sqrt {b} \sqrt {c-d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{\sqrt {b} \sqrt {d} \sqrt {(a+b x) (c-d x)}} \]
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Time = 1.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.85
method | result | size |
default | \(\frac {\arctan \left (\frac {\sqrt {b d}\, \left (x -\frac {-a d +b c}{2 b d}\right )}{\sqrt {a c +\left (-a d +b c \right ) x -b d \,x^{2}}}\right )}{\sqrt {b d}}\) | \(55\) |
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none
Time = 0.28 (sec) , antiderivative size = 202, normalized size of antiderivative = 3.11 \[ \int \frac {1}{\sqrt {(a+b x) (c-d x)}} \, dx=\left [-\frac {\sqrt {-b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2} - 4 \, \sqrt {-b d x^{2} + a c + {\left (b c - a d\right )} x} {\left (2 \, b d x - b c + a d\right )} \sqrt {-b d} - 8 \, {\left (b^{2} c d - a b d^{2}\right )} x\right )}{2 \, b d}, -\frac {\sqrt {b d} \arctan \left (\frac {\sqrt {-b d x^{2} + a c + {\left (b c - a d\right )} x} {\left (2 \, b d x - b c + a d\right )} \sqrt {b d}}{2 \, {\left (b^{2} d^{2} x^{2} - a b c d - {\left (b^{2} c d - a b d^{2}\right )} x\right )}}\right )}{b d}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (60) = 120\).
Time = 1.13 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.49 \[ \int \frac {1}{\sqrt {(a+b x) (c-d x)}} \, dx=\begin {cases} \frac {\log {\left (- a d + b c - 2 b d x + 2 \sqrt {- b d} \sqrt {a c - b d x^{2} + x \left (- a d + b c\right )} \right )}}{\sqrt {- b d}} & \text {for}\: b d \neq 0 \wedge a c + \frac {\left (a d - b c\right )^{2}}{4 b d} \neq 0 \\\frac {\left (x + \frac {a d - b c}{2 b d}\right ) \log {\left (x + \frac {a d - b c}{2 b d} \right )}}{\sqrt {- b d \left (x + \frac {a d - b c}{2 b d}\right )^{2}}} & \text {for}\: b d \neq 0 \\\frac {2 \sqrt {a c + x \left (- a d + b c\right )}}{- a d + b c} & \text {for}\: a d - b c \neq 0 \\\frac {x}{\sqrt {a c}} & \text {otherwise} \end {cases} \]
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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. 131 vs. \(2 (53) = 106\).
Time = 0.32 (sec) , antiderivative size = 131, 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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