Integrand size = 12, antiderivative size = 40 \[ \int \frac {1}{\sqrt {c x (a+b x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a c x+b c x^2}}\right )}{\sqrt {b} \sqrt {c}} \]
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Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1976, 634, 212} \[ \int \frac {1}{\sqrt {c x (a+b x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a c x+b c x^2}}\right )}{\sqrt {b} \sqrt {c}} \]
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Rule 212
Rule 634
Rule 1976
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {a c x+b c x^2}} \, dx \\ & = 2 \text {Subst}\left (\int \frac {1}{1-b c x^2} \, dx,x,\frac {x}{\sqrt {a c x+b c x^2}}\right ) \\ & = \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a c x+b c x^2}}\right )}{\sqrt {b} \sqrt {c}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.40 \[ \int \frac {1}{\sqrt {c x (a+b x)}} \, dx=-\frac {2 \sqrt {x} \sqrt {a+b x} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{\sqrt {b} \sqrt {c x (a+b x)}} \]
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Time = 1.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.70
method | result | size |
pseudoelliptic | \(\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {c x \left (b x +a \right )}}{x \sqrt {b c}}\right )}{\sqrt {b c}}\) | \(28\) |
default | \(\frac {\ln \left (\frac {\frac {1}{2} a c +b c x}{\sqrt {b c}}+\sqrt {b c \,x^{2}+a c x}\right )}{\sqrt {b c}}\) | \(37\) |
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none
Time = 0.34 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.18 \[ \int \frac {1}{\sqrt {c x (a+b x)}} \, dx=\left [\frac {\sqrt {b c} \log \left (2 \, b c x + a c + 2 \, \sqrt {b c x^{2} + a c x} \sqrt {b c}\right )}{b c}, -\frac {2 \, \sqrt {-b c} \arctan \left (\frac {\sqrt {b c x^{2} + a c x} \sqrt {-b c}}{b c x}\right )}{b c}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (39) = 78\).
Time = 0.71 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.42 \[ \int \frac {1}{\sqrt {c x (a+b x)}} \, dx=\begin {cases} \frac {\log {\left (a c + 2 b c x + 2 \sqrt {b c} \sqrt {a c x + b c x^{2}} \right )}}{\sqrt {b c}} & \text {for}\: b c \neq 0 \wedge \frac {a^{2} c}{b} \neq 0 \\\frac {\left (\frac {a}{2 b} + x\right ) \log {\left (\frac {a}{2 b} + x \right )}}{\sqrt {b c \left (\frac {a}{2 b} + x\right )^{2}}} & \text {for}\: b c \neq 0 \\\frac {2 \sqrt {a c x}}{a c} & \text {for}\: a c \neq 0 \\\tilde {\infty } x & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\sqrt {c x (a+b x)}} \, dx=\frac {\log \left (2 \, b c x + a c + 2 \, \sqrt {b c x^{2} + a c x} \sqrt {b c}\right )}{\sqrt {b c}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (30) = 60\).
Time = 0.30 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.90 \[ \int \frac {1}{\sqrt {c x (a+b x)}} \, dx=\frac {a^{2} c \log \left ({\left | -a c - 2 \, \sqrt {b c} {\left (\sqrt {b c} x - \sqrt {b c x^{2} + a c x}\right )} \right |}\right )}{8 \, \sqrt {b c} b} + \frac {1}{4} \, \sqrt {b c x^{2} + a c x} {\left (2 \, x + \frac {a}{b}\right )} \]
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Time = 22.70 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\sqrt {c x (a+b x)}} \, dx=\frac {\ln \left (a\,c+2\,\sqrt {b\,c}\,\sqrt {c\,x\,\left (a+b\,x\right )}+2\,b\,c\,x\right )}{\sqrt {b\,c}} \]
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