Integrand size = 23, antiderivative size = 39 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {-a d+b d x}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-a d+b d x}}\right )}{b \sqrt {d}} \]
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Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {65, 223, 212} \[ \int \frac {1}{\sqrt {a+b x} \sqrt {-a d+b d x}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b d x-a d}}\right )}{b \sqrt {d}} \]
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Rule 65
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {-2 a d+d x^2}} \, dx,x,\sqrt {a+b x}\right )}{b} \\ & = \frac {2 \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {-a d+b d x}}\right )}{b} \\ & = \frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-a d+b d x}}\right )}{b \sqrt {d}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {-a d+b d x}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-a d+b d x}}\right )}{b \sqrt {d}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(31)=62\).
Time = 0.39 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.95
method | result | size |
default | \(\frac {\sqrt {\left (b x +a \right ) \left (b d x -a d \right )}\, \ln \left (\frac {b^{2} d x}{\sqrt {b^{2} d}}+\sqrt {b^{2} d \,x^{2}-a^{2} d}\right )}{\sqrt {b x +a}\, \sqrt {b d x -a d}\, \sqrt {b^{2} d}}\) | \(76\) |
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none
Time = 0.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.77 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {-a d+b d x}} \, dx=\left [\frac {\log \left (2 \, b^{2} d x^{2} + 2 \, \sqrt {b d x - a d} \sqrt {b x + a} b \sqrt {d} x - a^{2} d\right )}{2 \, b \sqrt {d}}, -\frac {\sqrt {-d} \arctan \left (\frac {\sqrt {b d x - a d} \sqrt {b x + a} b \sqrt {-d} x}{b^{2} d x^{2} - a^{2} d}\right )}{b d}\right ] \]
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Result contains complex when optimal does not.
Time = 20.86 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.26 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {-a d+b d x}} \, dx=\frac {{G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} & \frac {1}{2}, \frac {1}{2}, 1, 1 \\0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 0 & \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} b \sqrt {d}} - \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1 & \\- \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, 0, 0, 0 \end {matrix} \middle | {\frac {a^{2} e^{2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} b \sqrt {d}} \]
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Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {-a d+b d x}} \, dx=\frac {\log \left (2 \, b^{2} d x + 2 \, \sqrt {b^{2} d x^{2} - a^{2} d} b \sqrt {d}\right )}{b \sqrt {d}} \]
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Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {-a d+b d x}} \, dx=-\frac {2 \, \log \left ({\left | -\sqrt {b x + a} \sqrt {d} + \sqrt {{\left (b x + a\right )} d - 2 \, a d} \right |}\right )}{b \sqrt {d}} \]
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Time = 0.32 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {-a d+b d x}} \, dx=-\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {b\,d\,x-a\,d}-\sqrt {-a\,d}\right )}{\sqrt {-b^2\,d}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}\right )}{\sqrt {-b^2\,d}} \]
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