Integrand size = 23, antiderivative size = 38 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{b \sqrt {c}} \]
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Time = 0.01 (sec) , antiderivative size = 38, 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, 209} \[ \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{b \sqrt {c}} \]
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Rule 65
Rule 209
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {2 a c-c x^2}} \, dx,x,\sqrt {a+b x}\right )}{b} \\ & = \frac {2 \text {Subst}\left (\int \frac {1}{1+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{b} \\ & = \frac {2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{b \sqrt {c}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {2 \sqrt {a-b x} \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )}{b \sqrt {c (a-b x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(70\) vs. \(2(30)=60\).
Time = 0.20 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.87
method | result | size |
default | \(\frac {\sqrt {\left (b x +a \right ) \left (-b c x +a c \right )}\, \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right )}{\sqrt {b x +a}\, \sqrt {-b c x +a c}\, \sqrt {b^{2} c}}\) | \(71\) |
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none
Time = 0.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.84 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\left [-\frac {\sqrt {-c} \log \left (2 \, b^{2} c x^{2} - 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {-c} x - a^{2} c\right )}{2 \, b c}, -\frac {\arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {c} x}{b^{2} c x^{2} - a^{2} c}\right )}{b \sqrt {c}}\right ] \]
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Result contains complex when optimal does not.
Time = 25.65 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.37 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=- \frac {i {G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} & \frac {1}{2}, \frac {1}{2}, 1, 1 \\0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 0 & \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} b \sqrt {c}} + \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1 & \\- \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, 0, 0, 0 \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} b \sqrt {c}} \]
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none
Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.37 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {\arcsin \left (\frac {b x}{a}\right )}{b \sqrt {c}} \]
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none
Time = 0.31 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {2 \, \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{b \sqrt {-c}} \]
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Time = 0.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.39 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}{\sqrt {b^2\,c}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}\right )}{\sqrt {b^2\,c}} \]
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