Integrand size = 20, antiderivative size = 43 \[ \int \frac {1}{\sqrt {a+a x} \sqrt {c-c x}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {c} \sqrt {a+a x}}{\sqrt {a} \sqrt {c-c x}}\right )}{\sqrt {a} \sqrt {c}} \]
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Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {65, 223, 209} \[ \int \frac {1}{\sqrt {a+a x} \sqrt {c-c x}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {c} \sqrt {a x+a}}{\sqrt {a} \sqrt {c-c x}}\right )}{\sqrt {a} \sqrt {c}} \]
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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 c-\frac {c x^2}{a}}} \, dx,x,\sqrt {a+a x}\right )}{a} \\ & = \frac {2 \text {Subst}\left (\int \frac {1}{1+\frac {c x^2}{a}} \, dx,x,\frac {\sqrt {a+a x}}{\sqrt {c-c x}}\right )}{a} \\ & = \frac {2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+a x}}{\sqrt {a} \sqrt {c-c x}}\right )}{\sqrt {a} \sqrt {c}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\sqrt {a+a x} \sqrt {c-c x}} \, dx=\frac {2 \sqrt {1+x} \arctan \left (\frac {\sqrt {c} \sqrt {1+x}}{\sqrt {c-c x}}\right )}{\sqrt {c} \sqrt {a (1+x)}} \]
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Time = 0.35 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.33
method | result | size |
default | \(\frac {\sqrt {\left (-c x +c \right ) \left (a x +a \right )}\, \arctan \left (\frac {\sqrt {a c}\, x}{\sqrt {-a c \,x^{2}+a c}}\right )}{\sqrt {a x +a}\, \sqrt {-c x +c}\, \sqrt {a c}}\) | \(57\) |
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none
Time = 0.24 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.35 \[ \int \frac {1}{\sqrt {a+a x} \sqrt {c-c x}} \, dx=\left [-\frac {\sqrt {-a c} \log \left (2 \, a c x^{2} - 2 \, \sqrt {-a c} \sqrt {a x + a} \sqrt {-c x + c} x - a c\right )}{2 \, a c}, -\frac {\sqrt {a c} \arctan \left (\frac {\sqrt {a c} \sqrt {a x + a} \sqrt {-c x + c} x}{a c x^{2} - a c}\right )}{a c}\right ] \]
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Result contains complex when optimal does not.
Time = 24.43 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.98 \[ \int \frac {1}{\sqrt {a+a x} \sqrt {c-c x}} \, dx=- \frac {i {G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} & \frac {1}{2}, \frac {1}{2}, 1, 1 \\0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 0 & \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} \sqrt {a} \sqrt {c}} + \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1 & \\- \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, 0, 0, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} \sqrt {a} \sqrt {c}} \]
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none
Time = 0.30 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.19 \[ \int \frac {1}{\sqrt {a+a x} \sqrt {c-c x}} \, dx=\frac {\arcsin \left (x\right )}{\sqrt {a c}} \]
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none
Time = 0.34 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\sqrt {a+a x} \sqrt {c-c x}} \, dx=-\frac {2 \, a \log \left ({\left | -\sqrt {-a c} \sqrt {a x + a} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt {-a c} {\left | a \right |}} \]
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Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\sqrt {a+a x} \sqrt {c-c x}} \, dx=-\frac {4\,\mathrm {atan}\left (\frac {a\,\left (\sqrt {c-c\,x}-\sqrt {c}\right )}{\sqrt {a\,c}\,\left (\sqrt {a+a\,x}-\sqrt {a}\right )}\right )}{\sqrt {a\,c}} \]
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