Integrand size = 22, antiderivative size = 35 \[ \int \frac {1}{x \sqrt {-2+3 x} \sqrt {3+5 x}} \, dx=\sqrt {\frac {2}{3}} \arctan \left (\frac {\sqrt {\frac {3}{2}} \sqrt {-2+3 x}}{\sqrt {3+5 x}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {95, 209} \[ \int \frac {1}{x \sqrt {-2+3 x} \sqrt {3+5 x}} \, dx=\sqrt {\frac {2}{3}} \arctan \left (\frac {\sqrt {\frac {3}{2}} \sqrt {3 x-2}}{\sqrt {5 x+3}}\right ) \]
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Rule 95
Rule 209
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{2+3 x^2} \, dx,x,\frac {\sqrt {-2+3 x}}{\sqrt {3+5 x}}\right ) \\ & = \sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {-2+3 x}}{\sqrt {3+5 x}}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \sqrt {-2+3 x} \sqrt {3+5 x}} \, dx=-\sqrt {\frac {2}{3}} \arctan \left (\frac {\sqrt {2+\frac {10 x}{3}}}{\sqrt {-2+3 x}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(52\) vs. \(2(25)=50\).
Time = 0.62 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.51
method | result | size |
default | \(-\frac {\sqrt {-2+3 x}\, \sqrt {3+5 x}\, \sqrt {6}\, \arctan \left (\frac {\left (12+x \right ) \sqrt {6}}{12 \sqrt {15 x^{2}-x -6}}\right )}{6 \sqrt {15 x^{2}-x -6}}\) | \(53\) |
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Time = 0.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.31 \[ \int \frac {1}{x \sqrt {-2+3 x} \sqrt {3+5 x}} \, dx=-\frac {1}{6} \, \sqrt {3} \sqrt {2} \arctan \left (\frac {\sqrt {3} \sqrt {2} \sqrt {5 \, x + 3} \sqrt {3 \, x - 2} {\left (x + 12\right )}}{12 \, {\left (15 \, x^{2} - x - 6\right )}}\right ) \]
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\[ \int \frac {1}{x \sqrt {-2+3 x} \sqrt {3+5 x}} \, dx=\int \frac {1}{x \sqrt {3 x - 2} \sqrt {5 x + 3}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x \sqrt {-2+3 x} \sqrt {3+5 x}} \, dx=-\frac {1}{6} \, \sqrt {6} \arcsin \left (\frac {x}{19 \, {\left | x \right |}} + \frac {12}{19 \, {\left | x \right |}}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x \sqrt {-2+3 x} \sqrt {3+5 x}} \, dx=-\frac {1}{15} \, \sqrt {10} \sqrt {5} \sqrt {3} \arctan \left (\frac {1}{60} \, \sqrt {10} {\left ({\left (\sqrt {3} \sqrt {5 \, x + 3} - \sqrt {15 \, x - 10}\right )}^{2} + 1\right )}\right ) \]
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Time = 5.46 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.37 \[ \int \frac {1}{x \sqrt {-2+3 x} \sqrt {3+5 x}} \, dx=-\frac {\sqrt {6}\,\left (\ln \left (\frac {{\left (-\sqrt {3\,x-2}+\sqrt {2}\,1{}\mathrm {i}\right )}^2}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {3}{5}-\frac {\sqrt {6}\,\left (-\sqrt {3\,x-2}+\sqrt {2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{30\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )-\ln \left (\frac {-\sqrt {3\,x-2}+\sqrt {2}\,1{}\mathrm {i}}{\sqrt {3}-\sqrt {5\,x+3}}\right )\right )\,1{}\mathrm {i}}{6} \]
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