Integrand size = 24, antiderivative size = 50 \[ \int \frac {2+3 x}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {3}{10} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {37 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{10 \sqrt {10}} \]
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Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {81, 56, 222} \[ \int \frac {2+3 x}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\frac {37 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{10 \sqrt {10}}-\frac {3}{10} \sqrt {1-2 x} \sqrt {5 x+3} \]
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Rule 56
Rule 81
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {3}{10} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {37}{20} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {3}{10} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {37 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{10 \sqrt {5}} \\ & = -\frac {3}{10} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {37 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{10 \sqrt {10}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.26 \[ \int \frac {2+3 x}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\frac {-30 \sqrt {1-2 x} (3+5 x)-37 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{100 \sqrt {3+5 x}} \]
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Time = 1.17 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.10
method | result | size |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (37 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-60 \sqrt {-10 x^{2}-x +3}\right )}{200 \sqrt {-10 x^{2}-x +3}}\) | \(55\) |
risch | \(\frac {3 \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{10 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {37 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{200 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(88\) |
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Time = 0.23 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.14 \[ \int \frac {2+3 x}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {37}{200} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac {3}{10} \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} \]
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\[ \int \frac {2+3 x}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\int \frac {3 x + 2}{\sqrt {1 - 2 x} \sqrt {5 x + 3}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.52 \[ \int \frac {2+3 x}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {37}{200} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {3}{10} \, \sqrt {-10 \, x^{2} - x + 3} \]
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Time = 0.32 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.80 \[ \int \frac {2+3 x}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\frac {1}{100} \, \sqrt {5} {\left (37 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - 6 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
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Time = 4.32 (sec) , antiderivative size = 220, normalized size of antiderivative = 4.40 \[ \int \frac {2+3 x}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {\frac {3\,{\left (\sqrt {1-2\,x}-1\right )}^3}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {6\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {24\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}}{\frac {4\,{\left (\sqrt {1-2\,x}-1\right )}^2}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^4}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {4}{25}}-\frac {3\,\sqrt {10}\,\left (\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {1-2\,x}-1\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )+\frac {40\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}{5\,\left (\sqrt {1-2\,x}-1\right )}\right )}{3}\right )}{50} \]
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