Integrand size = 26, antiderivative size = 62 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx=\frac {1}{3} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {2 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3 \sqrt {7}} \]
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Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {132, 56, 222, 12, 95, 210} \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx=\frac {1}{3} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {2 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}} \]
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Rule 12
Rule 56
Rule 95
Rule 132
Rule 210
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {5}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx+\int -\frac {1}{3 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = -\left (\frac {1}{3} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\right )+\frac {1}{3} \left (2 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right ) \\ & = \frac {1}{3} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {2}{3} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right ) \\ & = \frac {1}{3} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3 \sqrt {7}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(138\) vs. \(2(62)=124\).
Time = 0.84 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.23 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx=-\frac {2}{21} \left (\sqrt {7} \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )+7 \sqrt {10} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )+\sqrt {7} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right ) \]
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Time = 1.18 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.11
method | result | size |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (7 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-2 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )\right )}{42 \sqrt {-10 x^{2}-x +3}}\) | \(69\) |
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Time = 0.23 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx=\frac {1}{21} \, \sqrt {7} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac {1}{6} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
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\[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx=\int \frac {\sqrt {5 x + 3}}{\sqrt {1 - 2 x} \left (3 x + 2\right )}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx=\frac {1}{6} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {1}{21} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (44) = 88\).
Time = 0.34 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.27 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx=-\frac {1}{210} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{6} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} \]
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Time = 4.60 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.23 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx=-\frac {2\,\sqrt {7}\,\mathrm {atan}\left (\frac {5580\,\sqrt {21}\,x+2699\,\sqrt {21}-5489\,\sqrt {35\,x+21}+649\,\sqrt {21-42\,x}+2141\,\sqrt {7}\,\sqrt {1-2\,x}\,\sqrt {5\,x+3}}{7400\,x-5489\,\sqrt {1-2\,x}-4543\,\sqrt {15\,x+9}+3063\,\sqrt {3}\,\sqrt {1-2\,x}\,\sqrt {5\,x+3}+9929}\right )}{21}-\frac {2\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}-\sqrt {10-20\,x}}{2\,\sqrt {3}-2\,\sqrt {5\,x+3}}\right )}{3} \]
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