Integrand size = 26, antiderivative size = 84 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x} \, dx=\frac {1}{3} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {37 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{9 \sqrt {10}}+\frac {2}{9} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 84, 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 = {103, 163, 56, 222, 95, 210} \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x} \, dx=\frac {37 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{9 \sqrt {10}}+\frac {2}{9} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+\frac {1}{3} \sqrt {1-2 x} \sqrt {5 x+3} \]
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Rule 56
Rule 95
Rule 103
Rule 163
Rule 210
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{3} \int \frac {-10-\frac {37 x}{2}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = \frac {1}{3} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {7}{9} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx+\frac {37}{18} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = \frac {1}{3} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {14}{9} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {37 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{9 \sqrt {5}} \\ & = \frac {1}{3} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {37 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{9 \sqrt {10}}+\frac {2}{9} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x} \, dx=\frac {1}{90} \left (30 \sqrt {1-2 x} \sqrt {3+5 x}-37 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+20 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right ) \]
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Time = 1.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.99
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (20 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-37 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-60 \sqrt {-10 x^{2}-x +3}\right )}{180 \sqrt {-10 x^{2}-x +3}}\) | \(83\) |
risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{3 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {37 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{180}+\frac {\sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{9}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(121\) |
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Time = 0.23 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x} \, dx=\frac {1}{9} \, \sqrt {7} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac {37}{180} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + \frac {1}{3} \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} \]
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\[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x} \, dx=\int \frac {\sqrt {1 - 2 x} \sqrt {5 x + 3}}{3 x + 2}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x} \, dx=\frac {37}{180} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {1}{9} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {1}{3} \, \sqrt {-10 \, x^{2} - x + 3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (60) = 120\).
Time = 0.33 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.90 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x} \, dx=-\frac {1}{90} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {37}{180} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{15} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} \]
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Time = 5.06 (sec) , antiderivative size = 566, normalized size of antiderivative = 6.74 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x} \, dx=\frac {37\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {1-2\,x}-1\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )}{45}-\frac {2\,\sqrt {7}\,\mathrm {atan}\left (\frac {6645115904\,\sqrt {7}\,\left (\sqrt {1-2\,x}-1\right )}{3955078125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )\,\left (\frac {229677056\,{\left (\sqrt {1-2\,x}-1\right )}^2}{158203125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {3168922624\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{1318359375\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {459354112}{791015625}\right )}-\frac {192432128\,\sqrt {3}\,\sqrt {7}}{439453125\,\left (\frac {229677056\,{\left (\sqrt {1-2\,x}-1\right )}^2}{158203125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {3168922624\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{1318359375\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {459354112}{791015625}\right )}+\frac {96216064\,\sqrt {3}\,\sqrt {7}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{87890625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2\,\left (\frac {229677056\,{\left (\sqrt {1-2\,x}-1\right )}^2}{158203125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {3168922624\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{1318359375\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {459354112}{791015625}\right )}\right )}{9}+\frac {2\,{\left (\sqrt {1-2\,x}-1\right )}^3}{15\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3\,\left (\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}\right )}-\frac {4\,\left (\sqrt {1-2\,x}-1\right )}{75\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )\,\left (\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}\right )}+\frac {16\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{15\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2\,\left (\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}\right )} \]
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