Integrand size = 26, antiderivative size = 91 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)}-\frac {2}{9} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {37 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{9 \sqrt {7}} \]
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Time = 0.02 (sec) , antiderivative size = 91, 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 = {99, 163, 56, 222, 95, 210} \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=-\frac {2}{9} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {37 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{9 \sqrt {7}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)} \]
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Rule 56
Rule 95
Rule 99
Rule 163
Rule 210
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {1}{3} \int \frac {-\frac {1}{2}-10 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = -\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)}-\frac {10}{9} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx+\frac {37}{18} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = -\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {37}{9} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )-\frac {1}{9} \left (4 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right ) \\ & = -\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)}-\frac {2}{9} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {37 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{9 \sqrt {7}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{6+9 x}+\frac {2}{9} \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-\frac {37 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{9 \sqrt {7}} \]
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Time = 1.14 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.41
method | result | size |
risch | \(\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{3 \left (2+3 x \right ) \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (\frac {\sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{9}-\frac {37 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{126}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(128\) |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (42 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -111 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +28 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-74 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+42 \sqrt {-10 x^{2}-x +3}\right )}{126 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )}\) | \(131\) |
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Time = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=-\frac {37 \, \sqrt {7} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, \sqrt {10} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{126 \, {\left (3 \, x + 2\right )}} \]
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\[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=\int \frac {\sqrt {1 - 2 x} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{2}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=-\frac {1}{9} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {37}{126} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {\sqrt {-10 \, x^{2} - x + 3}}{3 \, {\left (3 \, x + 2\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 260 vs. \(2 (67) = 134\).
Time = 0.38 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.86 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=\frac {37}{1260} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1}{9} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {22 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{3 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
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Time = 7.22 (sec) , antiderivative size = 752, normalized size of antiderivative = 8.26 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=\frac {37\,\sqrt {7}\,\mathrm {atan}\left (\frac {31139252096\,\sqrt {7}\,\left (\sqrt {1-2\,x}-1\right )}{38759765625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )\,\left (\frac {1076274944\,{\left (\sqrt {1-2\,x}-1\right )}^2}{1550390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {14849685376\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{12919921875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {2152549888}{7751953125}\right )}-\frac {901743872\,\sqrt {3}\,\sqrt {7}}{4306640625\,\left (\frac {1076274944\,{\left (\sqrt {1-2\,x}-1\right )}^2}{1550390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {14849685376\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{12919921875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {2152549888}{7751953125}\right )}+\frac {450871936\,\sqrt {3}\,\sqrt {7}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{861328125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2\,\left (\frac {1076274944\,{\left (\sqrt {1-2\,x}-1\right )}^2}{1550390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {14849685376\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{12919921875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {2152549888}{7751953125}\right )}\right )}{63}-\frac {4\,\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 )}{9}-\frac {2\,{\left (\sqrt {1-2\,x}-1\right )}^3}{3\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3\,\left (\frac {14\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\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 {6\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^3}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {12\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{25\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {4}{25}\right )}+\frac {4\,\left (\sqrt {1-2\,x}-1\right )}{15\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )\,\left (\frac {14\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\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 {6\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^3}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {12\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{25\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {4}{25}\right )}-\frac {37\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{75\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2\,\left (\frac {14\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\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 {6\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^3}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {12\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{25\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {4}{25}\right )} \]
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