Integrand size = 24, antiderivative size = 94 \[ \int \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x} \, dx=\frac {407}{800} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {37}{80} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac {4477 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{800 \sqrt {10}} \]
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Time = 0.02 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {81, 52, 56, 222} \[ \int \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x} \, dx=\frac {4477 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{800 \sqrt {10}}-\frac {1}{10} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac {37}{80} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {407}{800} \sqrt {5 x+3} \sqrt {1-2 x} \]
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Rule 52
Rule 56
Rule 81
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{10} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac {37}{20} \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx \\ & = -\frac {37}{80} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac {407}{160} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx \\ & = \frac {407}{800} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {37}{80} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac {4477 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1600} \\ & = \frac {407}{800} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {37}{80} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac {4477 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{800 \sqrt {5}} \\ & = \frac {407}{800} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {37}{80} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac {4477 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{800 \sqrt {10}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.78 \[ \int \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x} \, dx=\frac {10 \sqrt {1-2 x} \left (-609+1445 x+6500 x^2+4000 x^3\right )-4477 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{8000 \sqrt {3+5 x}} \]
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Time = 1.12 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (16000 x^{2} \sqrt {-10 x^{2}-x +3}+4477 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+16400 x \sqrt {-10 x^{2}-x +3}-4060 \sqrt {-10 x^{2}-x +3}\right )}{16000 \sqrt {-10 x^{2}-x +3}}\) | \(87\) |
risch | \(-\frac {\left (800 x^{2}+820 x -203\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{800 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {4477 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{16000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(98\) |
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Time = 0.23 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.71 \[ \int \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x} \, dx=\frac {1}{800} \, {\left (800 \, x^{2} + 820 \, x - 203\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {4477}{16000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
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\[ \int \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x} \, dx=\int \sqrt {1 - 2 x} \left (3 x + 2\right ) \sqrt {5 x + 3}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.59 \[ \int \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x} \, dx=-\frac {1}{10} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {37}{40} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {4477}{16000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {37}{800} \, \sqrt {-10 \, x^{2} - x + 3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (67) = 134\).
Time = 0.31 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.49 \[ \int \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x} \, dx=\frac {1}{8000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {19}{2000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {3}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
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Time = 8.13 (sec) , antiderivative size = 588, normalized size of antiderivative = 6.26 \[ \int \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x} \, dx=2\,\sqrt {1-2\,x}\,\sqrt {5\,x+3}\,\left (\frac {x}{2}+\frac {1}{40}\right )-\frac {363\,\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 )}{4000}-\frac {\frac {7543\,{\left (\sqrt {1-2\,x}-1\right )}^3}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {726\,\left (\sqrt {1-2\,x}-1\right )}{390625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {19023\,{\left (\sqrt {1-2\,x}-1\right )}^5}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}+\frac {19023\,{\left (\sqrt {1-2\,x}-1\right )}^7}{6250\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}-\frac {7543\,{\left (\sqrt {1-2\,x}-1\right )}^9}{5000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}+\frac {363\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{2000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}+\frac {1152\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {11136\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {15936\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}-\frac {2784\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {72\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}}{\frac {192\,{\left (\sqrt {1-2\,x}-1\right )}^2}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {48\,{\left (\sqrt {1-2\,x}-1\right )}^4}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {32\,{\left (\sqrt {1-2\,x}-1\right )}^6}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {12\,{\left (\sqrt {1-2\,x}-1\right )}^8}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {12\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^{12}}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {64}{15625}}-\frac {\sqrt {2}\,\sqrt {5}\,\ln \left (x+\frac {1}{20}-\frac {\sqrt {10}\,\sqrt {1-2\,x}\,\sqrt {5\,x+3}\,1{}\mathrm {i}}{10}\right )\,121{}\mathrm {i}}{400} \]
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