Integrand size = 26, antiderivative size = 84 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {1}{10} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (5363+2220 x)}{1600}+\frac {44437 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1600 \sqrt {10}} \]
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Time = 0.02 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {102, 152, 56, 222} \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\frac {44437 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1600 \sqrt {10}}-\frac {1}{10} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2-\frac {\sqrt {1-2 x} \sqrt {5 x+3} (2220 x+5363)}{1600} \]
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Rule 56
Rule 102
Rule 152
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{10} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {1}{30} \int \frac {\left (-171-\frac {555 x}{2}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {1}{10} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (5363+2220 x)}{1600}+\frac {44437 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{3200} \\ & = -\frac {1}{10} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (5363+2220 x)}{1600}+\frac {44437 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1600 \sqrt {5}} \\ & = -\frac {1}{10} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (5363+2220 x)}{1600}+\frac {44437 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1600 \sqrt {10}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.87 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\frac {-90 \sqrt {1-2 x} \left (2001+4715 x+2780 x^2+800 x^3\right )-44437 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{16000 \sqrt {3+5 x}} \]
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Time = 1.18 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.04
method | result | size |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (-28800 x^{2} \sqrt {-10 x^{2}-x +3}+44437 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-82800 x \sqrt {-10 x^{2}-x +3}-120060 \sqrt {-10 x^{2}-x +3}\right )}{32000 \sqrt {-10 x^{2}-x +3}}\) | \(87\) |
risch | \(\frac {9 \left (160 x^{2}+460 x +667\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1600 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {44437 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{32000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(98\) |
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Time = 0.23 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.80 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {9}{1600} \, {\left (160 \, x^{2} + 460 \, x + 667\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {44437}{32000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
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\[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\int \frac {\left (3 x + 2\right )^{3}}{\sqrt {1 - 2 x} \sqrt {5 x + 3}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.69 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {9}{10} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {207}{80} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {44437}{32000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {6003}{1600} \, \sqrt {-10 \, x^{2} - x + 3} \]
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Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.64 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {1}{80000} \, \sqrt {5} {\left (18 \, {\left (4 \, {\left (40 \, x + 91\right )} {\left (5 \, x + 3\right )} + 2243\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 222185 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \]
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Time = 8.79 (sec) , antiderivative size = 534, normalized size of antiderivative = 6.36 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\frac {44437\,\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 )}{8000}-\frac {\frac {18837\,\left (\sqrt {1-2\,x}-1\right )}{390625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {154377\,{\left (\sqrt {1-2\,x}-1\right )}^3}{156250\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {226251\,{\left (\sqrt {1-2\,x}-1\right )}^5}{156250\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}+\frac {226251\,{\left (\sqrt {1-2\,x}-1\right )}^7}{62500\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {154377\,{\left (\sqrt {1-2\,x}-1\right )}^9}{10000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}-\frac {18837\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{4000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}+\frac {4608\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {59904\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {107136\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {14976\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {288\,\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}} \]
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