Integrand size = 26, antiderivative size = 113 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {259}{800} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (187559+77820 x)}{128000}+\frac {10866247 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{128000 \sqrt {10}} \]
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Time = 0.02 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {102, 158, 152, 56, 222} \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\frac {10866247 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{128000 \sqrt {10}}-\frac {3}{40} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^3-\frac {259}{800} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2-\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} (77820 x+187559)}{128000} \]
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Rule 56
Rule 102
Rule 152
Rule 158
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {3}{40} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}-\frac {1}{40} \int \frac {\left (-238-\frac {777 x}{2}\right ) (2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {259}{800} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}+\frac {\int \frac {(2+3 x) \left (\frac {41769}{2}+\frac {136185 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1200} \\ & = -\frac {259}{800} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (187559+77820 x)}{128000}+\frac {10866247 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{256000} \\ & = -\frac {259}{800} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (187559+77820 x)}{128000}+\frac {10866247 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{128000 \sqrt {5}} \\ & = -\frac {259}{800} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (187559+77820 x)}{128000}+\frac {10866247 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{128000 \sqrt {10}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.69 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\frac {-30 \sqrt {1-2 x} \left (1555473+3980075 x+3204060 x^2+1744800 x^3+432000 x^4\right )-10866247 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{1280000 \sqrt {3+5 x}} \]
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Time = 1.17 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.91
method | result | size |
risch | \(\frac {3 \left (86400 x^{3}+297120 x^{2}+462540 x +518491\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{128000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {10866247 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2560000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(103\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (-5184000 x^{3} \sqrt {-10 x^{2}-x +3}-17827200 x^{2} \sqrt {-10 x^{2}-x +3}+10866247 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-27752400 x \sqrt {-10 x^{2}-x +3}-31109460 \sqrt {-10 x^{2}-x +3}\right )}{2560000 \sqrt {-10 x^{2}-x +3}}\) | \(104\) |
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Time = 0.23 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.64 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {3}{128000} \, {\left (86400 \, x^{3} + 297120 \, x^{2} + 462540 \, x + 518491\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {10866247}{2560000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
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\[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\int \frac {\left (3 x + 2\right )^{4}}{\sqrt {1 - 2 x} \sqrt {5 x + 3}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.66 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {81}{40} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {5571}{800} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {69381}{6400} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {10866247}{2560000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {1555473}{128000} \, \sqrt {-10 \, x^{2} - x + 3} \]
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Time = 0.33 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.56 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {1}{6400000} \, \sqrt {5} {\left (6 \, {\left (12 \, {\left (8 \, {\left (180 \, x + 403\right )} {\left (5 \, x + 3\right )} + 16609\right )} {\left (5 \, x + 3\right )} + 1646339\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 54331235 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \]
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Time = 12.98 (sec) , antiderivative size = 708, normalized size of antiderivative = 6.27 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\frac {10866247\,\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 )}{640000}-\frac {\frac {6770247\,\left (\sqrt {1-2\,x}-1\right )}{195312500\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {33291573\,{\left (\sqrt {1-2\,x}-1\right )}^3}{78125000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {883182573\,{\left (\sqrt {1-2\,x}-1\right )}^5}{156250000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}+\frac {451883391\,{\left (\sqrt {1-2\,x}-1\right )}^7}{62500000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}-\frac {451883391\,{\left (\sqrt {1-2\,x}-1\right )}^9}{25000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}+\frac {883182573\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{10000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}+\frac {33291573\,{\left (\sqrt {1-2\,x}-1\right )}^{13}}{800000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{13}}-\frac {6770247\,{\left (\sqrt {1-2\,x}-1\right )}^{15}}{320000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{15}}+\frac {49152\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {258048\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {1032192\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {16147968\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {258048\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {16128\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {768\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{14}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{14}}}{\frac {1024\,{\left (\sqrt {1-2\,x}-1\right )}^2}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {1792\,{\left (\sqrt {1-2\,x}-1\right )}^4}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {1792\,{\left (\sqrt {1-2\,x}-1\right )}^6}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {224\,{\left (\sqrt {1-2\,x}-1\right )}^8}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {448\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {112\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {16\,{\left (\sqrt {1-2\,x}-1\right )}^{14}}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{14}}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^{16}}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{16}}+\frac {256}{390625}} \]
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