Integrand size = 37, antiderivative size = 391 \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x} \, dx=-\frac {13027 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{4800 \sqrt {-5+2 x}}+\frac {23}{240} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}-\frac {1}{9} (2-3 x)^{3/2} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}+\frac {13027 \sqrt {\frac {143}{3}} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{3200 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {1368371 \sqrt {\frac {11}{23}} \sqrt {7+5 x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{43200 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}-\frac {65750101 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{216000 \sqrt {429} \sqrt {-5+2 x} \sqrt {1+4 x}} \]
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Time = 0.27 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.270, Rules used = {167, 1614, 1616, 1612, 176, 429, 171, 551, 182, 435} \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x} \, dx=\frac {13027 \sqrt {\frac {143}{3}} \sqrt {\frac {5 x+7}{5-2 x}} \sqrt {2-3 x} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{3200 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {65750101 \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} (2-3 x) \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {5 x+7}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{216000 \sqrt {429} \sqrt {2 x-5} \sqrt {4 x+1}}-\frac {1368371 \sqrt {\frac {11}{23}} \sqrt {5 x+7} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{43200 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}-\frac {1}{9} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7} (2-3 x)^{3/2}+\frac {23}{240} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7} \sqrt {2-3 x}-\frac {13027 \sqrt {4 x+1} \sqrt {5 x+7} \sqrt {2-3 x}}{4800 \sqrt {2 x-5}} \]
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Rule 167
Rule 171
Rule 176
Rule 182
Rule 429
Rule 435
Rule 551
Rule 1612
Rule 1614
Rule 1616
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{9} (2-3 x)^{3/2} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}-\frac {1}{18} \int \frac {\sqrt {2-3 x} \left (617+1042 x-138 x^2\right )}{\sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx \\ & = \frac {23}{240} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}-\frac {1}{9} (2-3 x)^{3/2} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}-\frac {\int \frac {170254+143540 x-468972 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{2880} \\ & = -\frac {13027 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{4800 \sqrt {-5+2 x}}+\frac {23}{240} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}-\frac {1}{9} (2-3 x)^{3/2} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}+\frac {\int \frac {-154352184+50903304 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{691200}-\frac {1862861 \int \frac {\sqrt {2-3 x}}{(-5+2 x)^{3/2} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{3200} \\ & = -\frac {13027 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{4800 \sqrt {-5+2 x}}+\frac {23}{240} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}-\frac {1}{9} (2-3 x)^{3/2} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}-\frac {2120971 \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{86400}-\frac {15052081 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{86400}+\frac {\left (169351 \sqrt {\frac {11}{23}} \sqrt {2-3 x} \sqrt {-\frac {7+5 x}{-5+2 x}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sqrt {1-\frac {39 x^2}{23}}} \, dx,x,\frac {\sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )}{3200 \sqrt {-\frac {2-3 x}{-5+2 x}} \sqrt {7+5 x}} \\ & = -\frac {13027 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{4800 \sqrt {-5+2 x}}+\frac {23}{240} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}-\frac {1}{9} (2-3 x)^{3/2} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}+\frac {13027 \sqrt {\frac {143}{3}} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{3200 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {\left (65750101 (2-3 x) \sqrt {-\frac {-5+2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {11 x^2}{23}} \sqrt {1+\frac {11 x^2}{39}} \left (5+3 x^2\right )} \, dx,x,\frac {\sqrt {7+5 x}}{\sqrt {2-3 x}}\right )}{43200 \sqrt {897} \sqrt {-5+2 x} \sqrt {1+4 x}}-\frac {\left (1368371 \sqrt {\frac {11}{46}} \sqrt {-\frac {-5+2 x}{2-3 x}} \sqrt {7+5 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{2}} \sqrt {1+\frac {31 x^2}{23}}} \, dx,x,\frac {\sqrt {1+4 x}}{\sqrt {2-3 x}}\right )}{43200 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{2-3 x}}} \\ & = -\frac {13027 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{4800 \sqrt {-5+2 x}}+\frac {23}{240} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}-\frac {1}{9} (2-3 x)^{3/2} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}+\frac {13027 \sqrt {\frac {143}{3}} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{3200 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {1368371 \sqrt {\frac {11}{23}} \sqrt {7+5 x} F\left (\tan ^{-1}\left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right )|-\frac {39}{23}\right )}{43200 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}-\frac {65750101 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}} \Pi \left (-\frac {69}{55};\sin ^{-1}\left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right )|-\frac {23}{39}\right )}{216000 \sqrt {429} \sqrt {-5+2 x} \sqrt {1+4 x}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 12.20 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.43 \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x} \, dx=\frac {1}{720} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x} (-91+240 x)+\frac {\frac {7796073285 \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}}{\sqrt {2-3 x}}-\frac {2598691095 i \sqrt {253} \sqrt {\frac {-5+2 x}{-2+3 x}} \sqrt {1+4 x} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {11}{39}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right )|-\frac {39}{23}\right )}{\sqrt {-5+2 x} \sqrt {\frac {1+4 x}{-2+3 x}}}-\frac {310954490 \sqrt {429} \sqrt {\frac {-5+2 x}{-2+3 x}} \sqrt {1+4 x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{\sqrt {-5+2 x} \sqrt {\frac {1+4 x}{-2+3 x}}}-\frac {4481783026 \sqrt {429} \sqrt {\frac {-5+2 x}{-2+3 x}} \sqrt {1+4 x} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{\sqrt {-5+2 x} \sqrt {\frac {1+4 x}{-2+3 x}}}+\frac {290533815 i \sqrt {682} \sqrt {2-3 x} \sqrt {\frac {1+4 x}{-5+2 x}} \operatorname {EllipticPi}\left (-\frac {23}{55},i \text {arcsinh}\left (\frac {\sqrt {\frac {22}{23}} \sqrt {7+5 x}}{\sqrt {-5+2 x}}\right ),\frac {23}{62}\right )}{\sqrt {\frac {2-3 x}{5-2 x}} \sqrt {1+4 x}}-\frac {1958698170 \sqrt {682} \sqrt {2-3 x} \sqrt {\frac {-5+2 x}{1+4 x}} \operatorname {EllipticPi}\left (\frac {78}{55},\arcsin \left (\frac {\sqrt {\frac {22}{39}} \sqrt {7+5 x}}{\sqrt {1+4 x}}\right ),\frac {39}{62}\right )}{\sqrt {-5+2 x} \sqrt {\frac {-2+3 x}{1+4 x}}}}{1915056000} \]
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Time = 1.78 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.14
method | result | size |
elliptic | \(\frac {\sqrt {-\left (7+5 x \right ) \left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (\frac {x \sqrt {-120 x^{4}+182 x^{3}+385 x^{2}-197 x -70}}{3}-\frac {91 \sqrt {-120 x^{4}+182 x^{3}+385 x^{2}-197 x -70}}{720}-\frac {85127 \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{220231440 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}-\frac {7177 \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, \left (\frac {2 F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{3}-\frac {31 \Pi \left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right )}{15}\right )}{22023144 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}+\frac {\frac {13027 \left (x +\frac {7}{5}\right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}{160}-\frac {13027 \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, \left (\frac {181 F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{341}-\frac {117 E\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{62}+\frac {91 \Pi \left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right )}{55}\right )}{12916800}}{\sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {7+5 x}}\) | \(446\) |
risch | \(-\frac {\left (-91+240 x \right ) \sqrt {7+5 x}\, \left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {\left (7+5 x \right ) \left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{720 \sqrt {-\left (7+5 x \right ) \left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}-\frac {\left (-\frac {85127 \sqrt {1705}\, \sqrt {\frac {x +\frac {7}{5}}{x +\frac {1}{4}}}\, \left (x +\frac {1}{4}\right )^{2} \sqrt {1794}\, \sqrt {\frac {x -\frac {5}{2}}{x +\frac {1}{4}}}\, \sqrt {2139}\, \sqrt {\frac {-\frac {2}{3}+x}{x +\frac {1}{4}}}\, F\left (\frac {\sqrt {1705}\, \sqrt {\frac {x +\frac {7}{5}}{x +\frac {1}{4}}}}{62}, \frac {\sqrt {2418}}{39}\right )}{220231440 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}-\frac {7177 \sqrt {1705}\, \sqrt {\frac {x +\frac {7}{5}}{x +\frac {1}{4}}}\, \left (x +\frac {1}{4}\right )^{2} \sqrt {1794}\, \sqrt {\frac {x -\frac {5}{2}}{x +\frac {1}{4}}}\, \sqrt {2139}\, \sqrt {\frac {-\frac {2}{3}+x}{x +\frac {1}{4}}}\, \left (-\frac {F\left (\frac {\sqrt {1705}\, \sqrt {\frac {x +\frac {7}{5}}{x +\frac {1}{4}}}}{62}, \frac {\sqrt {2418}}{39}\right )}{4}-\frac {23 \Pi \left (\frac {\sqrt {1705}\, \sqrt {\frac {x +\frac {7}{5}}{x +\frac {1}{4}}}}{62}, \frac {124}{55}, \frac {\sqrt {2418}}{39}\right )}{20}\right )}{22023144 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}-\frac {13027 \left (\left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right )-\frac {\sqrt {1705}\, \sqrt {\frac {x +\frac {7}{5}}{x +\frac {1}{4}}}\, \left (x +\frac {1}{4}\right )^{2} \sqrt {1794}\, \sqrt {\frac {x -\frac {5}{2}}{x +\frac {1}{4}}}\, \sqrt {2139}\, \sqrt {\frac {-\frac {2}{3}+x}{x +\frac {1}{4}}}\, \left (\frac {283 F\left (\frac {\sqrt {1705}\, \sqrt {\frac {x +\frac {7}{5}}{x +\frac {1}{4}}}}{62}, \frac {\sqrt {2418}}{39}\right )}{253}-\frac {78 E\left (\frac {\sqrt {1705}\, \sqrt {\frac {x +\frac {7}{5}}{x +\frac {1}{4}}}}{62}, \frac {\sqrt {2418}}{39}\right )}{23}-\frac {91 \Pi \left (\frac {\sqrt {1705}\, \sqrt {\frac {x +\frac {7}{5}}{x +\frac {1}{4}}}}{62}, \frac {124}{55}, \frac {\sqrt {2418}}{39}\right )}{55}\right )}{145080}\right )}{160 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}\right ) \sqrt {\left (7+5 x \right ) \left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {7+5 x}}\) | \(497\) |
default | \(-\frac {\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {7+5 x}\, \left (2263376115 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x^{2} E\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )-1354687290 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x^{2} F\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )-1183501818 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x^{2} \Pi \left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right )-3017834820 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x E\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )+1806249720 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x F\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )+1578002424 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x \Pi \left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right )+1005944940 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, E\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )-602083240 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, F\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )-526000808 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, \Pi \left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right )-170501760000 x^{5}+323242920000 x^{4}+1143078136200 x^{3}-1077307970310 x^{2}-2613563017065 x -569627583525\right )}{4262544000 \left (120 x^{4}-182 x^{3}-385 x^{2}+197 x +70\right )}\) | \(831\) |
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\[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x} \, dx=\int { \sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} \,d x } \]
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\[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x} \, dx=\int \sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1} \sqrt {5 x + 7}\, dx \]
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\[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x} \, dx=\int { \sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} \,d x } \]
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\[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x} \, dx=\int { \sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} \,d x } \]
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Timed out. \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x} \, dx=\int \sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,\sqrt {5\,x+7} \,d x \]
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