Integrand size = 37, antiderivative size = 391 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^{3/2}}{\sqrt {-5+2 x}} \, dx=\frac {66377 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{1920 \sqrt {-5+2 x}}+\frac {977}{288} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}+\frac {1}{6} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}-\frac {66377 \sqrt {\frac {143}{3}} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{1280 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {2824441 \sqrt {\frac {11}{23}} \sqrt {7+5 x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{17280 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}+\frac {963142751 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{86400 \sqrt {429} \sqrt {-5+2 x} \sqrt {1+4 x}} \]
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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 = {168, 1614, 1616, 1612, 176, 429, 171, 551, 182, 435} \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^{3/2}}{\sqrt {-5+2 x}} \, dx=-\frac {66377 \sqrt {\frac {143}{3}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{1280 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {963142751 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {5 x+7}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{86400 \sqrt {429} \sqrt {2 x-5} \sqrt {4 x+1}}+\frac {2824441 \sqrt {\frac {11}{23}} \sqrt {5 x+7} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{17280 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}+\frac {1}{6} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^{3/2}+\frac {977}{288} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}+\frac {66377 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{1920 \sqrt {2 x-5}} \]
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Rule 168
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}{6} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}-\frac {1}{12} \int \frac {\sqrt {7+5 x} \left (-465+20 x+1954 x^2\right )}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx \\ & = \frac {977}{288} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}+\frac {1}{6} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}+\frac {\int \frac {697418-1294820 x-2389572 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{1152} \\ & = \frac {66377 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{1920 \sqrt {-5+2 x}}+\frac {977}{288} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}+\frac {1}{6} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}-\frac {\int \frac {-745656744+745658904 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{276480}+\frac {9491911 \int \frac {\sqrt {2-3 x}}{(-5+2 x)^{3/2} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{1280} \\ & = \frac {66377 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{1920 \sqrt {-5+2 x}}+\frac {977}{288} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}+\frac {1}{6} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}+\frac {31068851 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{34560}+\frac {31069121 \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{34560}-\frac {\left (862901 \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 )}{1280 \sqrt {-\frac {2-3 x}{-5+2 x}} \sqrt {7+5 x}} \\ & = \frac {66377 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{1920 \sqrt {-5+2 x}}+\frac {977}{288} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}+\frac {1}{6} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}-\frac {66377 \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 )}{1280 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {\left (963142751 (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 )}{17280 \sqrt {897} \sqrt {-5+2 x} \sqrt {1+4 x}}+\frac {\left (2824441 \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 )}{17280 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{2-3 x}}} \\ & = \frac {66377 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{1920 \sqrt {-5+2 x}}+\frac {977}{288} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}+\frac {1}{6} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}-\frac {66377 \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 )}{1280 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {2824441 \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 )}{17280 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}+\frac {963142751 (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 )}{86400 \sqrt {429} \sqrt {-5+2 x} \sqrt {1+4 x}} \\ \end{align*}
Time = 36.53 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^{3/2}}{\sqrt {-5+2 x}} \, dx=-\frac {\sqrt {-5+2 x} \sqrt {1+4 x} \left (-37038366 \sqrt {682} \sqrt {\frac {-5-18 x+8 x^2}{(2-3 x)^2}} \left (-14+11 x+15 x^2\right ) E\left (\arcsin \left (\sqrt {\frac {31}{39}} \sqrt {\frac {-5+2 x}{-2+3 x}}\right )|\frac {39}{62}\right )+31389484 \sqrt {682} \sqrt {\frac {-5-18 x+8 x^2}{(2-3 x)^2}} \left (-14+11 x+15 x^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {31}{39}} \sqrt {\frac {-5+2 x}{-2+3 x}}\right ),\frac {39}{62}\right )+\sqrt {\frac {7+5 x}{-2+3 x}} \left (186 \left (-17232355-79187903 x-38640362 x^2+10641080 x^3+4555200 x^4+1152000 x^5\right )+31069121 \sqrt {682} (2-3 x)^2 \sqrt {\frac {1+4 x}{-2+3 x}} \sqrt {\frac {-35-11 x+10 x^2}{(2-3 x)^2}} \operatorname {EllipticPi}\left (\frac {117}{62},\arcsin \left (\sqrt {\frac {31}{39}} \sqrt {\frac {-5+2 x}{-2+3 x}}\right ),\frac {39}{62}\right )\right )\right )}{2142720 \sqrt {2-3 x} \sqrt {7+5 x} \sqrt {\frac {7+5 x}{-2+3 x}} \left (-5-18 x+8 x^2\right )} \]
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Time = 1.72 (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 {5 x \sqrt {-120 x^{4}+182 x^{3}+385 x^{2}-197 x -70}}{6}+\frac {1313 \sqrt {-120 x^{4}+182 x^{3}+385 x^{2}-197 x -70}}{288}+\frac {348709 \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 )}{88092576 \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 {323705 \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 )}{44046288 \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 {66377 \left (\left (x +\frac {7}{5}\right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )-\frac {\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 )}{80730}\right )}{64 \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 (1313+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 )}}{288 \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 {348709 \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 )}{88092576 \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 {323705 \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 )}{44046288 \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 {66377 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right )}{64}-\frac {66377 \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 )}{9285120}}{\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 {7+5 x}\, \sqrt {2-3 x}\, \sqrt {1+4 x}\, \sqrt {-5+2 x}\, \left (2796196590 \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 )+17336569518 \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 )-11532671865 \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 )-3728262120 \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 )-23115426024 \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 )+15376895820 \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 )+1242754040 \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 )+7705142008 \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 )-5125631940 \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 )-170501760000 x^{5}-674192376000 x^{4}-1574933045400 x^{3}+5718966777810 x^{2}+11720205583515 x +2550474701775\right )}{1705017600 \left (120 x^{4}-182 x^{3}-385 x^{2}+197 x +70\right )}\) | \(831\) |
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\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^{3/2}}{\sqrt {-5+2 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{\frac {3}{2}} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}} \,d x } \]
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\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^{3/2}}{\sqrt {-5+2 x}} \, dx=\int \frac {\sqrt {2 - 3 x} \sqrt {4 x + 1} \left (5 x + 7\right )^{\frac {3}{2}}}{\sqrt {2 x - 5}}\, dx \]
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\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^{3/2}}{\sqrt {-5+2 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{\frac {3}{2}} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}} \,d x } \]
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\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^{3/2}}{\sqrt {-5+2 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{\frac {3}{2}} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^{3/2}}{\sqrt {-5+2 x}} \, dx=\int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,{\left (5\,x+7\right )}^{3/2}}{\sqrt {2\,x-5}} \,d x \]
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