Integrand size = 37, antiderivative size = 365 \[ \int \frac {\sqrt {2-3 x} \sqrt {7+5 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\frac {\sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{4 \sqrt {-5+2 x}}-\frac {\sqrt {429} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {39 \sqrt {\frac {11}{23}} \sqrt {7+5 x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{8 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}+\frac {179 \sqrt {\frac {11}{62}} \sqrt {2-3 x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {\frac {22}{23}} \sqrt {7+5 x}}{\sqrt {-5+2 x}}\right ),\frac {39}{62}\right )}{16 \sqrt {-\frac {2-3 x}{1+4 x}} \sqrt {1+4 x}}+\frac {4117 \sqrt {2-3 x} \operatorname {EllipticPi}\left (\frac {78}{55},\arctan \left (\frac {\sqrt {\frac {22}{23}} \sqrt {7+5 x}}{\sqrt {-5+2 x}}\right ),\frac {39}{62}\right )}{80 \sqrt {682} \sqrt {-\frac {2-3 x}{1+4 x}} \sqrt {1+4 x}} \]
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Time = 0.15 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {179, 182, 435, 171, 550, 429, 553, 176} \[ \int \frac {\sqrt {2-3 x} \sqrt {7+5 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=-\frac {\sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {39 \sqrt {\frac {11}{23}} \sqrt {5 x+7} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}+\frac {179 \sqrt {\frac {11}{62}} \sqrt {2-3 x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {\frac {22}{23}} \sqrt {5 x+7}}{\sqrt {2 x-5}}\right ),\frac {39}{62}\right )}{16 \sqrt {-\frac {2-3 x}{4 x+1}} \sqrt {4 x+1}}+\frac {4117 \sqrt {2-3 x} \operatorname {EllipticPi}\left (\frac {78}{55},\arctan \left (\frac {\sqrt {\frac {22}{23}} \sqrt {5 x+7}}{\sqrt {2 x-5}}\right ),\frac {39}{62}\right )}{80 \sqrt {682} \sqrt {-\frac {2-3 x}{4 x+1}} \sqrt {4 x+1}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}} \]
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Rule 171
Rule 176
Rule 179
Rule 182
Rule 429
Rule 435
Rule 550
Rule 553
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{4 \sqrt {-5+2 x}}-\frac {179}{16} \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx-\frac {429}{16} \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx+\frac {429}{8} \int \frac {\sqrt {2-3 x}}{(-5+2 x)^{3/2} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx \\ & = \frac {\sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{4 \sqrt {-5+2 x}}-\frac {\left (6981 \sqrt {-\frac {2-3 x}{-5+2 x}} (-5+2 x) \sqrt {\frac {1+4 x}{-5+2 x}}\right ) \text {Subst}\left (\int \frac {1}{\left (5-2 x^2\right ) \sqrt {1+\frac {11 x^2}{31}} \sqrt {1+\frac {22 x^2}{23}}} \, dx,x,\frac {\sqrt {7+5 x}}{\sqrt {-5+2 x}}\right )}{8 \sqrt {713} \sqrt {2-3 x} \sqrt {1+4 x}}-\frac {\left (39 \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 )}{8 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{2-3 x}}}-\frac {\left (39 \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 )}{8 \sqrt {-\frac {2-3 x}{-5+2 x}} \sqrt {7+5 x}} \\ & = \frac {\sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{4 \sqrt {-5+2 x}}-\frac {\sqrt {429} \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 )}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {39 \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 )}{8 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}-\frac {\left (179 \sqrt {\frac {23}{31}} \sqrt {-\frac {2-3 x}{-5+2 x}} (-5+2 x) \sqrt {\frac {1+4 x}{-5+2 x}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {22 x^2}{23}}}{\left (5-2 x^2\right ) \sqrt {1+\frac {11 x^2}{31}}} \, dx,x,\frac {\sqrt {7+5 x}}{\sqrt {-5+2 x}}\right )}{16 \sqrt {2-3 x} \sqrt {1+4 x}}-\frac {\left (1969 \sqrt {-\frac {2-3 x}{-5+2 x}} (-5+2 x) \sqrt {\frac {1+4 x}{-5+2 x}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {11 x^2}{31}} \sqrt {1+\frac {22 x^2}{23}}} \, dx,x,\frac {\sqrt {7+5 x}}{\sqrt {-5+2 x}}\right )}{16 \sqrt {713} \sqrt {2-3 x} \sqrt {1+4 x}} \\ & = \frac {\sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{4 \sqrt {-5+2 x}}-\frac {\sqrt {429} \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 )}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {39 \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 )}{8 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}+\frac {179 \sqrt {\frac {11}{62}} \sqrt {2-3 x} F\left (\tan ^{-1}\left (\frac {\sqrt {\frac {22}{23}} \sqrt {7+5 x}}{\sqrt {-5+2 x}}\right )|\frac {39}{62}\right )}{16 \sqrt {-\frac {2-3 x}{1+4 x}} \sqrt {1+4 x}}+\frac {4117 \sqrt {2-3 x} \Pi \left (\frac {78}{55};\tan ^{-1}\left (\frac {\sqrt {\frac {22}{23}} \sqrt {7+5 x}}{\sqrt {-5+2 x}}\right )|\frac {39}{62}\right )}{80 \sqrt {682} \sqrt {-\frac {2-3 x}{1+4 x}} \sqrt {1+4 x}} \\ \end{align*}
Time = 6.99 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {2-3 x} \sqrt {7+5 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=-\frac {6820 \sqrt {341} \sqrt {\frac {-2+3 x}{1+4 x}} \sqrt {\frac {7+5 x}{1+4 x}} \left (-5-18 x+8 x^2\right ) E\left (\arcsin \left (\sqrt {\frac {22}{39}} \sqrt {\frac {7+5 x}{1+4 x}}\right )|\frac {39}{62}\right )-1265 \sqrt {341} \sqrt {\frac {-2+3 x}{1+4 x}} \sqrt {\frac {7+5 x}{1+4 x}} \left (-5-18 x+8 x^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {22}{39}} \sqrt {\frac {7+5 x}{1+4 x}}\right ),\frac {39}{62}\right )+\sqrt {\frac {-5+2 x}{1+4 x}} \left (13640 \sqrt {2} \left (70-83 x-53 x^2+30 x^3\right )+4117 \sqrt {341} \sqrt {\frac {-2+3 x}{1+4 x}} (1+4 x)^2 \sqrt {\frac {-35-11 x+10 x^2}{(1+4 x)^2}} \operatorname {EllipticPi}\left (\frac {78}{55},\arcsin \left (\sqrt {\frac {22}{39}} \sqrt {\frac {7+5 x}{1+4 x}}\right ),\frac {39}{62}\right )\right )}{27280 \sqrt {2-3 x} \sqrt {-10+4 x} \sqrt {\frac {-5+2 x}{1+4 x}} \sqrt {1+4 x} \sqrt {7+5 x}} \]
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Time = 1.57 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.09
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 {28 \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 )}{305877 \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 {2 \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 )}{27807 \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 {15 \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 )}{2 \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}}\) | \(397\) |
default | \(-\frac {\sqrt {7+5 x}\, \sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (30690 \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 )+99882 \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 )-57915 \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 )-40920 \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 )-133176 \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 )+77220 \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 )+13640 \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 )+44392 \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 )-25740 \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 )-17760600 x^{3}+15096510 x^{2}+67046265 x +15540525\right )}{1184040 \left (120 x^{4}-182 x^{3}-385 x^{2}+197 x +70\right )}\) | \(821\) |
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\[ \int \frac {\sqrt {2-3 x} \sqrt {7+5 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {\sqrt {5 \, x + 7} \sqrt {-3 \, x + 2}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]
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\[ \int \frac {\sqrt {2-3 x} \sqrt {7+5 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int \frac {\sqrt {2 - 3 x} \sqrt {5 x + 7}}{\sqrt {2 x - 5} \sqrt {4 x + 1}}\, dx \]
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\[ \int \frac {\sqrt {2-3 x} \sqrt {7+5 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {\sqrt {5 \, x + 7} \sqrt {-3 \, x + 2}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]
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\[ \int \frac {\sqrt {2-3 x} \sqrt {7+5 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {\sqrt {5 \, x + 7} \sqrt {-3 \, x + 2}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {7+5 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int \frac {\sqrt {2-3\,x}\,\sqrt {5\,x+7}}{\sqrt {4\,x+1}\,\sqrt {2\,x-5}} \,d x \]
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