Integrand size = 37, antiderivative size = 351 \[ \int \frac {(7+5 x)^{5/2}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=-\frac {2135 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{192 \sqrt {-5+2 x}}-\frac {25}{48} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}+\frac {2135 \sqrt {\frac {143}{3}} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{128 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {29047 \sqrt {\frac {23}{11}} \sqrt {7+5 x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{576 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}-\frac {3431855 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{576 \sqrt {429} \sqrt {-5+2 x} \sqrt {1+4 x}} \]
-3431855/247104*(2-3*x)*EllipticPi(1/23*253^(1/2)*(7+5*x)^(1/2)/(2-3*x)^(1 /2),-69/55,1/39*I*897^(1/2))*((5-2*x)/(2-3*x))^(1/2)*((-1-4*x)/(2-3*x))^(1 /2)*429^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2)-2135/192*(2-3*x)^(1/2)*(1+4*x)^ (1/2)*(7+5*x)^(1/2)/(-5+2*x)^(1/2)-25/48*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4 *x)^(1/2)*(7+5*x)^(1/2)+29047/6336*(1/(4+2*(1+4*x)/(2-3*x)))^(1/2)*(4+2*(1 +4*x)/(2-3*x))^(1/2)*EllipticF((1+4*x)^(1/2)*2^(1/2)/(2-3*x)^(1/2)/(4+2*(1 +4*x)/(2-3*x))^(1/2),1/23*I*897^(1/2))*253^(1/2)*(7+5*x)^(1/2)/(-5+2*x)^(1 /2)/((7+5*x)/(5-2*x))^(1/2)+2135/384*EllipticE(1/23*897^(1/2)*(1+4*x)^(1/2 )/(-5+2*x)^(1/2),1/39*I*897^(1/2))*429^(1/2)*(2-3*x)^(1/2)*((7+5*x)/(5-2*x ))^(1/2)/((2-3*x)/(5-2*x))^(1/2)/(7+5*x)^(1/2)
Time = 24.32 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.99 \[ \int \frac {(7+5 x)^{5/2}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\frac {\sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x} \left (1227600 (-2+3 x)+\frac {-13104630 \sqrt {682} (-2+3 x) (7+5 x) \sqrt {\frac {-5-18 x+8 x^2}{(2-3 x)^2}} E\left (\arcsin \left (\sqrt {\frac {31}{39}} \sqrt {\frac {-5+2 x}{-2+3 x}}\right )|\frac {39}{62}\right )+17113116 \sqrt {682} (-2+3 x) (7+5 x) \sqrt {\frac {-5-18 x+8 x^2}{(2-3 x)^2}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {31}{39}} \sqrt {\frac {-5+2 x}{-2+3 x}}\right ),\frac {39}{62}\right )-385 \sqrt {\frac {7+5 x}{-2+3 x}} \left (-102114 \left (-35-151 x-34 x^2+40 x^3\right )-47445 \sqrt {682} (2-3 x)^2 \sqrt {\frac {1+4 x}{-2+3 x}} \sqrt {\frac {-35-11 x+10 x^2}{(2-3 x)^2}} \operatorname {EllipticPi}\left (\frac {117}{62},\arcsin \left (\sqrt {\frac {31}{39}} \sqrt {\frac {-5+2 x}{-2+3 x}}\right ),\frac {39}{62}\right )\right )}{(2-3 x) \left (\frac {7+5 x}{-2+3 x}\right )^{3/2} \left (5+18 x-8 x^2\right )}\right )}{2356992 \sqrt {2-3 x}} \]
(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x]*(1227600*(-2 + 3*x) + (-131046 30*Sqrt[682]*(-2 + 3*x)*(7 + 5*x)*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*El lipticE[ArcSin[Sqrt[31/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62] + 17113116 *Sqrt[682]*(-2 + 3*x)*(7 + 5*x)*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*Elli pticF[ArcSin[Sqrt[31/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62] - 385*Sqrt[( 7 + 5*x)/(-2 + 3*x)]*(-102114*(-35 - 151*x - 34*x^2 + 40*x^3) - 47445*Sqrt [682]*(2 - 3*x)^2*Sqrt[(1 + 4*x)/(-2 + 3*x)]*Sqrt[(-35 - 11*x + 10*x^2)/(2 - 3*x)^2]*EllipticPi[117/62, ArcSin[Sqrt[31/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x )]], 39/62]))/((2 - 3*x)*((7 + 5*x)/(-2 + 3*x))^(3/2)*(5 + 18*x - 8*x^2))) )/(2356992*Sqrt[2 - 3*x])
Time = 0.70 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.29, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, Rules used = {185, 2105, 27, 194, 27, 327, 2101, 183, 27, 188, 27, 320, 412}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(5 x+7)^{5/2}}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}} \, dx\) |
| \(\Big \downarrow \) 185 |
| \(\displaystyle \frac {1}{96} \int \frac {64050 x^2+89810 x+28003}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {25}{48} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\) |
| \(\Big \downarrow \) 2105 |
| \(\displaystyle \frac {1}{96} \left (-\frac {915915}{4} \int \frac {\sqrt {2-3 x}}{(2 x-5)^{3/2} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {1}{240} \int \frac {60 (146323-553525 x)}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {2135 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{2 \sqrt {2 x-5}}\right )-\frac {25}{48} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{96} \left (-\frac {915915}{4} \int \frac {\sqrt {2-3 x}}{(2 x-5)^{3/2} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {1}{4} \int \frac {146323-553525 x}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {2135 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{2 \sqrt {2 x-5}}\right )-\frac {25}{48} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\) |
| \(\Big \downarrow \) 194 |
| \(\displaystyle \frac {1}{96} \left (-\frac {1}{4} \int \frac {146323-553525 x}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx+\frac {83265 \sqrt {\frac {11}{23}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} \int \frac {\sqrt {23} \sqrt {\frac {4 x+1}{2 x-5}+1}}{\sqrt {23-\frac {39 (4 x+1)}{2 x-5}}}d\frac {\sqrt {4 x+1}}{\sqrt {2 x-5}}}{4 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {2135 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{2 \sqrt {2 x-5}}\right )-\frac {25}{48} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{96} \left (-\frac {1}{4} \int \frac {146323-553525 x}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx+\frac {83265 \sqrt {11} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} \int \frac {\sqrt {\frac {4 x+1}{2 x-5}+1}}{\sqrt {23-\frac {39 (4 x+1)}{2 x-5}}}d\frac {\sqrt {4 x+1}}{\sqrt {2 x-5}}}{4 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {2135 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{2 \sqrt {2 x-5}}\right )-\frac {25}{48} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1}{96} \left (-\frac {1}{4} \int \frac {146323-553525 x}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx+\frac {2135 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{4 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {2135 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{2 \sqrt {2 x-5}}\right )-\frac {25}{48} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\) |
| \(\Big \downarrow \) 2101 |
| \(\displaystyle \frac {1}{96} \left (\frac {1}{4} \left (\frac {668081}{3} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {553525}{3} \int \frac {\sqrt {2-3 x}}{\sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx\right )+\frac {2135 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{4 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {2135 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{2 \sqrt {2 x-5}}\right )-\frac {25}{48} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\) |
| \(\Big \downarrow \) 183 |
| \(\displaystyle \frac {1}{96} \left (\frac {1}{4} \left (\frac {668081}{3} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {34318550 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \int \frac {\sqrt {897}}{\sqrt {23-\frac {11 (5 x+7)}{2-3 x}} \left (\frac {3 (5 x+7)}{2-3 x}+5\right ) \sqrt {\frac {11 (5 x+7)}{2-3 x}+39}}d\frac {\sqrt {5 x+7}}{\sqrt {2-3 x}}}{3 \sqrt {897} \sqrt {2 x-5} \sqrt {4 x+1}}\right )+\frac {2135 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{4 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {2135 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{2 \sqrt {2 x-5}}\right )-\frac {25}{48} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{96} \left (\frac {1}{4} \left (\frac {668081}{3} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {34318550 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \int \frac {1}{\sqrt {23-\frac {11 (5 x+7)}{2-3 x}} \left (\frac {3 (5 x+7)}{2-3 x}+5\right ) \sqrt {\frac {11 (5 x+7)}{2-3 x}+39}}d\frac {\sqrt {5 x+7}}{\sqrt {2-3 x}}}{3 \sqrt {2 x-5} \sqrt {4 x+1}}\right )+\frac {2135 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{4 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {2135 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{2 \sqrt {2 x-5}}\right )-\frac {25}{48} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\) |
| \(\Big \downarrow \) 188 |
| \(\displaystyle \frac {1}{96} \left (\frac {1}{4} \left (\frac {29047 \sqrt {\frac {46}{11}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \int \frac {\sqrt {46}}{\sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23}}d\frac {\sqrt {4 x+1}}{\sqrt {2-3 x}}}{3 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}}}-\frac {34318550 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \int \frac {1}{\sqrt {23-\frac {11 (5 x+7)}{2-3 x}} \left (\frac {3 (5 x+7)}{2-3 x}+5\right ) \sqrt {\frac {11 (5 x+7)}{2-3 x}+39}}d\frac {\sqrt {5 x+7}}{\sqrt {2-3 x}}}{3 \sqrt {2 x-5} \sqrt {4 x+1}}\right )+\frac {2135 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{4 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {2135 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{2 \sqrt {2 x-5}}\right )-\frac {25}{48} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{96} \left (\frac {1}{4} \left (\frac {1336162 \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \int \frac {1}{\sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23}}d\frac {\sqrt {4 x+1}}{\sqrt {2-3 x}}}{3 \sqrt {11} \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}}}-\frac {34318550 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \int \frac {1}{\sqrt {23-\frac {11 (5 x+7)}{2-3 x}} \left (\frac {3 (5 x+7)}{2-3 x}+5\right ) \sqrt {\frac {11 (5 x+7)}{2-3 x}+39}}d\frac {\sqrt {5 x+7}}{\sqrt {2-3 x}}}{3 \sqrt {2 x-5} \sqrt {4 x+1}}\right )+\frac {2135 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{4 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {2135 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{2 \sqrt {2 x-5}}\right )-\frac {25}{48} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {1}{96} \left (\frac {1}{4} \left (\frac {58094 \sqrt {\frac {23}{11}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{3 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}} \sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {\frac {31 (4 x+1)}{2-3 x}+23}{\frac {4 x+1}{2-3 x}+2}}}-\frac {34318550 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \int \frac {1}{\sqrt {23-\frac {11 (5 x+7)}{2-3 x}} \left (\frac {3 (5 x+7)}{2-3 x}+5\right ) \sqrt {\frac {11 (5 x+7)}{2-3 x}+39}}d\frac {\sqrt {5 x+7}}{\sqrt {2-3 x}}}{3 \sqrt {2 x-5} \sqrt {4 x+1}}\right )+\frac {2135 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{4 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {2135 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{2 \sqrt {2 x-5}}\right )-\frac {25}{48} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {1}{96} \left (\frac {1}{4} \left (\frac {58094 \sqrt {\frac {23}{11}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{3 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}} \sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {\frac {31 (4 x+1)}{2-3 x}+23}{\frac {4 x+1}{2-3 x}+2}}}-\frac {6863710 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {5 x+7}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{3 \sqrt {429} \sqrt {2 x-5} \sqrt {4 x+1}}\right )+\frac {2135 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{4 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {2135 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{2 \sqrt {2 x-5}}\right )-\frac {25}{48} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\) |
(-25*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/48 + ((-213 5*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(2*Sqrt[-5 + 2*x]) + (2135*Sq rt[429]*Sqrt[2 - 3*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[39/ 23]*Sqrt[1 + 4*x])/Sqrt[-5 + 2*x]], -23/39])/(4*Sqrt[(2 - 3*x)/(5 - 2*x)]* Sqrt[7 + 5*x]) + ((58094*Sqrt[23/11]*Sqrt[(5 - 2*x)/(2 - 3*x)]*Sqrt[7 + 5* x]*Sqrt[23 + (31*(1 + 4*x))/(2 - 3*x)]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqr t[2]*Sqrt[2 - 3*x])], -39/23])/(3*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(2 - 3*x)] *Sqrt[2 + (1 + 4*x)/(2 - 3*x)]*Sqrt[(23 + (31*(1 + 4*x))/(2 - 3*x))/(2 + ( 1 + 4*x)/(2 - 3*x))]) - (6863710*(2 - 3*x)*Sqrt[(5 - 2*x)/(2 - 3*x)]*Sqrt[ -((1 + 4*x)/(2 - 3*x))]*EllipticPi[-69/55, ArcSin[(Sqrt[11/23]*Sqrt[7 + 5* x])/Sqrt[2 - 3*x]], -23/39])/(3*Sqrt[429]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]))/4 )/96
3.2.1.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_.) + (b_.)*(x_)]/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*( x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[2*(a + b*x)*Sqrt[(b*g - a*h)*(( c + d*x)/((d*g - c*h)*(a + b*x)))]*(Sqrt[(b*g - a*h)*((e + f*x)/((f*g - e*h )*(a + b*x)))]/(Sqrt[c + d*x]*Sqrt[e + f*x])) Subst[Int[1/((h - b*x^2)*Sq rt[1 + (b*c - a*d)*(x^2/(d*g - c*h))]*Sqrt[1 + (b*e - a*f)*(x^2/(f*g - e*h) )]), x], x, Sqrt[g + h*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[((a_.) + (b_.)*(x_))^(m_)/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)* (x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[2*b^2*(a + b*x)^(m - 2)*Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/(d*f*h*(2*m - 1))), x] - Simp[1/(d*f*h *(2*m - 1)) Int[((a + b*x)^(m - 3)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[a*b^2*(d*e*g + c*f*g + c*e*h) + 2*b^3*c*e*g*(m - 2) - a^3*d*f*h *(2*m - 1) + b*(2*a*b*(d*f*g + d*e*h + c*f*h) + b^2*(2*m - 3)*(d*e*g + c*f* g + c*e*h) - 3*a^2*d*f*h*(2*m - 1))*x - 2*b^2*(m - 1)*(3*a*d*f*h - b*(d*f*g + d*e*h + c*f*h))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IntegerQ[2*m] && GeQ[m, 2]
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.) *(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[2*Sqrt[g + h*x]*(Sqrt[(b*e - a*f)*((c + d*x)/((d*e - c*f)*(a + b*x)))]/((f*g - e*h)*Sqrt[c + d*x]*Sqrt[( -(b*e - a*f))*((g + h*x)/((f*g - e*h)*(a + b*x)))])) Subst[Int[1/(Sqrt[1 + (b*c - a*d)*(x^2/(d*e - c*f))]*Sqrt[1 - (b*g - a*h)*(x^2/(f*g - e*h))]), x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[Sqrt[(c_.) + (d_.)*(x_)]/(((a_.) + (b_.)*(x_))^(3/2)*Sqrt[(e_.) + (f_.) *(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2*Sqrt[c + d*x]*(Sqrt[(-(b*e - a*f))*((g + h*x)/((f*g - e*h)*(a + b*x)))]/((b*e - a*f)*Sqrt[g + h*x]*Sq rt[(b*e - a*f)*((c + d*x)/((d*e - c*f)*(a + b*x)))])) Subst[Int[Sqrt[1 + (b*c - a*d)*(x^2/(d*e - c*f))]/Sqrt[1 - (b*g - a*h)*(x^2/(f*g - e*h))], x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[((A_.) + (B_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)] *Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[(A*b - a*B)/b Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]) , x], x] + Simp[B/b Int[Sqrt[a + b*x]/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B}, x]
Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_. ) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbo l] :> Simp[C*Sqrt[a + b*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/(b*f*h*Sqrt[c + d*x ])), x] + (Simp[1/(2*b*d*f*h) Int[(1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[2*A*b*d*f*h - C*(b*d*e*g + a*c*f*h) + (2*b*B*d* f*h - C*(a*d*f*h + b*(d*f*g + d*e*h + c*f*h)))*x, x], x], x] + Simp[C*(d*e - c*f)*((d*g - c*h)/(2*b*d*f*h)) Int[Sqrt[a + b*x]/((c + d*x)^(3/2)*Sqrt[ e + f*x]*Sqrt[g + h*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C} , x]
Time = 1.73 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.20
| 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 {25 \sqrt {-120 x^{4}+182 x^{3}+385 x^{2}-197 x -70}}{48}+\frac {28003 \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 )}{14682096 \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 {44905 \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 )}{7341048 \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 {10675 \left (x +\frac {7}{5}\right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}{32}-\frac {2135 \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 )}{516672}}{\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}}\) | \(421\) |
| risch | \(\frac {25 \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 )}}{48 \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 {28003 \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 )}{14682096 \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 {44905 \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 )}{7341048 \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 {10675 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right )}{32}-\frac {2135 \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 )}{928512}}{\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}}\) | \(491\) |
| default | \(-\frac {\sqrt {7+5 x}\, \sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (12025458 \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 )-61773390 \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 )+24729705 \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 )-16033944 \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 )+82364520 \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 )-32972940 \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 )+5344648 \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 )-27454840 \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 )+10990980 \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 )+710424000 x^{4}+6506299800 x^{3}-8725486770 x^{2}-27462475755 x -6221390175\right )}{11366784 \left (120 x^{4}-182 x^{3}-385 x^{2}+197 x +70\right )}\) | \(826\) |
(-(7+5*x)*(-2+3*x)*(-5+2*x)*(1+4*x))^(1/2)/(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1 +4*x)^(1/2)/(7+5*x)^(1/2)*(-25/48*(-120*x^4+182*x^3+385*x^2-197*x-70)^(1/2 )+28003/14682096*(-3795*(x+7/5)/(-2/3+x))^(1/2)*(-2/3+x)^2*806^(1/2)*((x-5 /2)/(-2/3+x))^(1/2)*2139^(1/2)*((x+1/4)/(-2/3+x))^(1/2)/(-30*(x+7/5)*(-2/3 +x)*(x-5/2)*(x+1/4))^(1/2)*EllipticF(1/69*(-3795*(x+7/5)/(-2/3+x))^(1/2),1 /39*I*897^(1/2))+44905/7341048*(-3795*(x+7/5)/(-2/3+x))^(1/2)*(-2/3+x)^2*8 06^(1/2)*((x-5/2)/(-2/3+x))^(1/2)*2139^(1/2)*((x+1/4)/(-2/3+x))^(1/2)/(-30 *(x+7/5)*(-2/3+x)*(x-5/2)*(x+1/4))^(1/2)*(2/3*EllipticF(1/69*(-3795*(x+7/5 )/(-2/3+x))^(1/2),1/39*I*897^(1/2))-31/15*EllipticPi(1/69*(-3795*(x+7/5)/( -2/3+x))^(1/2),-69/55,1/39*I*897^(1/2)))+10675/32*((x+7/5)*(x-5/2)*(x+1/4) -1/80730*(-3795*(x+7/5)/(-2/3+x))^(1/2)*(-2/3+x)^2*806^(1/2)*((x-5/2)/(-2/ 3+x))^(1/2)*2139^(1/2)*((x+1/4)/(-2/3+x))^(1/2)*(181/341*EllipticF(1/69*(- 3795*(x+7/5)/(-2/3+x))^(1/2),1/39*I*897^(1/2))-117/62*EllipticE(1/69*(-379 5*(x+7/5)/(-2/3+x))^(1/2),1/39*I*897^(1/2))+91/55*EllipticPi(1/69*(-3795*( x+7/5)/(-2/3+x))^(1/2),-69/55,1/39*I*897^(1/2))))/(-30*(x+7/5)*(-2/3+x)*(x -5/2)*(x+1/4))^(1/2))
\[ \int \frac {(7+5 x)^{5/2}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{\frac {5}{2}}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
integral(-(25*x^2 + 70*x + 49)*sqrt(5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)*s qrt(-3*x + 2)/(24*x^3 - 70*x^2 + 21*x + 10), x)
Timed out. \[ \int \frac {(7+5 x)^{5/2}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\text {Timed out} \]
\[ \int \frac {(7+5 x)^{5/2}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{\frac {5}{2}}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
\[ \int \frac {(7+5 x)^{5/2}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{\frac {5}{2}}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
Timed out. \[ \int \frac {(7+5 x)^{5/2}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int \frac {{\left (5\,x+7\right )}^{5/2}}{\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}} \,d x \]