Integrand size = 37, antiderivative size = 391 \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{5/2}} \, dx=-\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{15 (7+5 x)^{3/2}}+\frac {17906 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{417105 \sqrt {7+5 x}}-\frac {35812 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{2085525 \sqrt {-5+2 x}}+\frac {17906 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{53475 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {496 \sqrt {\frac {11}{23}} \sqrt {7+5 x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{1725 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}+\frac {496 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{125 \sqrt {429} \sqrt {-5+2 x} \sqrt {1+4 x}} \]
-2/15*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(3/2)+496/53625*( 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)+17906/417105*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x )^(1/2)/(7+5*x)^(1/2)-35812/2085525*(2-3*x)^(1/2)*(1+4*x)^(1/2)*(7+5*x)^(1 /2)/(-5+2*x)^(1/2)-496/39675*(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)+17906/2085525*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)
Result contains complex when optimal does not.
Time = 17.09 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{5/2}} \, dx=\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (34864+44765 x)}{417105 (7+5 x)^{3/2}}+\frac {\frac {3571978410 \sqrt {1+4 x} \sqrt {7+5 x} \sqrt {-75+30 x}}{\sqrt {2-3 x}}-\frac {3571978410 \sqrt {715} \sqrt {-5+2 x} \sqrt {\frac {1+4 x}{-2+3 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right )|-\frac {23}{39}\right )}{\sqrt {\frac {5-2 x}{2-3 x}} \sqrt {1+4 x}}+\frac {5251113560 \sqrt {715} \sqrt {-5+2 x} \sqrt {\frac {1+4 x}{-2+3 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{\sqrt {\frac {5-2 x}{2-3 x}} \sqrt {1+4 x}}-\frac {6160344428 \sqrt {715} \sqrt {-5+2 x} \sqrt {\frac {1+4 x}{-2+3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{\sqrt {\frac {5-2 x}{2-3 x}} \sqrt {1+4 x}}-\frac {344407635 i \sqrt {10230} \sqrt {2-3 x} \sqrt {\frac {1+4 x}{-5+2 x}} \operatorname {EllipticPi}\left (-\frac {23}{55},i \text {arcsinh}\left (\frac {\sqrt {\frac {22}{23}} \sqrt {7+5 x}}{\sqrt {-5+2 x}}\right ),\frac {23}{62}\right )}{\sqrt {\frac {2-3 x}{5-2 x}} \sqrt {1+4 x}}-\frac {371344545 \sqrt {10230} \sqrt {2-3 x} \sqrt {\frac {-5+2 x}{1+4 x}} \operatorname {EllipticPi}\left (\frac {78}{55},\arcsin \left (\frac {\sqrt {\frac {22}{39}} \sqrt {7+5 x}}{\sqrt {1+4 x}}\right ),\frac {39}{62}\right )}{\sqrt {-5+2 x} \sqrt {\frac {-2+3 x}{1+4 x}}}}{138676984875 \sqrt {15}} \]
(2*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(34864 + 44765*x))/(417105*( 7 + 5*x)^(3/2)) + ((3571978410*Sqrt[1 + 4*x]*Sqrt[7 + 5*x]*Sqrt[-75 + 30*x ])/Sqrt[2 - 3*x] - (3571978410*Sqrt[715]*Sqrt[-5 + 2*x]*Sqrt[(1 + 4*x)/(-2 + 3*x)]*EllipticE[ArcSin[(Sqrt[11/23]*Sqrt[7 + 5*x])/Sqrt[2 - 3*x]], -23/ 39])/(Sqrt[(5 - 2*x)/(2 - 3*x)]*Sqrt[1 + 4*x]) + (5251113560*Sqrt[715]*Sqr t[-5 + 2*x]*Sqrt[(1 + 4*x)/(-2 + 3*x)]*EllipticF[ArcSin[(Sqrt[11/23]*Sqrt[ 7 + 5*x])/Sqrt[2 - 3*x]], -23/39])/(Sqrt[(5 - 2*x)/(2 - 3*x)]*Sqrt[1 + 4*x ]) - (6160344428*Sqrt[715]*Sqrt[-5 + 2*x]*Sqrt[(1 + 4*x)/(-2 + 3*x)]*Ellip ticPi[-69/55, ArcSin[(Sqrt[11/23]*Sqrt[7 + 5*x])/Sqrt[2 - 3*x]], -23/39])/ (Sqrt[(5 - 2*x)/(2 - 3*x)]*Sqrt[1 + 4*x]) - ((344407635*I)*Sqrt[10230]*Sqr t[2 - 3*x]*Sqrt[(1 + 4*x)/(-5 + 2*x)]*EllipticPi[-23/55, I*ArcSinh[(Sqrt[2 2/23]*Sqrt[7 + 5*x])/Sqrt[-5 + 2*x]], 23/62])/(Sqrt[(2 - 3*x)/(5 - 2*x)]*S qrt[1 + 4*x]) - (371344545*Sqrt[10230]*Sqrt[2 - 3*x]*Sqrt[(-5 + 2*x)/(1 + 4*x)]*EllipticPi[78/55, ArcSin[(Sqrt[22/39]*Sqrt[7 + 5*x])/Sqrt[1 + 4*x]], 39/62])/(Sqrt[-5 + 2*x]*Sqrt[(-2 + 3*x)/(1 + 4*x)]))/(138676984875*Sqrt[1 5])
Time = 0.87 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.27, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.432, Rules used = {178, 25, 2107, 27, 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.
| \(\displaystyle \int \frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{(5 x+7)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 178 |
| \(\displaystyle \frac {1}{15} \int -\frac {72 x^2-140 x+21}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^{3/2}}dx-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{15} \int \frac {72 x^2-140 x+21}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^{3/2}}dx-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^{3/2}}\) |
| \(\Big \downarrow \) 2107 |
| \(\displaystyle \frac {1}{15} \left (\frac {17906 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 \sqrt {5 x+7}}-\frac {\int -\frac {2 \left (214872 x^2-363155 x+20321\right )}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx}{27807}\right )-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (\frac {2 \int \frac {214872 x^2-363155 x+20321}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx}{27807}+\frac {17906 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 \sqrt {5 x+7}}\right )-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^{3/2}}\) |
| \(\Big \downarrow \) 2105 |
| \(\displaystyle \frac {1}{15} \left (\frac {2 \left (-\frac {3840837}{5} \int \frac {\sqrt {2-3 x}}{(2 x-5)^{3/2} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {1}{240} \int \frac {232128 (207 x+203)}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {17906 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{5 \sqrt {2 x-5}}\right )}{27807}+\frac {17906 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 \sqrt {5 x+7}}\right )-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (\frac {2 \left (-\frac {3840837}{5} \int \frac {\sqrt {2-3 x}}{(2 x-5)^{3/2} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {4836}{5} \int \frac {207 x+203}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {17906 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{5 \sqrt {2 x-5}}\right )}{27807}+\frac {17906 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 \sqrt {5 x+7}}\right )-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^{3/2}}\) |
| \(\Big \downarrow \) 194 |
| \(\displaystyle \frac {1}{15} \left (\frac {2 \left (-\frac {4836}{5} \int \frac {207 x+203}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx+\frac {349167 \sqrt {\frac {11}{23}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} \int \frac {\sqrt {23} \sqrt {\frac {4 x+1}{2 x-5}+1}}{\sqrt {23-\frac {39 (4 x+1)}{2 x-5}}}d\frac {\sqrt {4 x+1}}{\sqrt {2 x-5}}}{5 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {17906 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{5 \sqrt {2 x-5}}\right )}{27807}+\frac {17906 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 \sqrt {5 x+7}}\right )-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (\frac {2 \left (-\frac {4836}{5} \int \frac {207 x+203}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx+\frac {349167 \sqrt {11} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} \int \frac {\sqrt {\frac {4 x+1}{2 x-5}+1}}{\sqrt {23-\frac {39 (4 x+1)}{2 x-5}}}d\frac {\sqrt {4 x+1}}{\sqrt {2 x-5}}}{5 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {17906 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{5 \sqrt {2 x-5}}\right )}{27807}+\frac {17906 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 \sqrt {5 x+7}}\right )-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^{3/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1}{15} \left (\frac {2 \left (-\frac {4836}{5} \int \frac {207 x+203}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx+\frac {8953 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{5 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {17906 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{5 \sqrt {2 x-5}}\right )}{27807}+\frac {17906 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 \sqrt {5 x+7}}\right )-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^{3/2}}\) |
| \(\Big \downarrow \) 2101 |
| \(\displaystyle \frac {1}{15} \left (\frac {2 \left (-\frac {4836}{5} \left (341 \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx-69 \int \frac {\sqrt {2-3 x}}{\sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx\right )+\frac {8953 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{5 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {17906 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{5 \sqrt {2 x-5}}\right )}{27807}+\frac {17906 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 \sqrt {5 x+7}}\right )-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^{3/2}}\) |
| \(\Big \downarrow \) 183 |
| \(\displaystyle \frac {1}{15} \left (\frac {2 \left (-\frac {4836}{5} \left (341 \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {62 \sqrt {\frac {69}{13}} (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \int \frac {\sqrt {897}}{\sqrt {23-\frac {11 (5 x+7)}{2-3 x}} \left (\frac {3 (5 x+7)}{2-3 x}+5\right ) \sqrt {\frac {11 (5 x+7)}{2-3 x}+39}}d\frac {\sqrt {5 x+7}}{\sqrt {2-3 x}}}{\sqrt {2 x-5} \sqrt {4 x+1}}\right )+\frac {8953 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{5 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {17906 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{5 \sqrt {2 x-5}}\right )}{27807}+\frac {17906 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 \sqrt {5 x+7}}\right )-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (\frac {2 \left (-\frac {4836}{5} \left (341 \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {4278 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \int \frac {1}{\sqrt {23-\frac {11 (5 x+7)}{2-3 x}} \left (\frac {3 (5 x+7)}{2-3 x}+5\right ) \sqrt {\frac {11 (5 x+7)}{2-3 x}+39}}d\frac {\sqrt {5 x+7}}{\sqrt {2-3 x}}}{\sqrt {2 x-5} \sqrt {4 x+1}}\right )+\frac {8953 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{5 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {17906 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{5 \sqrt {2 x-5}}\right )}{27807}+\frac {17906 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 \sqrt {5 x+7}}\right )-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^{3/2}}\) |
| \(\Big \downarrow \) 188 |
| \(\displaystyle \frac {1}{15} \left (\frac {2 \left (-\frac {4836}{5} \left (\frac {31 \sqrt {\frac {22}{23}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \int \frac {\sqrt {46}}{\sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23}}d\frac {\sqrt {4 x+1}}{\sqrt {2-3 x}}}{\sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}}}-\frac {4278 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \int \frac {1}{\sqrt {23-\frac {11 (5 x+7)}{2-3 x}} \left (\frac {3 (5 x+7)}{2-3 x}+5\right ) \sqrt {\frac {11 (5 x+7)}{2-3 x}+39}}d\frac {\sqrt {5 x+7}}{\sqrt {2-3 x}}}{\sqrt {2 x-5} \sqrt {4 x+1}}\right )+\frac {8953 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{5 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {17906 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{5 \sqrt {2 x-5}}\right )}{27807}+\frac {17906 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 \sqrt {5 x+7}}\right )-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (\frac {2 \left (-\frac {4836}{5} \left (\frac {62 \sqrt {11} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \int \frac {1}{\sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23}}d\frac {\sqrt {4 x+1}}{\sqrt {2-3 x}}}{\sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}}}-\frac {4278 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \int \frac {1}{\sqrt {23-\frac {11 (5 x+7)}{2-3 x}} \left (\frac {3 (5 x+7)}{2-3 x}+5\right ) \sqrt {\frac {11 (5 x+7)}{2-3 x}+39}}d\frac {\sqrt {5 x+7}}{\sqrt {2-3 x}}}{\sqrt {2 x-5} \sqrt {4 x+1}}\right )+\frac {8953 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{5 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {17906 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{5 \sqrt {2 x-5}}\right )}{27807}+\frac {17906 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 \sqrt {5 x+7}}\right )-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^{3/2}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {1}{15} \left (\frac {2 \left (-\frac {4836}{5} \left (\frac {62 \sqrt {\frac {11}{23}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{\sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}} \sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {\frac {31 (4 x+1)}{2-3 x}+23}{\frac {4 x+1}{2-3 x}+2}}}-\frac {4278 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \int \frac {1}{\sqrt {23-\frac {11 (5 x+7)}{2-3 x}} \left (\frac {3 (5 x+7)}{2-3 x}+5\right ) \sqrt {\frac {11 (5 x+7)}{2-3 x}+39}}d\frac {\sqrt {5 x+7}}{\sqrt {2-3 x}}}{\sqrt {2 x-5} \sqrt {4 x+1}}\right )+\frac {8953 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{5 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {17906 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{5 \sqrt {2 x-5}}\right )}{27807}+\frac {17906 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 \sqrt {5 x+7}}\right )-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^{3/2}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {1}{15} \left (\frac {2 \left (-\frac {4836}{5} \left (\frac {62 \sqrt {\frac {11}{23}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{\sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}} \sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {\frac {31 (4 x+1)}{2-3 x}+23}{\frac {4 x+1}{2-3 x}+2}}}-\frac {1426 \sqrt {\frac {3}{143}} (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {5 x+7}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{5 \sqrt {2 x-5} \sqrt {4 x+1}}\right )+\frac {8953 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{5 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {17906 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{5 \sqrt {2 x-5}}\right )}{27807}+\frac {17906 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 \sqrt {5 x+7}}\right )-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^{3/2}}\) |
(-2*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(15*(7 + 5*x)^(3/2)) + ((1 7906*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(27807*Sqrt[7 + 5*x]) + ( 2*((-17906*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(5*Sqrt[-5 + 2*x]) + (8953*Sqrt[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])/(5*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) - (4836*((62*Sqrt[11/23]*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]/(Sqrt[2]*Sqrt[2 - 3*x])], -39/23])/(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))]) - (1426*Sqrt[3/143]*(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])/(5*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])))/5))/27807)/15
3.1.82.3.1 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[((a_.) + (b_.)*(x_))^(m_)*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*( x_)]*Sqrt[(g_.) + (h_.)*(x_)], x_] :> Simp[(a + b*x)^(m + 1)*Sqrt[c + d*x]* Sqrt[e + f*x]*(Sqrt[g + h*x]/(b*(m + 1))), x] - Simp[1/(2*b*(m + 1)) Int[ ((a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[d*e*g + c*f*g + c*e*h + 2*(d*f*g + d*e*h + c*f*h)*x + 3*d*f*h*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IntegerQ[2*m] && LtQ[m, -1]
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[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]
Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2))/(Sqrt[( c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sy mbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[ e + f*x]*(Sqrt[g + h*x]/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h))), x] - Simp[1/(2*(m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)) Int[((a + b*x)^( m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[A*(2*a^2*d*f*h*(m + 1) - 2*a*b*(m + 1)*(d*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h)) - (b*B - a*C)*(a*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*(m + 1)) - 2*((A*b - a*B)*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h)) - C*(a ^2*(d*f*g + d*e*h + c*f*h) - b^2*c*e*g*(m + 1) + a*b*(m + 1)*(d*e*g + c*f*g + c*e*h)))*x + d*f*h*(2*m + 5)*(A*b^2 - a*b*B + a^2*C)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x] && IntegerQ[2*m] && LtQ[m, -1]
Time = 1.61 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.19
| 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 {2 \sqrt {-120 x^{4}+182 x^{3}+385 x^{2}-197 x -70}}{375 \left (x +\frac {7}{5}\right )^{2}}+\frac {-\frac {143248}{139035} x^{3}+\frac {250684}{83421} x^{2}-\frac {125342}{139035} x -\frac {35812}{83421}}{\sqrt {\left (x +\frac {7}{5}\right ) \left (-120 x^{3}+350 x^{2}-105 x -50\right )}}+\frac {81284 \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 )}{127582826085 \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 {22348 \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 )}{1962812709 \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 {71624 \left (x +\frac {7}{5}\right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}{139035}-\frac {35812 \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 )}{5612147775}}{\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}}\) | \(464\) |
| default | \(\text {Expression too large to display}\) | \(1077\) |
(-(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)*(-2/375*(-120*x^4+182*x^3+385*x^2-197*x-70)^(1/2 )/(x+7/5)^2+17906/2085525*(-120*x^3+350*x^2-105*x-50)/((x+7/5)*(-120*x^3+3 50*x^2-105*x-50))^(1/2)+81284/127582826085*(-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))-22348/1962812709*(-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)*(2/3 *EllipticF(1/69*(-3795*(x+7/5)/(-2/3+x))^(1/2),1/39*I*897^(1/2))-31/15*Ell ipticPi(1/69*(-3795*(x+7/5)/(-2/3+x))^(1/2),-69/55,1/39*I*897^(1/2)))+7162 4/139035*((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*(-3795*(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))
\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{5/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {5}{2}}} \,d x } \]
integral(sqrt(5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(125*x^3 + 525*x^2 + 735*x + 343), x)
\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{5/2}} \, dx=\int \frac {\sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1}}{\left (5 x + 7\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{5/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{5/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{5/2}} \, dx=\int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}}{{\left (5\,x+7\right )}^{5/2}} \,d x \]