3.1.95 \(\int \frac {\sqrt {2-3 x} \sqrt {7+5 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx\) [95]

3.1.95.1 Optimal result

Integrand size = 37, antiderivative size = 365 \[ \int \frac {\sqrt {2-3 x} \sqrt {7+5 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\frac {\sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{4 \sqrt {-5+2 x}}-\frac {\sqrt {429} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {39 \sqrt {\frac {11}{23}} \sqrt {7+5 x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{8 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}+\frac {179 \sqrt {\frac {11}{62}} \sqrt {2-3 x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {\frac {22}{23}} \sqrt {7+5 x}}{\sqrt {-5+2 x}}\right ),\frac {39}{62}\right )}{16 \sqrt {-\frac {2-3 x}{1+4 x}} \sqrt {1+4 x}}+\frac {4117 \sqrt {2-3 x} \operatorname {EllipticPi}\left (\frac {78}{55},\arctan \left (\frac {\sqrt {\frac {22}{23}} \sqrt {7+5 x}}{\sqrt {-5+2 x}}\right ),\frac {39}{62}\right )}{80 \sqrt {682} \sqrt {-\frac {2-3 x}{1+4 x}} \sqrt {1+4 x}} \]

output

179/992*(1/(529+506*(7+5*x)/(-5+2*x)))^(1/2)*(529+506*(7+5*x)/(-5+2*x))^(1 
/2)*EllipticF(506^(1/2)*(7+5*x)^(1/2)/(-5+2*x)^(1/2)/(529+506*(7+5*x)/(-5+ 
2*x))^(1/2),1/62*2418^(1/2))*682^(1/2)*(2-3*x)^(1/2)/((-2+3*x)/(1+4*x))^(1 
/2)/(1+4*x)^(1/2)+4117/54560*(1/(529+506*(7+5*x)/(-5+2*x)))^(1/2)*(529+506 
*(7+5*x)/(-5+2*x))^(1/2)*EllipticPi(506^(1/2)*(7+5*x)^(1/2)/(-5+2*x)^(1/2) 
/(529+506*(7+5*x)/(-5+2*x))^(1/2),78/55,1/62*2418^(1/2))*(2-3*x)^(1/2)*682 
^(1/2)/((-2+3*x)/(1+4*x))^(1/2)/(1+4*x)^(1/2)+1/4*(2-3*x)^(1/2)*(1+4*x)^(1 
/2)*(7+5*x)^(1/2)/(-5+2*x)^(1/2)-39/184*(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)-1/8*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)
 

3.1.95.2 Mathematica [A] (warning: unable to verify)

Time = 6.99 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {2-3 x} \sqrt {7+5 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=-\frac {6820 \sqrt {341} \sqrt {\frac {-2+3 x}{1+4 x}} \sqrt {\frac {7+5 x}{1+4 x}} \left (-5-18 x+8 x^2\right ) E\left (\arcsin \left (\sqrt {\frac {22}{39}} \sqrt {\frac {7+5 x}{1+4 x}}\right )|\frac {39}{62}\right )-1265 \sqrt {341} \sqrt {\frac {-2+3 x}{1+4 x}} \sqrt {\frac {7+5 x}{1+4 x}} \left (-5-18 x+8 x^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {22}{39}} \sqrt {\frac {7+5 x}{1+4 x}}\right ),\frac {39}{62}\right )+\sqrt {\frac {-5+2 x}{1+4 x}} \left (13640 \sqrt {2} \left (70-83 x-53 x^2+30 x^3\right )+4117 \sqrt {341} \sqrt {\frac {-2+3 x}{1+4 x}} (1+4 x)^2 \sqrt {\frac {-35-11 x+10 x^2}{(1+4 x)^2}} \operatorname {EllipticPi}\left (\frac {78}{55},\arcsin \left (\sqrt {\frac {22}{39}} \sqrt {\frac {7+5 x}{1+4 x}}\right ),\frac {39}{62}\right )\right )}{27280 \sqrt {2-3 x} \sqrt {-10+4 x} \sqrt {\frac {-5+2 x}{1+4 x}} \sqrt {1+4 x} \sqrt {7+5 x}} \]

input

Integrate[(Sqrt[2 - 3*x]*Sqrt[7 + 5*x])/(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]),x]
 

output

-1/27280*(6820*Sqrt[341]*Sqrt[(-2 + 3*x)/(1 + 4*x)]*Sqrt[(7 + 5*x)/(1 + 4* 
x)]*(-5 - 18*x + 8*x^2)*EllipticE[ArcSin[Sqrt[22/39]*Sqrt[(7 + 5*x)/(1 + 4 
*x)]], 39/62] - 1265*Sqrt[341]*Sqrt[(-2 + 3*x)/(1 + 4*x)]*Sqrt[(7 + 5*x)/( 
1 + 4*x)]*(-5 - 18*x + 8*x^2)*EllipticF[ArcSin[Sqrt[22/39]*Sqrt[(7 + 5*x)/ 
(1 + 4*x)]], 39/62] + Sqrt[(-5 + 2*x)/(1 + 4*x)]*(13640*Sqrt[2]*(70 - 83*x 
 - 53*x^2 + 30*x^3) + 4117*Sqrt[341]*Sqrt[(-2 + 3*x)/(1 + 4*x)]*(1 + 4*x)^ 
2*Sqrt[(-35 - 11*x + 10*x^2)/(1 + 4*x)^2]*EllipticPi[78/55, ArcSin[Sqrt[22 
/39]*Sqrt[(7 + 5*x)/(1 + 4*x)]], 39/62]))/(Sqrt[2 - 3*x]*Sqrt[-10 + 4*x]*S 
qrt[(-5 + 2*x)/(1 + 4*x)]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])
 

3.1.95.3 Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 608, normalized size of antiderivative = 1.67, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {191, 183, 27, 188, 27, 194, 27, 320, 327, 411, 320, 414}

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.

\(\displaystyle \int \frac {\sqrt {2-3 x} \sqrt {5 x+7}}{\sqrt {2 x-5} \sqrt {4 x+1}} \, dx\)

\(\Big \downarrow \) 191

\(\displaystyle \frac {429}{8} \int \frac {\sqrt {2-3 x}}{(2 x-5)^{3/2} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {429}{16} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {179}{16} \int \frac {\sqrt {2 x-5}}{\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}dx+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\)

\(\Big \downarrow \) 183

\(\displaystyle \frac {429}{8} \int \frac {\sqrt {2-3 x}}{(2 x-5)^{3/2} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {429}{16} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx+\frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \int \frac {\sqrt {713}}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}}{8 \sqrt {713} \sqrt {2-3 x} \sqrt {4 x+1}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {429}{8} \int \frac {\sqrt {2-3 x}}{(2 x-5)^{3/2} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {429}{16} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx+\frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \int \frac {1}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}}{8 \sqrt {2-3 x} \sqrt {4 x+1}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\)

\(\Big \downarrow \) 188

\(\displaystyle \frac {429}{8} \int \frac {\sqrt {2-3 x}}{(2 x-5)^{3/2} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {39 \sqrt {\frac {11}{46}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \int \frac {\sqrt {46}}{\sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23}}d\frac {\sqrt {4 x+1}}{\sqrt {2-3 x}}}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}}}+\frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \int \frac {1}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}}{8 \sqrt {2-3 x} \sqrt {4 x+1}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {429}{8} \int \frac {\sqrt {2-3 x}}{(2 x-5)^{3/2} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {39 \sqrt {11} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \int \frac {1}{\sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23}}d\frac {\sqrt {4 x+1}}{\sqrt {2-3 x}}}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}}}+\frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \int \frac {1}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}}{8 \sqrt {2-3 x} \sqrt {4 x+1}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\)

\(\Big \downarrow \) 194

\(\displaystyle -\frac {39 \sqrt {11} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \int \frac {1}{\sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23}}d\frac {\sqrt {4 x+1}}{\sqrt {2-3 x}}}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}}}-\frac {39 \sqrt {\frac {11}{23}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} \int \frac {\sqrt {23} \sqrt {\frac {4 x+1}{2 x-5}+1}}{\sqrt {23-\frac {39 (4 x+1)}{2 x-5}}}d\frac {\sqrt {4 x+1}}{\sqrt {2 x-5}}}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \int \frac {1}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}}{8 \sqrt {2-3 x} \sqrt {4 x+1}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {39 \sqrt {11} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \int \frac {1}{\sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23}}d\frac {\sqrt {4 x+1}}{\sqrt {2-3 x}}}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}}}-\frac {39 \sqrt {11} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} \int \frac {\sqrt {\frac {4 x+1}{2 x-5}+1}}{\sqrt {23-\frac {39 (4 x+1)}{2 x-5}}}d\frac {\sqrt {4 x+1}}{\sqrt {2 x-5}}}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \int \frac {1}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}}{8 \sqrt {2-3 x} \sqrt {4 x+1}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\)

\(\Big \downarrow \) 320

\(\displaystyle -\frac {39 \sqrt {11} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} \int \frac {\sqrt {\frac {4 x+1}{2 x-5}+1}}{\sqrt {23-\frac {39 (4 x+1)}{2 x-5}}}d\frac {\sqrt {4 x+1}}{\sqrt {2 x-5}}}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \int \frac {1}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}}{8 \sqrt {2-3 x} \sqrt {4 x+1}}-\frac {39 \sqrt {\frac {11}{23}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}} \sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {\frac {31 (4 x+1)}{2-3 x}+23}{\frac {4 x+1}{2-3 x}+2}}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \int \frac {1}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}}{8 \sqrt {2-3 x} \sqrt {4 x+1}}-\frac {\sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {39 \sqrt {\frac {11}{23}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}} \sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {\frac {31 (4 x+1)}{2-3 x}+23}{\frac {4 x+1}{2-3 x}+2}}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\)

\(\Big \downarrow \) 411

\(\displaystyle \frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \left (\frac {11}{78} \int \frac {1}{\sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}+\frac {1}{78} \int \frac {\sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}\right )}{8 \sqrt {2-3 x} \sqrt {4 x+1}}-\frac {\sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {39 \sqrt {\frac {11}{23}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}} \sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {\frac {31 (4 x+1)}{2-3 x}+23}{\frac {4 x+1}{2-3 x}+2}}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \left (\frac {1}{78} \int \frac {\sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}+\frac {\sqrt {\frac {11}{62}} \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {\frac {22}{23}} \sqrt {5 x+7}}{\sqrt {2 x-5}}\right ),\frac {39}{62}\right )}{78 \sqrt {\frac {\frac {11 (5 x+7)}{2 x-5}+31}{\frac {22 (5 x+7)}{2 x-5}+23}} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}\right )}{8 \sqrt {2-3 x} \sqrt {4 x+1}}-\frac {\sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {39 \sqrt {\frac {11}{23}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}} \sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {\frac {31 (4 x+1)}{2-3 x}+23}{\frac {4 x+1}{2-3 x}+2}}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\)

\(\Big \downarrow \) 414

\(\displaystyle -\frac {\sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {39 \sqrt {\frac {11}{23}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}} \sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {\frac {31 (4 x+1)}{2-3 x}+23}{\frac {4 x+1}{2-3 x}+2}}}+\frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \left (\frac {\sqrt {\frac {11}{62}} \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {\frac {22}{23}} \sqrt {5 x+7}}{\sqrt {2 x-5}}\right ),\frac {39}{62}\right )}{78 \sqrt {\frac {\frac {11 (5 x+7)}{2 x-5}+31}{\frac {22 (5 x+7)}{2 x-5}+23}} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}+\frac {23 \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \operatorname {EllipticPi}\left (\frac {78}{55},\arctan \left (\frac {\sqrt {\frac {22}{23}} \sqrt {5 x+7}}{\sqrt {2 x-5}}\right ),\frac {39}{62}\right )}{390 \sqrt {682} \sqrt {\frac {\frac {11 (5 x+7)}{2 x-5}+31}{\frac {22 (5 x+7)}{2 x-5}+23}} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}\right )}{8 \sqrt {2-3 x} \sqrt {4 x+1}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\)

input

Int[(Sqrt[2 - 3*x]*Sqrt[7 + 5*x])/(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]),x]
 

output

(Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(4*Sqrt[-5 + 2*x]) - (Sqrt[429 
]*Sqrt[2 - 3*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[39/23]*Sq 
rt[1 + 4*x])/Sqrt[-5 + 2*x]], -23/39])/(8*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 
 + 5*x]) - (39*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])/(8*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))]) + (6981*Sqrt[(2 - 3*x)/(5 - 2*x)]*(5 - 2*x)*Sqrt[-((1 + 4*x)/( 
5 - 2*x))]*((Sqrt[11/62]*Sqrt[31 + (11*(7 + 5*x))/(-5 + 2*x)]*EllipticF[Ar 
cTan[(Sqrt[22/23]*Sqrt[7 + 5*x])/Sqrt[-5 + 2*x]], 39/62])/(78*Sqrt[(31 + ( 
11*(7 + 5*x))/(-5 + 2*x))/(23 + (22*(7 + 5*x))/(-5 + 2*x))]*Sqrt[23 + (22* 
(7 + 5*x))/(-5 + 2*x)]) + (23*Sqrt[31 + (11*(7 + 5*x))/(-5 + 2*x)]*Ellipti 
cPi[78/55, ArcTan[(Sqrt[22/23]*Sqrt[7 + 5*x])/Sqrt[-5 + 2*x]], 39/62])/(39 
0*Sqrt[682]*Sqrt[(31 + (11*(7 + 5*x))/(-5 + 2*x))/(23 + (22*(7 + 5*x))/(-5 
 + 2*x))]*Sqrt[23 + (22*(7 + 5*x))/(-5 + 2*x)])))/(8*Sqrt[2 - 3*x]*Sqrt[1 
+ 4*x])
 

3.1.95.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 183

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]
 

rule 188

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]
 

rule 191

Int[(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)])/(Sqrt[(e_.) + (f_.) 
*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[Sqrt[a + b*x]*Sqrt[c + d*x]*( 
Sqrt[g + h*x]/(h*Sqrt[e + f*x])), x] + (-Simp[(d*e - c*f)*((f*g - e*h)/(2*f 
*h))   Int[Sqrt[a + b*x]/(Sqrt[c + d*x]*(e + f*x)^(3/2)*Sqrt[g + h*x]), x], 
 x] + Simp[(a*d*f*h - b*(d*f*g + d*e*h - c*f*h))/(2*f^2*h)   Int[Sqrt[e + f 
*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[g + h*x]), x], x] + Simp[(d*e - c*f)* 
((b*f*g + b*e*h - 2*a*f*h)/(2*f^2*h))   Int[1/(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}, x]
 

rule 194

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]
 

rule 320

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]
 

rule 327

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]
 

rule 411

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[-f/(b*e - a*f)   Int[1/(Sqrt[c + d*x^2]*Sqrt[e + 
f*x^2]), x], x] + Simp[b/(b*e - a*f)   Int[Sqrt[e + f*x^2]/((a + b*x^2)*Sqr 
t[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[d/c, 0] && GtQ 
[f/e, 0] &&  !SimplerSqrtQ[d/c, f/e]
 

rule 414

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_) 
^2]), x_Symbol] :> Simp[c*(Sqrt[e + f*x^2]/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]* 
Sqrt[c*((e + f*x^2)/(e*(c + d*x^2)))]))*EllipticPi[1 - b*(c/(a*d)), ArcTan[ 
Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ 
[d/c]
 

3.1.95.4 Maple [A] (verified)

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}[phone] x^{2}[phone] x [phone]\right )}{1184040 \left (120 x^{4}-182 x^{3}-385 x^{2}+197 x +70\right )}\)	\(821\)

input

int((7+5*x)^(1/2)*(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x,method=_RET 
URNVERBOSE)
 

output

(-(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)*(28/305877*(-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))-2/27807*(-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*EllipticPi(1/69*(- 
3795*(x+7/5)/(-2/3+x))^(1/2),-69/55,1/39*I*897^(1/2)))-15/2*((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*Ellipti 
cF(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(1/6 
9*(-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))
 

3.1.95.5 Fricas [F]

\[ \int \frac {\sqrt {2-3 x} \sqrt {7+5 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {\sqrt {5 \, x + 7} \sqrt {-3 \, x + 2}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]

input

integrate((7+5*x)^(1/2)*(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algo 
rithm="fricas")
 

output

integral(sqrt(5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(8*x^2 - 
 18*x - 5), x)
 

3.1.95.6 Sympy [F]

\[ \int \frac {\sqrt {2-3 x} \sqrt {7+5 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int \frac {\sqrt {2 - 3 x} \sqrt {5 x + 7}}{\sqrt {2 x - 5} \sqrt {4 x + 1}}\, dx \]

input

integrate((7+5*x)**(1/2)*(2-3*x)**(1/2)/(-5+2*x)**(1/2)/(1+4*x)**(1/2),x)
 

output

Integral(sqrt(2 - 3*x)*sqrt(5*x + 7)/(sqrt(2*x - 5)*sqrt(4*x + 1)), x)
 

3.1.95.7 Maxima [F]

\[ \int \frac {\sqrt {2-3 x} \sqrt {7+5 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {\sqrt {5 \, x + 7} \sqrt {-3 \, x + 2}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]

input

integrate((7+5*x)^(1/2)*(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algo 
rithm="maxima")
 

output

integrate(sqrt(5*x + 7)*sqrt(-3*x + 2)/(sqrt(4*x + 1)*sqrt(2*x - 5)), x)
 

3.1.95.8 Giac [F]

\[ \int \frac {\sqrt {2-3 x} \sqrt {7+5 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {\sqrt {5 \, x + 7} \sqrt {-3 \, x + 2}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]

input

integrate((7+5*x)^(1/2)*(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algo 
rithm="giac")
 

output

integrate(sqrt(5*x + 7)*sqrt(-3*x + 2)/(sqrt(4*x + 1)*sqrt(2*x - 5)), x)
 

3.1.95.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {7+5 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int \frac {\sqrt {2-3\,x}\,\sqrt {5\,x+7}}{\sqrt {4\,x+1}\,\sqrt {2\,x-5}} \,d x \]

input

int(((2 - 3*x)^(1/2)*(5*x + 7)^(1/2))/((4*x + 1)^(1/2)*(2*x - 5)^(1/2)),x)
 

output

int(((2 - 3*x)^(1/2)*(5*x + 7)^(1/2))/((4*x + 1)^(1/2)*(2*x - 5)^(1/2)), x 
)