Integrand size = 37, antiderivative size = 469 \[ \int \frac {(7+5 x)^{3/2}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=-\frac {5 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{12 \sqrt {-5+2 x}}+\frac {5 \sqrt {\frac {143}{3}} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {65 \sqrt {\frac {11}{23}} \sqrt {7+5 x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{8 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}-\frac {895 \sqrt {\frac {11}{62}} \sqrt {2-3 x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {\frac {22}{23}} \sqrt {7+5 x}}{\sqrt {-5+2 x}}\right ),\frac {39}{62}\right )}{48 \sqrt {-\frac {2-3 x}{1+4 x}} \sqrt {1+4 x}}+\frac {23 \sqrt {\frac {31}{22}} \sqrt {\frac {2-3 x}{7+5 x}} \sqrt {\frac {5-2 x}{7+5 x}} (7+5 x) \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {\sqrt {\frac {31}{11}} \sqrt {1+4 x}}{\sqrt {7+5 x}}\right ),\frac {39}{62}\right )}{6 \sqrt {2-3 x} \sqrt {-5+2 x}}-\frac {4117 \sqrt {2-3 x} \operatorname {EllipticPi}\left (\frac {78}{55},\arctan \left (\frac {\sqrt {\frac {22}{23}} \sqrt {7+5 x}}{\sqrt {-5+2 x}}\right ),\frac {39}{62}\right )}{48 \sqrt {682} \sqrt {-\frac {2-3 x}{1+4 x}} \sqrt {1+4 x}} \]
-895/2976*(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)-4117/32736*(1/(529+506*(7+5*x)/(-5+2*x)))^(1/2)*(529+5 06*(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)*6 82^(1/2)/((-2+3*x)/(1+4*x))^(1/2)/(1+4*x)^(1/2)+23/132*(7+5*x)*EllipticPi( 1/11*341^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(1/2),55/124,1/62*2418^(1/2))*682^(1/ 2)*((2-3*x)/(7+5*x))^(1/2)*((5-2*x)/(7+5*x))^(1/2)/(2-3*x)^(1/2)/(-5+2*x)^ (1/2)-5/12*(2-3*x)^(1/2)*(1+4*x)^(1/2)*(7+5*x)^(1/2)/(-5+2*x)^(1/2)+65/184 *(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)+5/24*E llipticE(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 = 9.72 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.74 \[ \int \frac {(7+5 x)^{3/2}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\frac {\sqrt {-5+2 x} \left (6820 \sqrt {341} \sqrt {\frac {-2+3 x}{1+4 x}} \sqrt {\frac {7+5 x}{1+4 x}} \left (-5-18 x+8 x^2\right ) E\left (\arcsin \left (\sqrt {\frac {22}{39}} \sqrt {\frac {7+5 x}{1+4 x}}\right )|\frac {39}{62}\right )-6969 \sqrt {341} \sqrt {\frac {-2+3 x}{1+4 x}} \sqrt {\frac {7+5 x}{1+4 x}} \left (-5-18 x+8 x^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {22}{39}} \sqrt {\frac {7+5 x}{1+4 x}}\right ),\frac {39}{62}\right )+\sqrt {\frac {-5+2 x}{1+4 x}} \left (13640 \sqrt {2} \left (70-83 x-53 x^2+30 x^3\right )+9821 \sqrt {341} \sqrt {\frac {-2+3 x}{1+4 x}} (1+4 x)^2 \sqrt {\frac {-35-11 x+10 x^2}{(1+4 x)^2}} \operatorname {EllipticPi}\left (\frac {78}{55},\arcsin \left (\sqrt {\frac {22}{39}} \sqrt {\frac {7+5 x}{1+4 x}}\right ),\frac {39}{62}\right )\right )\right )}{16368 \sqrt {4-6 x} \left (\frac {-5+2 x}{1+4 x}\right )^{3/2} (1+4 x)^{3/2} \sqrt {7+5 x}} \]
(Sqrt[-5 + 2*x]*(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] - 6969*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) + 9821*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])))/(16368*Sqrt[4 - 6*x]*((- 5 + 2*x)/(1 + 4*x))^(3/2)*(1 + 4*x)^(3/2)*Sqrt[7 + 5*x])
Time = 0.70 (sec) , antiderivative size = 713, normalized size of antiderivative = 1.52, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.432, Rules used = {184, 183, 27, 191, 183, 27, 188, 27, 194, 27, 320, 327, 411, 320, 412, 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.
| \(\displaystyle \int \frac {(5 x+7)^{3/2}}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}} \, dx\) |
| \(\Big \downarrow \) 184 |
| \(\displaystyle \frac {31}{3} \int \frac {\sqrt {5 x+7}}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-\frac {5}{3} \int \frac {\sqrt {2-3 x} \sqrt {5 x+7}}{\sqrt {2 x-5} \sqrt {4 x+1}}dx\) |
| \(\Big \downarrow \) 183 |
| \(\displaystyle \frac {713 \sqrt {2} \sqrt {\frac {2-3 x}{5 x+7}} \sqrt {\frac {5-2 x}{5 x+7}} (5 x+7) \int \frac {11 \sqrt {2}}{\sqrt {22-\frac {39 (4 x+1)}{5 x+7}} \sqrt {11-\frac {31 (4 x+1)}{5 x+7}} \left (4-\frac {5 (4 x+1)}{5 x+7}\right )}d\frac {\sqrt {4 x+1}}{\sqrt {5 x+7}}}{33 \sqrt {2-3 x} \sqrt {2 x-5}}-\frac {5}{3} \int \frac {\sqrt {2-3 x} \sqrt {5 x+7}}{\sqrt {2 x-5} \sqrt {4 x+1}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1426 \sqrt {\frac {2-3 x}{5 x+7}} \sqrt {\frac {5-2 x}{5 x+7}} (5 x+7) \int \frac {1}{\sqrt {22-\frac {39 (4 x+1)}{5 x+7}} \sqrt {11-\frac {31 (4 x+1)}{5 x+7}} \left (4-\frac {5 (4 x+1)}{5 x+7}\right )}d\frac {\sqrt {4 x+1}}{\sqrt {5 x+7}}}{3 \sqrt {2-3 x} \sqrt {2 x-5}}-\frac {5}{3} \int \frac {\sqrt {2-3 x} \sqrt {5 x+7}}{\sqrt {2 x-5} \sqrt {4 x+1}}dx\) |
| \(\Big \downarrow \) 191 |
| \(\displaystyle \frac {1426 \sqrt {\frac {2-3 x}{5 x+7}} \sqrt {\frac {5-2 x}{5 x+7}} (5 x+7) \int \frac {1}{\sqrt {22-\frac {39 (4 x+1)}{5 x+7}} \sqrt {11-\frac {31 (4 x+1)}{5 x+7}} \left (4-\frac {5 (4 x+1)}{5 x+7}\right )}d\frac {\sqrt {4 x+1}}{\sqrt {5 x+7}}}{3 \sqrt {2-3 x} \sqrt {2 x-5}}-\frac {5}{3} \left (\frac {429}{8} \int \frac {\sqrt {2-3 x}}{(2 x-5)^{3/2} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {429}{16} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {179}{16} \int \frac {\sqrt {2 x-5}}{\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}dx+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\right )\) |
| \(\Big \downarrow \) 183 |
| \(\displaystyle \frac {1426 \sqrt {\frac {2-3 x}{5 x+7}} \sqrt {\frac {5-2 x}{5 x+7}} (5 x+7) \int \frac {1}{\sqrt {22-\frac {39 (4 x+1)}{5 x+7}} \sqrt {11-\frac {31 (4 x+1)}{5 x+7}} \left (4-\frac {5 (4 x+1)}{5 x+7}\right )}d\frac {\sqrt {4 x+1}}{\sqrt {5 x+7}}}{3 \sqrt {2-3 x} \sqrt {2 x-5}}-\frac {5}{3} \left (\frac {429}{8} \int \frac {\sqrt {2-3 x}}{(2 x-5)^{3/2} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {429}{16} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx+\frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \int \frac {\sqrt {713}}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}}{8 \sqrt {713} \sqrt {2-3 x} \sqrt {4 x+1}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1426 \sqrt {\frac {2-3 x}{5 x+7}} \sqrt {\frac {5-2 x}{5 x+7}} (5 x+7) \int \frac {1}{\sqrt {22-\frac {39 (4 x+1)}{5 x+7}} \sqrt {11-\frac {31 (4 x+1)}{5 x+7}} \left (4-\frac {5 (4 x+1)}{5 x+7}\right )}d\frac {\sqrt {4 x+1}}{\sqrt {5 x+7}}}{3 \sqrt {2-3 x} \sqrt {2 x-5}}-\frac {5}{3} \left (\frac {429}{8} \int \frac {\sqrt {2-3 x}}{(2 x-5)^{3/2} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {429}{16} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx+\frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \int \frac {1}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}}{8 \sqrt {2-3 x} \sqrt {4 x+1}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\right )\) |
| \(\Big \downarrow \) 188 |
| \(\displaystyle \frac {1426 \sqrt {\frac {2-3 x}{5 x+7}} \sqrt {\frac {5-2 x}{5 x+7}} (5 x+7) \int \frac {1}{\sqrt {22-\frac {39 (4 x+1)}{5 x+7}} \sqrt {11-\frac {31 (4 x+1)}{5 x+7}} \left (4-\frac {5 (4 x+1)}{5 x+7}\right )}d\frac {\sqrt {4 x+1}}{\sqrt {5 x+7}}}{3 \sqrt {2-3 x} \sqrt {2 x-5}}-\frac {5}{3} \left (\frac {429}{8} \int \frac {\sqrt {2-3 x}}{(2 x-5)^{3/2} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {39 \sqrt {\frac {11}{46}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \int \frac {\sqrt {46}}{\sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23}}d\frac {\sqrt {4 x+1}}{\sqrt {2-3 x}}}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}}}+\frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \int \frac {1}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}}{8 \sqrt {2-3 x} \sqrt {4 x+1}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1426 \sqrt {\frac {2-3 x}{5 x+7}} \sqrt {\frac {5-2 x}{5 x+7}} (5 x+7) \int \frac {1}{\sqrt {22-\frac {39 (4 x+1)}{5 x+7}} \sqrt {11-\frac {31 (4 x+1)}{5 x+7}} \left (4-\frac {5 (4 x+1)}{5 x+7}\right )}d\frac {\sqrt {4 x+1}}{\sqrt {5 x+7}}}{3 \sqrt {2-3 x} \sqrt {2 x-5}}-\frac {5}{3} \left (\frac {429}{8} \int \frac {\sqrt {2-3 x}}{(2 x-5)^{3/2} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {39 \sqrt {11} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \int \frac {1}{\sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23}}d\frac {\sqrt {4 x+1}}{\sqrt {2-3 x}}}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}}}+\frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \int \frac {1}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}}{8 \sqrt {2-3 x} \sqrt {4 x+1}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\right )\) |
| \(\Big \downarrow \) 194 |
| \(\displaystyle \frac {1426 \sqrt {\frac {2-3 x}{5 x+7}} \sqrt {\frac {5-2 x}{5 x+7}} (5 x+7) \int \frac {1}{\sqrt {22-\frac {39 (4 x+1)}{5 x+7}} \sqrt {11-\frac {31 (4 x+1)}{5 x+7}} \left (4-\frac {5 (4 x+1)}{5 x+7}\right )}d\frac {\sqrt {4 x+1}}{\sqrt {5 x+7}}}{3 \sqrt {2-3 x} \sqrt {2 x-5}}-\frac {5}{3} \left (-\frac {39 \sqrt {11} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \int \frac {1}{\sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23}}d\frac {\sqrt {4 x+1}}{\sqrt {2-3 x}}}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}}}-\frac {39 \sqrt {\frac {11}{23}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} \int \frac {\sqrt {23} \sqrt {\frac {4 x+1}{2 x-5}+1}}{\sqrt {23-\frac {39 (4 x+1)}{2 x-5}}}d\frac {\sqrt {4 x+1}}{\sqrt {2 x-5}}}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \int \frac {1}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}}{8 \sqrt {2-3 x} \sqrt {4 x+1}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1426 \sqrt {\frac {2-3 x}{5 x+7}} \sqrt {\frac {5-2 x}{5 x+7}} (5 x+7) \int \frac {1}{\sqrt {22-\frac {39 (4 x+1)}{5 x+7}} \sqrt {11-\frac {31 (4 x+1)}{5 x+7}} \left (4-\frac {5 (4 x+1)}{5 x+7}\right )}d\frac {\sqrt {4 x+1}}{\sqrt {5 x+7}}}{3 \sqrt {2-3 x} \sqrt {2 x-5}}-\frac {5}{3} \left (-\frac {39 \sqrt {11} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \int \frac {1}{\sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23}}d\frac {\sqrt {4 x+1}}{\sqrt {2-3 x}}}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}}}-\frac {39 \sqrt {11} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} \int \frac {\sqrt {\frac {4 x+1}{2 x-5}+1}}{\sqrt {23-\frac {39 (4 x+1)}{2 x-5}}}d\frac {\sqrt {4 x+1}}{\sqrt {2 x-5}}}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \int \frac {1}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}}{8 \sqrt {2-3 x} \sqrt {4 x+1}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\right )\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {1426 \sqrt {\frac {2-3 x}{5 x+7}} \sqrt {\frac {5-2 x}{5 x+7}} (5 x+7) \int \frac {1}{\sqrt {22-\frac {39 (4 x+1)}{5 x+7}} \sqrt {11-\frac {31 (4 x+1)}{5 x+7}} \left (4-\frac {5 (4 x+1)}{5 x+7}\right )}d\frac {\sqrt {4 x+1}}{\sqrt {5 x+7}}}{3 \sqrt {2-3 x} \sqrt {2 x-5}}-\frac {5}{3} \left (-\frac {39 \sqrt {11} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} \int \frac {\sqrt {\frac {4 x+1}{2 x-5}+1}}{\sqrt {23-\frac {39 (4 x+1)}{2 x-5}}}d\frac {\sqrt {4 x+1}}{\sqrt {2 x-5}}}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \int \frac {1}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}}{8 \sqrt {2-3 x} \sqrt {4 x+1}}-\frac {39 \sqrt {\frac {11}{23}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}} \sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {\frac {31 (4 x+1)}{2-3 x}+23}{\frac {4 x+1}{2-3 x}+2}}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\right )\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1426 \sqrt {\frac {2-3 x}{5 x+7}} \sqrt {\frac {5-2 x}{5 x+7}} (5 x+7) \int \frac {1}{\sqrt {22-\frac {39 (4 x+1)}{5 x+7}} \sqrt {11-\frac {31 (4 x+1)}{5 x+7}} \left (4-\frac {5 (4 x+1)}{5 x+7}\right )}d\frac {\sqrt {4 x+1}}{\sqrt {5 x+7}}}{3 \sqrt {2-3 x} \sqrt {2 x-5}}-\frac {5}{3} \left (\frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \int \frac {1}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}}{8 \sqrt {2-3 x} \sqrt {4 x+1}}-\frac {\sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {39 \sqrt {\frac {11}{23}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}} \sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {\frac {31 (4 x+1)}{2-3 x}+23}{\frac {4 x+1}{2-3 x}+2}}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\right )\) |
| \(\Big \downarrow \) 411 |
| \(\displaystyle \frac {1426 \sqrt {\frac {2-3 x}{5 x+7}} \sqrt {\frac {5-2 x}{5 x+7}} (5 x+7) \int \frac {1}{\sqrt {22-\frac {39 (4 x+1)}{5 x+7}} \sqrt {11-\frac {31 (4 x+1)}{5 x+7}} \left (4-\frac {5 (4 x+1)}{5 x+7}\right )}d\frac {\sqrt {4 x+1}}{\sqrt {5 x+7}}}{3 \sqrt {2-3 x} \sqrt {2 x-5}}-\frac {5}{3} \left (\frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \left (\frac {11}{78} \int \frac {1}{\sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}+\frac {1}{78} \int \frac {\sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}\right )}{8 \sqrt {2-3 x} \sqrt {4 x+1}}-\frac {\sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {39 \sqrt {\frac {11}{23}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}} \sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {\frac {31 (4 x+1)}{2-3 x}+23}{\frac {4 x+1}{2-3 x}+2}}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\right )\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {1426 \sqrt {\frac {2-3 x}{5 x+7}} \sqrt {\frac {5-2 x}{5 x+7}} (5 x+7) \int \frac {1}{\sqrt {22-\frac {39 (4 x+1)}{5 x+7}} \sqrt {11-\frac {31 (4 x+1)}{5 x+7}} \left (4-\frac {5 (4 x+1)}{5 x+7}\right )}d\frac {\sqrt {4 x+1}}{\sqrt {5 x+7}}}{3 \sqrt {2-3 x} \sqrt {2 x-5}}-\frac {5}{3} \left (\frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \left (\frac {1}{78} \int \frac {\sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}+\frac {\sqrt {\frac {11}{62}} \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {\frac {22}{23}} \sqrt {5 x+7}}{\sqrt {2 x-5}}\right ),\frac {39}{62}\right )}{78 \sqrt {\frac {\frac {11 (5 x+7)}{2 x-5}+31}{\frac {22 (5 x+7)}{2 x-5}+23}} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}\right )}{8 \sqrt {2-3 x} \sqrt {4 x+1}}-\frac {\sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {39 \sqrt {\frac {11}{23}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}} \sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {\frac {31 (4 x+1)}{2-3 x}+23}{\frac {4 x+1}{2-3 x}+2}}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\right )\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {713 \sqrt {\frac {2-3 x}{5 x+7}} \sqrt {\frac {5-2 x}{5 x+7}} (5 x+7) \operatorname {EllipticPi}\left (\frac {55}{78},\arcsin \left (\frac {\sqrt {\frac {39}{22}} \sqrt {4 x+1}}{\sqrt {5 x+7}}\right ),\frac {62}{39}\right )}{6 \sqrt {429} \sqrt {2-3 x} \sqrt {2 x-5}}-\frac {5}{3} \left (\frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \left (\frac {1}{78} \int \frac {\sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}{\left (5-\frac {2 (5 x+7)}{2 x-5}\right ) \sqrt {\frac {11 (5 x+7)}{2 x-5}+31}}d\frac {\sqrt {5 x+7}}{\sqrt {2 x-5}}+\frac {\sqrt {\frac {11}{62}} \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {\frac {22}{23}} \sqrt {5 x+7}}{\sqrt {2 x-5}}\right ),\frac {39}{62}\right )}{78 \sqrt {\frac {\frac {11 (5 x+7)}{2 x-5}+31}{\frac {22 (5 x+7)}{2 x-5}+23}} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}\right )}{8 \sqrt {2-3 x} \sqrt {4 x+1}}-\frac {\sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {39 \sqrt {\frac {11}{23}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}} \sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {\frac {31 (4 x+1)}{2-3 x}+23}{\frac {4 x+1}{2-3 x}+2}}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\right )\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {713 \sqrt {\frac {2-3 x}{5 x+7}} \sqrt {\frac {5-2 x}{5 x+7}} (5 x+7) \operatorname {EllipticPi}\left (\frac {55}{78},\arcsin \left (\frac {\sqrt {\frac {39}{22}} \sqrt {4 x+1}}{\sqrt {5 x+7}}\right ),\frac {62}{39}\right )}{6 \sqrt {429} \sqrt {2-3 x} \sqrt {2 x-5}}-\frac {5}{3} \left (-\frac {\sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {39 \sqrt {\frac {11}{23}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}} \sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {\frac {31 (4 x+1)}{2-3 x}+23}{\frac {4 x+1}{2-3 x}+2}}}+\frac {6981 \sqrt {\frac {2-3 x}{5-2 x}} (5-2 x) \sqrt {-\frac {4 x+1}{5-2 x}} \left (\frac {\sqrt {\frac {11}{62}} \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {\frac {22}{23}} \sqrt {5 x+7}}{\sqrt {2 x-5}}\right ),\frac {39}{62}\right )}{78 \sqrt {\frac {\frac {11 (5 x+7)}{2 x-5}+31}{\frac {22 (5 x+7)}{2 x-5}+23}} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}+\frac {23 \sqrt {\frac {11 (5 x+7)}{2 x-5}+31} \operatorname {EllipticPi}\left (\frac {78}{55},\arctan \left (\frac {\sqrt {\frac {22}{23}} \sqrt {5 x+7}}{\sqrt {2 x-5}}\right ),\frac {39}{62}\right )}{390 \sqrt {682} \sqrt {\frac {\frac {11 (5 x+7)}{2 x-5}+31}{\frac {22 (5 x+7)}{2 x-5}+23}} \sqrt {\frac {22 (5 x+7)}{2 x-5}+23}}\right )}{8 \sqrt {2-3 x} \sqrt {4 x+1}}+\frac {\sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{4 \sqrt {2 x-5}}\right )\) |
(713*Sqrt[(2 - 3*x)/(7 + 5*x)]*Sqrt[(5 - 2*x)/(7 + 5*x)]*(7 + 5*x)*Ellipti cPi[55/78, ArcSin[(Sqrt[39/22]*Sqrt[1 + 4*x])/Sqrt[7 + 5*x]], 62/39])/(6*S qrt[429]*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]) - (5*((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]*Sqrt[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]*Sq rt[(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[ArcTan[(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)]*EllipticPi[78/55, ArcTan[(Sqrt[22/23 ]*Sqrt[7 + 5*x])/Sqrt[-5 + 2*x]], 39/62])/(390*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
3.2.2.3.1 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_))^(3/2)/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.) *(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[b/d Int[Sqrt[a + b*x]*(Sqrt [c + d*x]/(Sqrt[e + f*x]*Sqrt[g + h*x])), x], x] - Simp[(b*c - a*d)/d 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}, 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[(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]
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[-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]
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[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]
Time = 1.60 (sec) , antiderivative size = 397, normalized size of antiderivative = 0.85
| 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 {98 \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 {140 \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 )}{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 {\frac {25 \left (x +\frac {7}{5}\right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}{2}-\frac {5 \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 )}{32292}}{\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 (107694 \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 )-238266 \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 )-143592 \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 )+317688 \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 )+47864 \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 )-105896 \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 )}{710424 \left (120 x^{4}-182 x^{3}-385 x^{2}+197 x +70\right )}\) | \(821\) |
(-(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)*(98/305877*(-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))+140/305877*(-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)*(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)))+25/2*((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*Elli pticF(1/69*(-3795*(x+7/5)/(-2/3+x))^(1/2),1/39*I*897^(1/2))-117/62*Ellipti cE(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))
\[ \int \frac {(7+5 x)^{3/2}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{\frac {3}{2}}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
integral(-(5*x + 7)^(3/2)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(24*x ^3 - 70*x^2 + 21*x + 10), x)
\[ \int \frac {(7+5 x)^{3/2}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int \frac {\left (5 x + 7\right )^{\frac {3}{2}}}{\sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1}}\, dx \]
\[ \int \frac {(7+5 x)^{3/2}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{\frac {3}{2}}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
\[ \int \frac {(7+5 x)^{3/2}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{\frac {3}{2}}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
Timed out. \[ \int \frac {(7+5 x)^{3/2}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int \frac {{\left (5\,x+7\right )}^{3/2}}{\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}} \,d x \]