Integrand size = 37, antiderivative size = 391 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^{3/2}}{\sqrt {-5+2 x}} \, dx=\frac {66377 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{1920 \sqrt {-5+2 x}}+\frac {977}{288} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}+\frac {1}{6} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}-\frac {66377 \sqrt {\frac {143}{3}} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{1280 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {2824441 \sqrt {\frac {11}{23}} \sqrt {7+5 x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{17280 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}+\frac {963142751 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{86400 \sqrt {429} \sqrt {-5+2 x} \sqrt {1+4 x}} \]
1/6*(7+5*x)^(3/2)*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)+963142751/370 65600*(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)+66377/1920*(2-3*x)^(1/2)*(1+4*x)^(1/2)*(7 +5*x)^(1/2)/(-5+2*x)^(1/2)+977/288*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1 /2)*(7+5*x)^(1/2)+2824441/397440*(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)-66377/3840*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 = 36.53 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^{3/2}}{\sqrt {-5+2 x}} \, dx=-\frac {\sqrt {-5+2 x} \sqrt {1+4 x} \left (-37038366 \sqrt {682} \sqrt {\frac {-5-18 x+8 x^2}{(2-3 x)^2}} \left (-14+11 x+15 x^2\right ) E\left (\arcsin \left (\sqrt {\frac {31}{39}} \sqrt {\frac {-5+2 x}{-2+3 x}}\right )|\frac {39}{62}\right )+31389484 \sqrt {682} \sqrt {\frac {-5-18 x+8 x^2}{(2-3 x)^2}} \left (-14+11 x+15 x^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {31}{39}} \sqrt {\frac {-5+2 x}{-2+3 x}}\right ),\frac {39}{62}\right )+\sqrt {\frac {7+5 x}{-2+3 x}} \left (186 \left (-17232355-79187903 x-38640362 x^2+10641080 x^3+4555200 x^4+1152000 x^5\right )+31069121 \sqrt {682} (2-3 x)^2 \sqrt {\frac {1+4 x}{-2+3 x}} \sqrt {\frac {-35-11 x+10 x^2}{(2-3 x)^2}} \operatorname {EllipticPi}\left (\frac {117}{62},\arcsin \left (\sqrt {\frac {31}{39}} \sqrt {\frac {-5+2 x}{-2+3 x}}\right ),\frac {39}{62}\right )\right )\right )}{2142720 \sqrt {2-3 x} \sqrt {7+5 x} \sqrt {\frac {7+5 x}{-2+3 x}} \left (-5-18 x+8 x^2\right )} \]
-1/2142720*(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(-37038366*Sqrt[682]*Sqrt[(-5 - 1 8*x + 8*x^2)/(2 - 3*x)^2]*(-14 + 11*x + 15*x^2)*EllipticE[ArcSin[Sqrt[31/3 9]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62] + 31389484*Sqrt[682]*Sqrt[(-5 - 18 *x + 8*x^2)/(2 - 3*x)^2]*(-14 + 11*x + 15*x^2)*EllipticF[ArcSin[Sqrt[31/39 ]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62] + Sqrt[(7 + 5*x)/(-2 + 3*x)]*(186*( -17232355 - 79187903*x - 38640362*x^2 + 10641080*x^3 + 4555200*x^4 + 11520 00*x^5) + 31069121*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]*Sq rt[(-5 + 2*x)/(-2 + 3*x)]], 39/62])))/(Sqrt[2 - 3*x]*Sqrt[7 + 5*x]*Sqrt[(7 + 5*x)/(-2 + 3*x)]*(-5 - 18*x + 8*x^2))
Time = 0.88 (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 = {180, 25, 2103, 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.
\(\displaystyle \int \frac {\sqrt {2-3 x} \sqrt {4 x+1} (5 x+7)^{3/2}}{\sqrt {2 x-5}} \, dx\) |
\(\Big \downarrow \) 180 |
\(\displaystyle \frac {1}{6} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^{3/2}-\frac {1}{12} \int -\frac {\sqrt {5 x+7} \left (-1954 x^2-20 x+465\right )}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{12} \int \frac {\sqrt {5 x+7} \left (-1954 x^2-20 x+465\right )}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {1}{6} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^{3/2}\) |
\(\Big \downarrow \) 2103 |
\(\displaystyle \frac {1}{12} \left (\frac {977}{24} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}-\frac {1}{96} \int -\frac {2 \left (-1194786 x^2-647410 x+348709\right )}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx\right )+\frac {1}{6} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^{3/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{48} \int \frac {-1194786 x^2-647410 x+348709}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx+\frac {977}{24} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\right )+\frac {1}{6} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^{3/2}\) |
\(\Big \downarrow \) 2105 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{48} \left (\frac {85427199}{20} \int \frac {\sqrt {2-3 x}}{(2 x-5)^{3/2} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {1}{240} \int -\frac {12 (31069031-31069121 x)}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx+\frac {199131 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{10 \sqrt {2 x-5}}\right )+\frac {977}{24} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\right )+\frac {1}{6} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^{3/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{48} \left (\frac {85427199}{20} \int \frac {\sqrt {2-3 x}}{(2 x-5)^{3/2} \sqrt {4 x+1} \sqrt {5 x+7}}dx+\frac {1}{20} \int \frac {31069031-31069121 x}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx+\frac {199131 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{10 \sqrt {2 x-5}}\right )+\frac {977}{24} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\right )+\frac {1}{6} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^{3/2}\) |
\(\Big \downarrow \) 194 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{48} \left (\frac {1}{20} \int \frac {31069031-31069121 x}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {7766109 \sqrt {\frac {11}{23}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} \int \frac {\sqrt {23} \sqrt {\frac {4 x+1}{2 x-5}+1}}{\sqrt {23-\frac {39 (4 x+1)}{2 x-5}}}d\frac {\sqrt {4 x+1}}{\sqrt {2 x-5}}}{20 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {199131 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{10 \sqrt {2 x-5}}\right )+\frac {977}{24} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\right )+\frac {1}{6} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^{3/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{48} \left (\frac {1}{20} \int \frac {31069031-31069121 x}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {7766109 \sqrt {11} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} \int \frac {\sqrt {\frac {4 x+1}{2 x-5}+1}}{\sqrt {23-\frac {39 (4 x+1)}{2 x-5}}}d\frac {\sqrt {4 x+1}}{\sqrt {2 x-5}}}{20 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {199131 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{10 \sqrt {2 x-5}}\right )+\frac {977}{24} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\right )+\frac {1}{6} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^{3/2}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{48} \left (\frac {1}{20} \int \frac {31069031-31069121 x}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx-\frac {199131 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{20 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {199131 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{10 \sqrt {2 x-5}}\right )+\frac {977}{24} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\right )+\frac {1}{6} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^{3/2}\) |
\(\Big \downarrow \) 2101 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{48} \left (\frac {1}{20} \left (\frac {31068851}{3} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx+\frac {31069121}{3} \int \frac {\sqrt {2-3 x}}{\sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx\right )-\frac {199131 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{20 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {199131 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{10 \sqrt {2 x-5}}\right )+\frac {977}{24} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\right )+\frac {1}{6} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^{3/2}\) |
\(\Big \downarrow \) 183 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{48} \left (\frac {1}{20} \left (\frac {31068851}{3} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx+\frac {1926285502 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \int \frac {\sqrt {897}}{\sqrt {23-\frac {11 (5 x+7)}{2-3 x}} \left (\frac {3 (5 x+7)}{2-3 x}+5\right ) \sqrt {\frac {11 (5 x+7)}{2-3 x}+39}}d\frac {\sqrt {5 x+7}}{\sqrt {2-3 x}}}{3 \sqrt {897} \sqrt {2 x-5} \sqrt {4 x+1}}\right )-\frac {199131 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{20 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {199131 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{10 \sqrt {2 x-5}}\right )+\frac {977}{24} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\right )+\frac {1}{6} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^{3/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{48} \left (\frac {1}{20} \left (\frac {31068851}{3} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}}dx+\frac {1926285502 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \int \frac {1}{\sqrt {23-\frac {11 (5 x+7)}{2-3 x}} \left (\frac {3 (5 x+7)}{2-3 x}+5\right ) \sqrt {\frac {11 (5 x+7)}{2-3 x}+39}}d\frac {\sqrt {5 x+7}}{\sqrt {2-3 x}}}{3 \sqrt {2 x-5} \sqrt {4 x+1}}\right )-\frac {199131 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{20 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {199131 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{10 \sqrt {2 x-5}}\right )+\frac {977}{24} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\right )+\frac {1}{6} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^{3/2}\) |
\(\Big \downarrow \) 188 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{48} \left (\frac {1}{20} \left (\frac {2824441 \sqrt {\frac {22}{23}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \int \frac {\sqrt {46}}{\sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23}}d\frac {\sqrt {4 x+1}}{\sqrt {2-3 x}}}{3 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}}}+\frac {1926285502 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \int \frac {1}{\sqrt {23-\frac {11 (5 x+7)}{2-3 x}} \left (\frac {3 (5 x+7)}{2-3 x}+5\right ) \sqrt {\frac {11 (5 x+7)}{2-3 x}+39}}d\frac {\sqrt {5 x+7}}{\sqrt {2-3 x}}}{3 \sqrt {2 x-5} \sqrt {4 x+1}}\right )-\frac {199131 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{20 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {199131 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{10 \sqrt {2 x-5}}\right )+\frac {977}{24} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\right )+\frac {1}{6} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^{3/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{48} \left (\frac {1}{20} \left (\frac {5648882 \sqrt {11} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \int \frac {1}{\sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23}}d\frac {\sqrt {4 x+1}}{\sqrt {2-3 x}}}{3 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}}}+\frac {1926285502 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \int \frac {1}{\sqrt {23-\frac {11 (5 x+7)}{2-3 x}} \left (\frac {3 (5 x+7)}{2-3 x}+5\right ) \sqrt {\frac {11 (5 x+7)}{2-3 x}+39}}d\frac {\sqrt {5 x+7}}{\sqrt {2-3 x}}}{3 \sqrt {2 x-5} \sqrt {4 x+1}}\right )-\frac {199131 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{20 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {199131 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{10 \sqrt {2 x-5}}\right )+\frac {977}{24} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\right )+\frac {1}{6} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^{3/2}\) |
\(\Big \downarrow \) 320 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{48} \left (\frac {1}{20} \left (\frac {1926285502 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \int \frac {1}{\sqrt {23-\frac {11 (5 x+7)}{2-3 x}} \left (\frac {3 (5 x+7)}{2-3 x}+5\right ) \sqrt {\frac {11 (5 x+7)}{2-3 x}+39}}d\frac {\sqrt {5 x+7}}{\sqrt {2-3 x}}}{3 \sqrt {2 x-5} \sqrt {4 x+1}}+\frac {5648882 \sqrt {\frac {11}{23}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{3 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}} \sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {\frac {31 (4 x+1)}{2-3 x}+23}{\frac {4 x+1}{2-3 x}+2}}}\right )-\frac {199131 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{20 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {199131 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{10 \sqrt {2 x-5}}\right )+\frac {977}{24} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\right )+\frac {1}{6} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^{3/2}\) |
\(\Big \downarrow \) 412 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{48} \left (\frac {1}{20} \left (\frac {1926285502 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {5 x+7}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{15 \sqrt {429} \sqrt {2 x-5} \sqrt {4 x+1}}+\frac {5648882 \sqrt {\frac {11}{23}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{3 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}} \sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {\frac {31 (4 x+1)}{2-3 x}+23}{\frac {4 x+1}{2-3 x}+2}}}\right )-\frac {199131 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{20 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {199131 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{10 \sqrt {2 x-5}}\right )+\frac {977}{24} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}\right )+\frac {1}{6} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^{3/2}\) |
(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^(3/2))/6 + ((977*Sqr t[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/24 + ((199131*Sqrt[ 2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(10*Sqrt[-5 + 2*x]) - (199131*Sqrt[4 29]*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])/(20*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqr t[7 + 5*x]) + ((5648882*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])/(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))]) + (1926285502*(2 - 3*x)*Sqrt[(5 - 2*x)/(2 - 3*x)]*Sqr t[-((1 + 4*x)/(2 - 3*x))]*EllipticPi[-69/55, ArcSin[(Sqrt[11/23]*Sqrt[7 + 5*x])/Sqrt[2 - 3*x]], -23/39])/(15*Sqrt[429]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]) )/20)/48)/12
3.1.86.3.1 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[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)* (x_)])/Sqrt[(c_.) + (d_.)*(x_)], x_] :> Simp[2*(a + b*x)^m*Sqrt[c + d*x]*Sq rt[e + f*x]*(Sqrt[g + h*x]/(d*(2*m + 3))), x] - Simp[1/(d*(2*m + 3)) Int[ ((a + b*x)^(m - 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[2*b*c* e*g*m + a*(c*(f*g + e*h) - 2*d*e*g*(m + 1)) - (b*(2*d*e*g - c*(f*g + e*h)*( 2*m + 1)) - a*(2*c*f*h - d*(2*m + 1)*(f*g + e*h)))*x - (2*a*d*f*h*m + b*(d* (f*g + e*h) - 2*c*f*h*(m + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IntegerQ[2*m] && GtQ[m, 0]
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_))^(m_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2))/(Sqrt[ (c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_S ymbol] :> Simp[2*C*(a + b*x)^m*Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/( d*f*h*(2*m + 3))), x] + Simp[1/(d*f*h*(2*m + 3)) Int[((a + b*x)^(m - 1)/( Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[a*A*d*f*h*(2*m + 3) - C*(a *(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*m) + ((A*b + a*B)*d*f*h*(2*m + 3) - C* (2*a*(d*f*g + d*e*h + c*f*h) + b*(2*m + 1)*(d*e*g + c*f*g + c*e*h)))*x + (b *B*d*f*h*(2*m + 3) + 2*C*(a*d*f*h*m - b*(m + 1)*(d*f*g + d*e*h + c*f*h)))*x ^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x] && IntegerQ[2 *m] && GtQ[m, 0]
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 = 1.72 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.14
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 {5 x \sqrt {-120 x^{4}+182 x^{3}+385 x^{2}-197 x -70}}{6}+\frac {1313 \sqrt {-120 x^{4}+182 x^{3}+385 x^{2}-197 x -70}}{288}+\frac {348709 \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 )}{88092576 \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 {323705 \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 )}{44046288 \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 {66377 \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 )}{64 \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}}\) | \(446\) |
risch | \(-\frac {\left (1313+240 x \right ) \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 )}}{288 \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 {348709 \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 )}{88092576 \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 {323705 \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 )}{44046288 \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 {66377 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right )}{64}-\frac {66377 \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 )}{9285120}}{\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}}\) | \(497\) |
default | \(-\frac {\sqrt {7+5 x}\, \sqrt {2-3 x}\, \sqrt {1+4 x}\, \sqrt {-5+2 x}\, \left (2796196590 \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 )+17336569518 \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 )-11532671865 \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 )-3728262120 \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 )-23115426024 \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 )+15376895820 \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 )+1242754040 \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 )+7705142008 \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 )-5125631940 \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 )-170501760000 x^{5}-674192376000 x^{4}-1574933045400 x^{3}+5718966777810 x^{2}+11720205583515 x +2550474701775\right )}{1705017600 \left (120 x^{4}-182 x^{3}-385 x^{2}+197 x +70\right )}\) | \(831\) |
(-(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)*(5/6*x*(-120*x^4+182*x^3+385*x^2-197*x-70)^(1/2) +1313/288*(-120*x^4+182*x^3+385*x^2-197*x-70)^(1/2)+348709/88092576*(-3795 *(x+7/5)/(-2/3+x))^(1/2)*(-2/3+x)^2*806^(1/2)*((x-5/2)/(-2/3+x))^(1/2)*213 9^(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))-323705 /44046288*(-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),1/3 9*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)))-66377/64*((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))
\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^{3/2}}{\sqrt {-5+2 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{\frac {3}{2}} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}} \,d x } \]
\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^{3/2}}{\sqrt {-5+2 x}} \, dx=\int \frac {\sqrt {2 - 3 x} \sqrt {4 x + 1} \left (5 x + 7\right )^{\frac {3}{2}}}{\sqrt {2 x - 5}}\, dx \]
\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^{3/2}}{\sqrt {-5+2 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{\frac {3}{2}} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}} \,d x } \]
\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^{3/2}}{\sqrt {-5+2 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{\frac {3}{2}} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^{3/2}}{\sqrt {-5+2 x}} \, dx=\int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,{\left (5\,x+7\right )}^{3/2}}{\sqrt {2\,x-5}} \,d x \]