Integrand size = 25, antiderivative size = 253 \[ \int \frac {1}{x \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=-\frac {57+94 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )}{10 \left (1+2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )^2\right )^2}-\frac {31+330 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )}{20 \left (1+2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )^2\right )}-\frac {65}{8} \arctan \left (\frac {1}{2} \left (1+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )\right )+\frac {1}{8} \log \left (7+3 x-20 x^2-\sqrt {-3-2 x+4 x^2}-10 x \sqrt {-3-2 x+4 x^2}\right )-\frac {1}{8} \log \left (x-4 x^2-2 x \sqrt {-3-2 x+4 x^2}\right ) \] Output:
-1/10*(57+188*x+94*(4*x^2-2*x-3)^(1/2))/(1+4*x+2*(4*x^2-2*x-3)^(1/2)+5*(2* x+(4*x^2-2*x-3)^(1/2))^2)^2-(31+660*x+330*(4*x^2-2*x-3)^(1/2))/(20+80*x+40 *(4*x^2-2*x-3)^(1/2)+100*(2*x+(4*x^2-2*x-3)^(1/2))^2)-65/8*arctan(1/2+5*x+ 5/2*(4*x^2-2*x-3)^(1/2))+1/8*ln(7+3*x-20*x^2-(4*x^2-2*x-3)^(1/2)-10*x*(4*x ^2-2*x-3)^(1/2))-1/8*ln(x-4*x^2-2*x*(4*x^2-2*x-3)^(1/2))
Time = 0.53 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.62 \[ \int \frac {1}{x \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {1}{8} \left (\frac {\sqrt {-3-2 x+4 x^2} \left (138+398 x+455 x^2+200 x^3\right )}{\left (4+8 x+5 x^2\right )^2}+\frac {210+612 x+722 x^2+305 x^3}{\left (4+8 x+5 x^2\right )^2}+65 \arctan \left (\frac {3}{2}+x-\frac {1}{2} \sqrt {-3-2 x+4 x^2}\right )+2 \text {arctanh}\left (\frac {-5-6 x+3 \sqrt {-3-2 x+4 x^2}}{-5-4 x-8 x^2+(3+4 x) \sqrt {-3-2 x+4 x^2}}\right )\right ) \] Input:
Integrate[1/(x*(1 + 3*x + Sqrt[-3 - 2*x + 4*x^2])^3),x]
Output:
((Sqrt[-3 - 2*x + 4*x^2]*(138 + 398*x + 455*x^2 + 200*x^3))/(4 + 8*x + 5*x ^2)^2 + (210 + 612*x + 722*x^2 + 305*x^3)/(4 + 8*x + 5*x^2)^2 + 65*ArcTan[ 3/2 + x - Sqrt[-3 - 2*x + 4*x^2]/2] + 2*ArcTanh[(-5 - 6*x + 3*Sqrt[-3 - 2* x + 4*x^2])/(-5 - 4*x - 8*x^2 + (3 + 4*x)*Sqrt[-3 - 2*x + 4*x^2])])/8
Time = 0.76 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.13, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {7293, 2009}
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.
| \(\displaystyle \int \frac {1}{x \left (\sqrt {4 x^2-2 x-3}+3 x+1\right )^3} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {-349 x-292}{5 \left (5 x^2+8 x+4\right )^3}+\frac {5 x+8}{8 \left (5 x^2+8 x+4\right )}+\frac {25 x+166}{10 \left (5 x^2+8 x+4\right )^2}-\frac {31 x \sqrt {4 x^2-2 x-3}}{\left (5 x^2+8 x+4\right )^3}-\frac {16 \sqrt {4 x^2-2 x-3}}{\left (5 x^2+8 x+4\right )^3}-\frac {1}{8 x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {65}{16} \arctan \left (\frac {7 x+8}{2 \sqrt {4 x^2-2 x-3}}\right )+\frac {65}{16} \arctan \left (\frac {5 x}{2}+2\right )+\frac {1}{8} \text {arctanh}\left (\frac {3 x+1}{\sqrt {4 x^2-2 x-3}}\right )+\frac {57-16 x}{20 \left (5 x^2+8 x+4\right )^2}-\frac {3 (5 x+4)}{10 \left (5 x^2+8 x+4\right )}+\frac {365 x+282}{40 \left (5 x^2+8 x+4\right )}+\frac {31 (965 x+701) \sqrt {4 x^2-2 x-3}}{2312 \left (5 x^2+8 x+4\right )}-\frac {(18355 x+13928) \sqrt {4 x^2-2 x-3}}{2312 \left (5 x^2+8 x+4\right )}+\frac {31 (x+1) \sqrt {4 x^2-2 x-3}}{4 \left (5 x^2+8 x+4\right )^2}-\frac {(5 x+4) \sqrt {4 x^2-2 x-3}}{\left (5 x^2+8 x+4\right )^2}+\frac {1}{16} \log \left (5 x^2+8 x+4\right )-\frac {\log (x)}{8}\) |
Input:
Int[1/(x*(1 + 3*x + Sqrt[-3 - 2*x + 4*x^2])^3),x]
Output:
(57 - 16*x)/(20*(4 + 8*x + 5*x^2)^2) + (31*(1 + x)*Sqrt[-3 - 2*x + 4*x^2]) /(4*(4 + 8*x + 5*x^2)^2) - ((4 + 5*x)*Sqrt[-3 - 2*x + 4*x^2])/(4 + 8*x + 5 *x^2)^2 - (3*(4 + 5*x))/(10*(4 + 8*x + 5*x^2)) + (282 + 365*x)/(40*(4 + 8* x + 5*x^2)) + (31*(701 + 965*x)*Sqrt[-3 - 2*x + 4*x^2])/(2312*(4 + 8*x + 5 *x^2)) - ((13928 + 18355*x)*Sqrt[-3 - 2*x + 4*x^2])/(2312*(4 + 8*x + 5*x^2 )) + (65*ArcTan[2 + (5*x)/2])/16 + (65*ArcTan[(8 + 7*x)/(2*Sqrt[-3 - 2*x + 4*x^2])])/16 + ArcTanh[(1 + 3*x)/Sqrt[-3 - 2*x + 4*x^2]]/8 - Log[x]/8 + L og[4 + 8*x + 5*x^2]/16
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.55 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.35
| method | result | size |
| trager | \(-\frac {\left (x -1\right ) \left (46225 x^{3}+106000 x^{2}+89638 x +31106\right )}{2312 \left (5 x^{2}+8 x +4\right )^{2}}+\frac {\left (200 x^{3}+455 x^{2}+398 x +138\right ) \sqrt {4 x^{2}-2 x -3}}{8 \left (5 x^{2}+8 x +4\right )^{2}}+\frac {289 \operatorname {RootOf}\left (334084 \textit {\_Z}^{2}-2312 \textit {\_Z} +4229\right ) \ln \left (\frac {1+3 x +\sqrt {4 x^{2}-2 x -3}}{x}\right )}{8}+\frac {\ln \left (-\frac {-578299404 \operatorname {RootOf}\left (334084 \textit {\_Z}^{2}-2312 \textit {\_Z} +4229\right )^{2} x +135147960 \sqrt {4 x^{2}-2 x -3}\, \operatorname {RootOf}\left (334084 \textit {\_Z}^{2}-2312 \textit {\_Z} +4229\right )+578299404 \operatorname {RootOf}\left (334084 \textit {\_Z}^{2}-2312 \textit {\_Z} +4229\right )^{2}+368591756 \operatorname {RootOf}\left (334084 \textit {\_Z}^{2}-2312 \textit {\_Z} +4229\right ) x +6980645 \sqrt {4 x^{2}-2 x -3}+161682784 \operatorname {RootOf}\left (334084 \textit {\_Z}^{2}-2312 \textit {\_Z} +4229\right )-4010960 x -1861440}{x}\right )}{8}-\frac {289 \ln \left (-\frac {-578299404 \operatorname {RootOf}\left (334084 \textit {\_Z}^{2}-2312 \textit {\_Z} +4229\right )^{2} x +135147960 \sqrt {4 x^{2}-2 x -3}\, \operatorname {RootOf}\left (334084 \textit {\_Z}^{2}-2312 \textit {\_Z} +4229\right )+578299404 \operatorname {RootOf}\left (334084 \textit {\_Z}^{2}-2312 \textit {\_Z} +4229\right )^{2}+368591756 \operatorname {RootOf}\left (334084 \textit {\_Z}^{2}-2312 \textit {\_Z} +4229\right ) x +6980645 \sqrt {4 x^{2}-2 x -3}+161682784 \operatorname {RootOf}\left (334084 \textit {\_Z}^{2}-2312 \textit {\_Z} +4229\right )-4010960 x -1861440}{x}\right ) \operatorname {RootOf}\left (334084 \textit {\_Z}^{2}-2312 \textit {\_Z} +4229\right )}{8}\) | \(341\) |
| default | \(\text {Expression too large to display}\) | \(8172\) |
Input:
int(1/x/(1+3*x+(4*x^2-2*x-3)^(1/2))^3,x,method=_RETURNVERBOSE)
Output:
-1/2312*(x-1)*(46225*x^3+106000*x^2+89638*x+31106)/(5*x^2+8*x+4)^2+1/8*(20 0*x^3+455*x^2+398*x+138)/(5*x^2+8*x+4)^2*(4*x^2-2*x-3)^(1/2)+289/8*RootOf( 334084*_Z^2-2312*_Z+4229)*ln((1+3*x+(4*x^2-2*x-3)^(1/2))/x)+1/8*ln(-(-5782 99404*RootOf(334084*_Z^2-2312*_Z+4229)^2*x+135147960*(4*x^2-2*x-3)^(1/2)*R ootOf(334084*_Z^2-2312*_Z+4229)+578299404*RootOf(334084*_Z^2-2312*_Z+4229) ^2+368591756*RootOf(334084*_Z^2-2312*_Z+4229)*x+6980645*(4*x^2-2*x-3)^(1/2 )+161682784*RootOf(334084*_Z^2-2312*_Z+4229)-4010960*x-1861440)/x)-289/8*l n(-(-578299404*RootOf(334084*_Z^2-2312*_Z+4229)^2*x+135147960*(4*x^2-2*x-3 )^(1/2)*RootOf(334084*_Z^2-2312*_Z+4229)+578299404*RootOf(334084*_Z^2-2312 *_Z+4229)^2+368591756*RootOf(334084*_Z^2-2312*_Z+4229)*x+6980645*(4*x^2-2* x-3)^(1/2)+161682784*RootOf(334084*_Z^2-2312*_Z+4229)-4010960*x-1861440)/x )*RootOf(334084*_Z^2-2312*_Z+4229)
Time = 0.08 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.36 \[ \int \frac {1}{x \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {800 \, x^{4} + 3170 \, x^{3} + 4772 \, x^{2} + 65 \, {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )} \arctan \left (\frac {5}{2} \, x + 2\right ) - 65 \, {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )} \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) - 65 \, {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )} \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) - {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )} \log \left (20 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (10 \, x + 1\right )} - 3 \, x - 7\right ) + {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )} \log \left (5 \, x^{2} + 8 \, x + 4\right ) + {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )} \log \left (4 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (2 \, x + 3\right )} + 5 \, x + 5\right ) - 2 \, {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )} \log \left (x\right ) + 2 \, {\left (200 \, x^{3} + 455 \, x^{2} + 398 \, x + 138\right )} \sqrt {4 \, x^{2} - 2 \, x - 3} + 3272 \, x + 932}{16 \, {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )}} \] Input:
integrate(1/x/(1+3*x+(4*x^2-2*x-3)^(1/2))^3,x, algorithm="fricas")
Output:
1/16*(800*x^4 + 3170*x^3 + 4772*x^2 + 65*(25*x^4 + 80*x^3 + 104*x^2 + 64*x + 16)*arctan(5/2*x + 2) - 65*(25*x^4 + 80*x^3 + 104*x^2 + 64*x + 16)*arct an(-x + 1/2*sqrt(4*x^2 - 2*x - 3) - 3/2) - 65*(25*x^4 + 80*x^3 + 104*x^2 + 64*x + 16)*arctan(-5*x + 5/2*sqrt(4*x^2 - 2*x - 3) - 1/2) - (25*x^4 + 80* x^3 + 104*x^2 + 64*x + 16)*log(20*x^2 - sqrt(4*x^2 - 2*x - 3)*(10*x + 1) - 3*x - 7) + (25*x^4 + 80*x^3 + 104*x^2 + 64*x + 16)*log(5*x^2 + 8*x + 4) + (25*x^4 + 80*x^3 + 104*x^2 + 64*x + 16)*log(4*x^2 - sqrt(4*x^2 - 2*x - 3) *(2*x + 3) + 5*x + 5) - 2*(25*x^4 + 80*x^3 + 104*x^2 + 64*x + 16)*log(x) + 2*(200*x^3 + 455*x^2 + 398*x + 138)*sqrt(4*x^2 - 2*x - 3) + 3272*x + 932) /(25*x^4 + 80*x^3 + 104*x^2 + 64*x + 16)
\[ \int \frac {1}{x \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\int \frac {1}{x \left (3 x + \sqrt {4 x^{2} - 2 x - 3} + 1\right )^{3}}\, dx \] Input:
integrate(1/x/(1+3*x+(4*x**2-2*x-3)**(1/2))**3,x)
Output:
Integral(1/(x*(3*x + sqrt(4*x**2 - 2*x - 3) + 1)**3), x)
\[ \int \frac {1}{x \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\int { \frac {1}{{\left (3 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right )}^{3} x} \,d x } \] Input:
integrate(1/x/(1+3*x+(4*x^2-2*x-3)^(1/2))^3,x, algorithm="maxima")
Output:
integrate(1/((3*x + sqrt(4*x^2 - 2*x - 3) + 1)^3*x), x)
Time = 0.16 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.68 \[ \int \frac {1}{x \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {470 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{7} + 4797 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{6} + 20915 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{5} + 46933 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{4} + 45116 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{3} + 24799 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 19238 \, x - 9619 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1999}{2 \, {\left (5 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{4} + 32 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{3} + 78 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 64 \, x - 32 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 13\right )}^{2}} + \frac {305 \, x^{3} + 722 \, x^{2} + 612 \, x + 210}{8 \, {\left (5 \, x^{2} + 8 \, x + 4\right )}^{2}} + \frac {65}{16} \, \arctan \left (\frac {5}{2} \, x + 2\right ) - \frac {65}{16} \, \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) - \frac {65}{16} \, \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) - \frac {1}{16} \, \log \left (5 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 4 \, x - 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) + \frac {1}{16} \, \log \left ({\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 12 \, x - 6 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 13\right ) + \frac {1}{16} \, \log \left (5 \, x^{2} + 8 \, x + 4\right ) - \frac {1}{8} \, \log \left ({\left | x \right |}\right ) \] Input:
integrate(1/x/(1+3*x+(4*x^2-2*x-3)^(1/2))^3,x, algorithm="giac")
Output:
1/2*(470*(2*x - sqrt(4*x^2 - 2*x - 3))^7 + 4797*(2*x - sqrt(4*x^2 - 2*x - 3))^6 + 20915*(2*x - sqrt(4*x^2 - 2*x - 3))^5 + 46933*(2*x - sqrt(4*x^2 - 2*x - 3))^4 + 45116*(2*x - sqrt(4*x^2 - 2*x - 3))^3 + 24799*(2*x - sqrt(4* x^2 - 2*x - 3))^2 + 19238*x - 9619*sqrt(4*x^2 - 2*x - 3) + 1999)/(5*(2*x - sqrt(4*x^2 - 2*x - 3))^4 + 32*(2*x - sqrt(4*x^2 - 2*x - 3))^3 + 78*(2*x - sqrt(4*x^2 - 2*x - 3))^2 + 64*x - 32*sqrt(4*x^2 - 2*x - 3) + 13)^2 + 1/8* (305*x^3 + 722*x^2 + 612*x + 210)/(5*x^2 + 8*x + 4)^2 + 65/16*arctan(5/2*x + 2) - 65/16*arctan(-x + 1/2*sqrt(4*x^2 - 2*x - 3) - 3/2) - 65/16*arctan( -5*x + 5/2*sqrt(4*x^2 - 2*x - 3) - 1/2) - 1/16*log(5*(2*x - sqrt(4*x^2 - 2 *x - 3))^2 + 4*x - 2*sqrt(4*x^2 - 2*x - 3) + 1) + 1/16*log((2*x - sqrt(4*x ^2 - 2*x - 3))^2 + 12*x - 6*sqrt(4*x^2 - 2*x - 3) + 13) + 1/16*log(5*x^2 + 8*x + 4) - 1/8*log(abs(x))
Timed out. \[ \int \frac {1}{x \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\int \frac {1}{x\,{\left (3\,x+\sqrt {4\,x^2-2\,x-3}+1\right )}^3} \,d x \] Input:
int(1/(x*(3*x + (4*x^2 - 2*x - 3)^(1/2) + 1)^3),x)
Output:
int(1/(x*(3*x + (4*x^2 - 2*x - 3)^(1/2) + 1)^3), x)
Time = 1.10 (sec) , antiderivative size = 947, normalized size of antiderivative = 3.74 \[ \int \frac {1}{x \left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx =\text {Too large to display} \] Input:
int(1/x/(1+3*x+(4*x^2-2*x-3)^(1/2))^3,x)
Output:
( - 33257184675000*atan((sqrt(4*x**2 - 2*x - 3) + 2*x + 3)/2)*x**4 - 10642 2990960000*atan((sqrt(4*x**2 - 2*x - 3) + 2*x + 3)/2)*x**3 - 1383498882480 00*atan((sqrt(4*x**2 - 2*x - 3) + 2*x + 3)/2)*x**2 - 85138392768000*atan(( sqrt(4*x**2 - 2*x - 3) + 2*x + 3)/2)*x - 21284598192000*atan((sqrt(4*x**2 - 2*x - 3) + 2*x + 3)/2) - 33257184675000*atan((5*sqrt(4*x**2 - 2*x - 3) + 10*x + 1)/2)*x**4 - 106422990960000*atan((5*sqrt(4*x**2 - 2*x - 3) + 10*x + 1)/2)*x**3 - 138349888248000*atan((5*sqrt(4*x**2 - 2*x - 3) + 10*x + 1) /2)*x**2 - 85138392768000*atan((5*sqrt(4*x**2 - 2*x - 3) + 10*x + 1)/2)*x - 21284598192000*atan((5*sqrt(4*x**2 - 2*x - 3) + 10*x + 1)/2) + 332571846 75000*atan((5*x + 4)/2)*x**4 + 106422990960000*atan((5*x + 4)/2)*x**3 + 13 8349888248000*atan((5*x + 4)/2)*x**2 + 85138392768000*atan((5*x + 4)/2)*x + 21284598192000*atan((5*x + 4)/2) + 8186383920000*sqrt(4*x**2 - 2*x - 3)* x**3 + 18624023418000*sqrt(4*x**2 - 2*x - 3)*x**2 + 16290904000800*sqrt(4* x**2 - 2*x - 3)*x + 5648604904800*sqrt(4*x**2 - 2*x - 3) + 511648995000*lo g(5*x**2 + 8*x + 4)*x**4 + 1637276784000*log(5*x**2 + 8*x + 4)*x**3 + 2128 459819200*log(5*x**2 + 8*x + 4)*x**2 + 1309821427200*log(5*x**2 + 8*x + 4) *x + 327455356800*log(5*x**2 + 8*x + 4) + 511648995000*log((80*sqrt(4*x**2 - 2*x - 3)*x + 8*sqrt(4*x**2 - 2*x - 3) + 160*x**2 - 24*x - 56)/sqrt(13)) *x**4 + 1637276784000*log((80*sqrt(4*x**2 - 2*x - 3)*x + 8*sqrt(4*x**2 - 2 *x - 3) + 160*x**2 - 24*x - 56)/sqrt(13))*x**3 + 2128459819200*log((80*...