Integrand size = 94, antiderivative size = 595 \[ \int \frac {x \left (2-5 \sqrt {3} x+12 x^2-3 \sqrt {3} x^3-20 x^4+10 \sqrt {3} x^5\right )}{1-4 \sqrt {3} x+18 x^2-12 \sqrt {3} x^3+8 x^4+2 \sqrt {3} x^5-3 x^6+5 x^8} \, dx=-\sqrt {-\frac {1}{4}+\frac {\sqrt {5}}{2}} \arctan \left (\frac {1}{3 \sqrt {\frac {19}{209+76 \sqrt {5}-12 \sqrt {5 \left (61+46 \sqrt {5}\right )}}}}-\sqrt {\frac {20}{57}+\frac {40 \sqrt {5}}{57}} x\right )+\sqrt {-\frac {1}{4}+\frac {\sqrt {5}}{2}} \arctan \left (\frac {1}{3 \sqrt {\frac {19}{209+76 \sqrt {5}+12 \sqrt {5 \left (61+46 \sqrt {5}\right )}}}}+\sqrt {\frac {20}{57}+\frac {40 \sqrt {5}}{57}} x\right )+\sqrt {-\frac {1}{4}+\frac {\sqrt {5}}{2}} \arctan \left (\sqrt {\frac {1}{19} \left (57+76 \sqrt {5}-12 \sqrt {61+46 \sqrt {5}}\right )}-\sqrt {\frac {108}{19}+\frac {216 \sqrt {5}}{19}} x+2 \sqrt {\frac {2}{19} \left (97+23 \sqrt {5}-3 \sqrt {5 \left (61+46 \sqrt {5}\right )}\right )} x^2-\sqrt {\frac {60}{19}+\frac {120 \sqrt {5}}{19}} x^3\right )+\sqrt {-\frac {1}{4}+\frac {\sqrt {5}}{2}} \arctan \left (\sqrt {\frac {1}{19} \left (57+76 \sqrt {5}+12 \sqrt {61+46 \sqrt {5}}\right )}-\sqrt {\frac {108}{19}+\frac {216 \sqrt {5}}{19}} x-2 \sqrt {\frac {2}{19} \left (97+23 \sqrt {5}+3 \sqrt {5 \left (61+46 \sqrt {5}\right )}\right )} x^2-\sqrt {\frac {60}{19}+\frac {120 \sqrt {5}}{19}} x^3\right )+\sqrt {\frac {1}{16}+\frac {\sqrt {5}}{8}} \log \left (\sqrt {5}-2 \sqrt {15} x+\left (3 \sqrt {5}+\sqrt {5 \left (1+2 \sqrt {5}\right )}\right ) x^2-\sqrt {15 \left (1+2 \sqrt {5}\right )} x^3+5 x^4\right )-\sqrt {\frac {1}{16}+\frac {\sqrt {5}}{8}} \log \left (\sqrt {5}-2 \sqrt {15} x+\left (3 \sqrt {5}-\sqrt {5 \left (1+2 \sqrt {5}\right )}\right ) x^2+\sqrt {15 \left (1+2 \sqrt {5}\right )} x^3+5 x^4\right ) \] Output:
1/2*(-1+2*5^(1/2))^(1/2)*arctan(-1/57*19^(1/2)*(209+76*5^(1/2)-12*(305+230 *5^(1/2))^(1/2))^(1/2)+2/57*(285+570*5^(1/2))^(1/2)*x)+1/2*(-1+2*5^(1/2))^ (1/2)*arctan(1/57*19^(1/2)*(209+76*5^(1/2)+12*(305+230*5^(1/2))^(1/2))^(1/ 2)+2/57*(285+570*5^(1/2))^(1/2)*x)-1/2*(-1+2*5^(1/2))^(1/2)*arctan(-1/19*( 1083+1444*5^(1/2)-228*(61+46*5^(1/2))^(1/2))^(1/2)+6/19*(57+114*5^(1/2))^( 1/2)*x-2/19*(3686+874*5^(1/2)-114*(305+230*5^(1/2))^(1/2))^(1/2)*x^2+2/19* (285+570*5^(1/2))^(1/2)*x^3)-1/2*(-1+2*5^(1/2))^(1/2)*arctan(-1/19*(1083+1 444*5^(1/2)+228*(61+46*5^(1/2))^(1/2))^(1/2)+6/19*(57+114*5^(1/2))^(1/2)*x +2/19*(3686+874*5^(1/2)+114*(305+230*5^(1/2))^(1/2))^(1/2)*x^2+2/19*(285+5 70*5^(1/2))^(1/2)*x^3)+1/4*(1+2*5^(1/2))^(1/2)*ln(5^(1/2)-2*15^(1/2)*x+(3* 5^(1/2)+(5+10*5^(1/2))^(1/2))*x^2-(15+30*5^(1/2))^(1/2)*x^3+5*x^4)-1/4*(1+ 2*5^(1/2))^(1/2)*ln(5^(1/2)-2*15^(1/2)*x+(3*5^(1/2)-(5+10*5^(1/2))^(1/2))* x^2+(15+30*5^(1/2))^(1/2)*x^3+5*x^4)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.09 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.35 \[ \int \frac {x \left (2-5 \sqrt {3} x+12 x^2-3 \sqrt {3} x^3-20 x^4+10 \sqrt {3} x^5\right )}{1-4 \sqrt {3} x+18 x^2-12 \sqrt {3} x^3+8 x^4+2 \sqrt {3} x^5-3 x^6+5 x^8} \, dx=\frac {1}{2} \text {RootSum}\left [1-4 \sqrt {3} \text {$\#$1}+18 \text {$\#$1}^2-12 \sqrt {3} \text {$\#$1}^3+8 \text {$\#$1}^4+2 \sqrt {3} \text {$\#$1}^5-3 \text {$\#$1}^6+5 \text {$\#$1}^8\&,\frac {2 \log (x-\text {$\#$1}) \text {$\#$1}-5 \sqrt {3} \log (x-\text {$\#$1}) \text {$\#$1}^2+12 \log (x-\text {$\#$1}) \text {$\#$1}^3-3 \sqrt {3} \log (x-\text {$\#$1}) \text {$\#$1}^4-20 \log (x-\text {$\#$1}) \text {$\#$1}^5+10 \sqrt {3} \log (x-\text {$\#$1}) \text {$\#$1}^6}{-2 \sqrt {3}+18 \text {$\#$1}-18 \sqrt {3} \text {$\#$1}^2+16 \text {$\#$1}^3+5 \sqrt {3} \text {$\#$1}^4-9 \text {$\#$1}^5+20 \text {$\#$1}^7}\&\right ] \] Input:
Integrate[(x*(2 - 5*Sqrt[3]*x + 12*x^2 - 3*Sqrt[3]*x^3 - 20*x^4 + 10*Sqrt[ 3]*x^5))/(1 - 4*Sqrt[3]*x + 18*x^2 - 12*Sqrt[3]*x^3 + 8*x^4 + 2*Sqrt[3]*x^ 5 - 3*x^6 + 5*x^8),x]
Output:
RootSum[1 - 4*Sqrt[3]*#1 + 18*#1^2 - 12*Sqrt[3]*#1^3 + 8*#1^4 + 2*Sqrt[3]* #1^5 - 3*#1^6 + 5*#1^8 & , (2*Log[x - #1]*#1 - 5*Sqrt[3]*Log[x - #1]*#1^2 + 12*Log[x - #1]*#1^3 - 3*Sqrt[3]*Log[x - #1]*#1^4 - 20*Log[x - #1]*#1^5 + 10*Sqrt[3]*Log[x - #1]*#1^6)/(-2*Sqrt[3] + 18*#1 - 18*Sqrt[3]*#1^2 + 16*# 1^3 + 5*Sqrt[3]*#1^4 - 9*#1^5 + 20*#1^7) & ]/2
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.
| \(\displaystyle \int \frac {x \left (10 \sqrt {3} x^5-20 x^4-3 \sqrt {3} x^3+12 x^2-5 \sqrt {3} x+2\right )}{5 x^8-3 x^6+2 \sqrt {3} x^5+8 x^4-12 \sqrt {3} x^3+18 x^2-4 \sqrt {3} x+1} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {10 \sqrt {3} x^6}{5 x^8-3 x^6+2 \sqrt {3} x^5+8 x^4-12 \sqrt {3} x^3+18 x^2-4 \sqrt {3} x+1}-\frac {20 x^5}{5 x^8-3 x^6+2 \sqrt {3} x^5+8 x^4-12 \sqrt {3} x^3+18 x^2-4 \sqrt {3} x+1}-\frac {3 \sqrt {3} x^4}{5 x^8-3 x^6+2 \sqrt {3} x^5+8 x^4-12 \sqrt {3} x^3+18 x^2-4 \sqrt {3} x+1}+\frac {12 x^3}{5 x^8-3 x^6+2 \sqrt {3} x^5+8 x^4-12 \sqrt {3} x^3+18 x^2-4 \sqrt {3} x+1}-\frac {5 \sqrt {3} x^2}{5 x^8-3 x^6+2 \sqrt {3} x^5+8 x^4-12 \sqrt {3} x^3+18 x^2-4 \sqrt {3} x+1}+\frac {2 x}{5 x^8-3 x^6+2 \sqrt {3} x^5+8 x^4-12 \sqrt {3} x^3+18 x^2-4 \sqrt {3} x+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \int \frac {x}{5 x^8-3 x^6+2 \sqrt {3} x^5+8 x^4-12 \sqrt {3} x^3+18 x^2-4 \sqrt {3} x+1}dx-5 \sqrt {3} \int \frac {x^2}{5 x^8-3 x^6+2 \sqrt {3} x^5+8 x^4-12 \sqrt {3} x^3+18 x^2-4 \sqrt {3} x+1}dx+12 \int \frac {x^3}{5 x^8-3 x^6+2 \sqrt {3} x^5+8 x^4-12 \sqrt {3} x^3+18 x^2-4 \sqrt {3} x+1}dx-3 \sqrt {3} \int \frac {x^4}{5 x^8-3 x^6+2 \sqrt {3} x^5+8 x^4-12 \sqrt {3} x^3+18 x^2-4 \sqrt {3} x+1}dx-20 \int \frac {x^5}{5 x^8-3 x^6+2 \sqrt {3} x^5+8 x^4-12 \sqrt {3} x^3+18 x^2-4 \sqrt {3} x+1}dx+10 \sqrt {3} \int \frac {x^6}{5 x^8-3 x^6+2 \sqrt {3} x^5+8 x^4-12 \sqrt {3} x^3+18 x^2-4 \sqrt {3} x+1}dx\) |
Input:
Int[(x*(2 - 5*Sqrt[3]*x + 12*x^2 - 3*Sqrt[3]*x^3 - 20*x^4 + 10*Sqrt[3]*x^5 ))/(1 - 4*Sqrt[3]*x + 18*x^2 - 12*Sqrt[3]*x^3 + 8*x^4 + 2*Sqrt[3]*x^5 - 3* x^6 + 5*x^8),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.47 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.08
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{2}+5\right )}{\sum }\textit {\_R} \ln \left (5 x^{2}+\left (\sqrt {3}\, \textit {\_R}^{3}-\sqrt {3}\, \textit {\_R} \right ) x -\textit {\_R}^{3}+\textit {\_R} \right )\right )}{2}\) | \(48\) |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (5 \textit {\_Z}^{8}+2 \sqrt {3}\, \textit {\_Z}^{5}-3 \textit {\_Z}^{6}-12 \sqrt {3}\, \textit {\_Z}^{3}+8 \textit {\_Z}^{4}-4 \sqrt {3}\, \textit {\_Z} +18 \textit {\_Z}^{2}+1\right )}{\sum }\frac {\left (-20 \textit {\_R}^{5}+12 \textit {\_R}^{3}+2 \textit {\_R} +\sqrt {3}\, \left (10 \textit {\_R}^{6}-3 \textit {\_R}^{4}-5 \textit {\_R}^{2}\right )\right ) \ln \left (x -\textit {\_R} \right )}{20 \textit {\_R}^{7}-9 \textit {\_R}^{5}+16 \textit {\_R}^{3}+18 \textit {\_R} +\sqrt {3}\, \left (5 \textit {\_R}^{4}-18 \textit {\_R}^{2}-2\right )}\right )}{2}\) | \(129\) |
Input:
int(x*(2-5*3^(1/2)*x+12*x^2-3*3^(1/2)*x^3-20*x^4+10*3^(1/2)*x^5)/(1-4*3^(1 /2)*x+18*x^2-12*3^(1/2)*x^3+8*x^4+2*3^(1/2)*x^5-3*x^6+5*x^8),x,method=_RET URNVERBOSE)
Output:
1/2*sum(_R*ln(5*x^2+(3^(1/2)*_R^3-3^(1/2)*_R)*x-_R^3+_R),_R=RootOf(_Z^4-_Z ^2+5))
Time = 0.10 (sec) , antiderivative size = 456, normalized size of antiderivative = 0.77 \[ \int \frac {x \left (2-5 \sqrt {3} x+12 x^2-3 \sqrt {3} x^3-20 x^4+10 \sqrt {3} x^5\right )}{1-4 \sqrt {3} x+18 x^2-12 \sqrt {3} x^3+8 x^4+2 \sqrt {3} x^5-3 x^6+5 x^8} \, dx=-\frac {1}{2} \, \sqrt {2 \, \sqrt {5} - 1} \arctan \left (\frac {1}{19} \, {\left (6 \, x^{2} + 2 \, \sqrt {5} {\left (6 \, x^{2} - 1\right )} - 1\right )} \sqrt {2 \, \sqrt {5} + 1} \sqrt {2 \, \sqrt {5} - 1} + \frac {2}{19} \, {\left (10 \, x^{2} + \sqrt {3} {\left (10 \, x^{3} + 3 \, x\right )} + \sqrt {5} {\left (x^{2} + \sqrt {3} {\left (x^{3} + 6 \, x\right )} - 6\right )} - 3\right )} \sqrt {2 \, \sqrt {5} - 1}\right ) + \frac {1}{2} \, \sqrt {2 \, \sqrt {5} - 1} \arctan \left (\frac {1}{19} \, {\left (6 \, x^{2} + 2 \, \sqrt {5} {\left (6 \, x^{2} - 1\right )} - 1\right )} \sqrt {2 \, \sqrt {5} + 1} \sqrt {2 \, \sqrt {5} - 1} - \frac {2}{19} \, {\left (10 \, x^{2} + \sqrt {3} {\left (10 \, x^{3} + 3 \, x\right )} + \sqrt {5} {\left (x^{2} + \sqrt {3} {\left (x^{3} + 6 \, x\right )} - 6\right )} - 3\right )} \sqrt {2 \, \sqrt {5} - 1}\right ) + \frac {1}{2} \, \sqrt {2 \, \sqrt {5} - 1} \arctan \left (\frac {1}{19} \, {\left (2 \, \sqrt {5} + 1\right )}^{\frac {3}{2}} \sqrt {2 \, \sqrt {5} - 1} + \frac {2}{57} \, {\left (\sqrt {5} {\left (\sqrt {3} x + 1\right )} + 10 \, \sqrt {3} x + 10\right )} \sqrt {2 \, \sqrt {5} - 1}\right ) - \frac {1}{2} \, \sqrt {2 \, \sqrt {5} - 1} \arctan \left (\frac {1}{19} \, {\left (2 \, \sqrt {5} + 1\right )}^{\frac {3}{2}} \sqrt {2 \, \sqrt {5} - 1} - \frac {2}{57} \, {\left (\sqrt {5} {\left (\sqrt {3} x + 1\right )} + 10 \, \sqrt {3} x + 10\right )} \sqrt {2 \, \sqrt {5} - 1}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {5} + 1} \log \left (5 \, x^{4} + \sqrt {5} {\left (\sqrt {3} x^{3} - x^{2}\right )} \sqrt {2 \, \sqrt {5} + 1} + \sqrt {5} {\left (3 \, x^{2} - 2 \, \sqrt {3} x + 1\right )}\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {5} + 1} \log \left (5 \, x^{4} - \sqrt {5} {\left (\sqrt {3} x^{3} - x^{2}\right )} \sqrt {2 \, \sqrt {5} + 1} + \sqrt {5} {\left (3 \, x^{2} - 2 \, \sqrt {3} x + 1\right )}\right ) \] Input:
integrate(x*(2-5*3^(1/2)*x+12*x^2-3*3^(1/2)*x^3-20*x^4+10*3^(1/2)*x^5)/(1- 4*3^(1/2)*x+18*x^2-12*3^(1/2)*x^3+8*x^4+2*3^(1/2)*x^5-3*x^6+5*x^8),x, algo rithm="fricas")
Output:
-1/2*sqrt(2*sqrt(5) - 1)*arctan(1/19*(6*x^2 + 2*sqrt(5)*(6*x^2 - 1) - 1)*s qrt(2*sqrt(5) + 1)*sqrt(2*sqrt(5) - 1) + 2/19*(10*x^2 + sqrt(3)*(10*x^3 + 3*x) + sqrt(5)*(x^2 + sqrt(3)*(x^3 + 6*x) - 6) - 3)*sqrt(2*sqrt(5) - 1)) + 1/2*sqrt(2*sqrt(5) - 1)*arctan(1/19*(6*x^2 + 2*sqrt(5)*(6*x^2 - 1) - 1)*s qrt(2*sqrt(5) + 1)*sqrt(2*sqrt(5) - 1) - 2/19*(10*x^2 + sqrt(3)*(10*x^3 + 3*x) + sqrt(5)*(x^2 + sqrt(3)*(x^3 + 6*x) - 6) - 3)*sqrt(2*sqrt(5) - 1)) + 1/2*sqrt(2*sqrt(5) - 1)*arctan(1/19*(2*sqrt(5) + 1)^(3/2)*sqrt(2*sqrt(5) - 1) + 2/57*(sqrt(5)*(sqrt(3)*x + 1) + 10*sqrt(3)*x + 10)*sqrt(2*sqrt(5) - 1)) - 1/2*sqrt(2*sqrt(5) - 1)*arctan(1/19*(2*sqrt(5) + 1)^(3/2)*sqrt(2*sq rt(5) - 1) - 2/57*(sqrt(5)*(sqrt(3)*x + 1) + 10*sqrt(3)*x + 10)*sqrt(2*sqr t(5) - 1)) - 1/4*sqrt(2*sqrt(5) + 1)*log(5*x^4 + sqrt(5)*(sqrt(3)*x^3 - x^ 2)*sqrt(2*sqrt(5) + 1) + sqrt(5)*(3*x^2 - 2*sqrt(3)*x + 1)) + 1/4*sqrt(2*s qrt(5) + 1)*log(5*x^4 - sqrt(5)*(sqrt(3)*x^3 - x^2)*sqrt(2*sqrt(5) + 1) + sqrt(5)*(3*x^2 - 2*sqrt(3)*x + 1))
Leaf count of result is larger than twice the leaf count of optimal. 4038 vs. \(2 (491) = 982\).
Time = 3.73 (sec) , antiderivative size = 4038, normalized size of antiderivative = 6.79 \[ \int \frac {x \left (2-5 \sqrt {3} x+12 x^2-3 \sqrt {3} x^3-20 x^4+10 \sqrt {3} x^5\right )}{1-4 \sqrt {3} x+18 x^2-12 \sqrt {3} x^3+8 x^4+2 \sqrt {3} x^5-3 x^6+5 x^8} \, dx=\text {Too large to display} \] Input:
integrate(x*(2-5*3**(1/2)*x+12*x**2-3*3**(1/2)*x**3-20*x**4+10*3**(1/2)*x* *5)/(1-4*3**(1/2)*x+18*x**2-12*3**(1/2)*x**3+8*x**4+2*3**(1/2)*x**5-3*x**6 +5*x**8),x)
Output:
(2*atan(20*sqrt(3)*x/(-3*sqrt(-2*sqrt(4*sqrt(5) + 21) + 1 + 6*sqrt(5)) + 3 *sqrt(4*sqrt(5) + 21)*sqrt(-2*sqrt(4*sqrt(5) + 21) + 1 + 6*sqrt(5))) - 24* sqrt(5)*sqrt(1 + 2*sqrt(5))/(-3*sqrt(-2*sqrt(4*sqrt(5) + 21) + 1 + 6*sqrt( 5)) + 3*sqrt(4*sqrt(5) + 21)*sqrt(-2*sqrt(4*sqrt(5) + 21) + 1 + 6*sqrt(5)) ) - 9*sqrt(1 + 2*sqrt(5))/(-3*sqrt(-2*sqrt(4*sqrt(5) + 21) + 1 + 6*sqrt(5) ) + 3*sqrt(4*sqrt(5) + 21)*sqrt(-2*sqrt(4*sqrt(5) + 21) + 1 + 6*sqrt(5))) + 20/(-3*sqrt(-2*sqrt(4*sqrt(5) + 21) + 1 + 6*sqrt(5)) + 3*sqrt(4*sqrt(5) + 21)*sqrt(-2*sqrt(4*sqrt(5) + 21) + 1 + 6*sqrt(5))) + 9*sqrt(1 + 2*sqrt(5 ))*sqrt(4*sqrt(5) + 21)/(-3*sqrt(-2*sqrt(4*sqrt(5) + 21) + 1 + 6*sqrt(5)) + 3*sqrt(4*sqrt(5) + 21)*sqrt(-2*sqrt(4*sqrt(5) + 21) + 1 + 6*sqrt(5)))) - 2*atan(20*sqrt(3)*x**3/(-sqrt(-2*sqrt(4*sqrt(5) + 21) + 1 + 6*sqrt(5)) + sqrt(4*sqrt(5) + 21)*sqrt(-2*sqrt(4*sqrt(5) + 21) + 1 + 6*sqrt(5))) - x**2 *(-18*sqrt(1 + 2*sqrt(5))*sqrt(4*sqrt(5) + 21)/(-sqrt(-2*sqrt(4*sqrt(5) + 21) + 1 + 6*sqrt(5)) + sqrt(4*sqrt(5) + 21)*sqrt(-2*sqrt(4*sqrt(5) + 21) + 1 + 6*sqrt(5))) - 20/(-sqrt(-2*sqrt(4*sqrt(5) + 21) + 1 + 6*sqrt(5)) + sq rt(4*sqrt(5) + 21)*sqrt(-2*sqrt(4*sqrt(5) + 21) + 1 + 6*sqrt(5))) + 18*sqr t(1 + 2*sqrt(5))/(-sqrt(-2*sqrt(4*sqrt(5) + 21) + 1 + 6*sqrt(5)) + sqrt(4* sqrt(5) + 21)*sqrt(-2*sqrt(4*sqrt(5) + 21) + 1 + 6*sqrt(5))) + 48*sqrt(5)* sqrt(1 + 2*sqrt(5))/(-sqrt(-2*sqrt(4*sqrt(5) + 21) + 1 + 6*sqrt(5)) + sqrt (4*sqrt(5) + 21)*sqrt(-2*sqrt(4*sqrt(5) + 21) + 1 + 6*sqrt(5)))) - x*(-...
\[ \int \frac {x \left (2-5 \sqrt {3} x+12 x^2-3 \sqrt {3} x^3-20 x^4+10 \sqrt {3} x^5\right )}{1-4 \sqrt {3} x+18 x^2-12 \sqrt {3} x^3+8 x^4+2 \sqrt {3} x^5-3 x^6+5 x^8} \, dx=\int { \frac {{\left (10 \, \sqrt {3} x^{5} - 20 \, x^{4} - 3 \, \sqrt {3} x^{3} + 12 \, x^{2} - 5 \, \sqrt {3} x + 2\right )} x}{5 \, x^{8} - 3 \, x^{6} + 2 \, \sqrt {3} x^{5} + 8 \, x^{4} - 12 \, \sqrt {3} x^{3} + 18 \, x^{2} - 4 \, \sqrt {3} x + 1} \,d x } \] Input:
integrate(x*(2-5*3^(1/2)*x+12*x^2-3*3^(1/2)*x^3-20*x^4+10*3^(1/2)*x^5)/(1- 4*3^(1/2)*x+18*x^2-12*3^(1/2)*x^3+8*x^4+2*3^(1/2)*x^5-3*x^6+5*x^8),x, algo rithm="maxima")
Output:
integrate((10*sqrt(3)*x^5 - 20*x^4 - 3*sqrt(3)*x^3 + 12*x^2 - 5*sqrt(3)*x + 2)*x/(5*x^8 - 3*x^6 + 2*sqrt(3)*x^5 + 8*x^4 - 12*sqrt(3)*x^3 + 18*x^2 - 4*sqrt(3)*x + 1), x)
Timed out. \[ \int \frac {x \left (2-5 \sqrt {3} x+12 x^2-3 \sqrt {3} x^3-20 x^4+10 \sqrt {3} x^5\right )}{1-4 \sqrt {3} x+18 x^2-12 \sqrt {3} x^3+8 x^4+2 \sqrt {3} x^5-3 x^6+5 x^8} \, dx=\text {Timed out} \] Input:
integrate(x*(2-5*3^(1/2)*x+12*x^2-3*3^(1/2)*x^3-20*x^4+10*3^(1/2)*x^5)/(1- 4*3^(1/2)*x+18*x^2-12*3^(1/2)*x^3+8*x^4+2*3^(1/2)*x^5-3*x^6+5*x^8),x, algo rithm="giac")
Output:
Timed out
Time = 10.83 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.45 \[ \int \frac {x \left (2-5 \sqrt {3} x+12 x^2-3 \sqrt {3} x^3-20 x^4+10 \sqrt {3} x^5\right )}{1-4 \sqrt {3} x+18 x^2-12 \sqrt {3} x^3+8 x^4+2 \sqrt {3} x^5-3 x^6+5 x^8} \, dx=\sum _{k=1}^8\ln \left (-\frac {\left (16\,{\mathrm {root}\left (z^8-\frac {z^6}{2}+\frac {11\,z^4}{16}-\frac {5\,z^2}{32}+\frac {25}{256},z,k\right )}^4-4\,{\mathrm {root}\left (z^8-\frac {z^6}{2}+\frac {11\,z^4}{16}-\frac {5\,z^2}{32}+\frac {25}{256},z,k\right )}^2+5\right )\,\left (8559440\,x+156497\,\sqrt {3}\,\mathrm {root}\left (z^8-\frac {z^6}{2}+\frac {11\,z^4}{16}-\frac {5\,z^2}{32}+\frac {25}{256},z,k\right )-\mathrm {root}\left (z^8-\frac {z^6}{2}+\frac {11\,z^4}{16}-\frac {5\,z^2}{32}+\frac {25}{256},z,k\right )\,x\,2573211-1767270\,\sqrt {3}+5079720\,\sqrt {3}\,{\mathrm {root}\left (z^8-\frac {z^6}{2}+\frac {11\,z^4}{16}-\frac {5\,z^2}{32}+\frac {25}{256},z,k\right )}^2-7464368\,\sqrt {3}\,{\mathrm {root}\left (z^8-\frac {z^6}{2}+\frac {11\,z^4}{16}-\frac {5\,z^2}{32}+\frac {25}{256},z,k\right )}^3-{\mathrm {root}\left (z^8-\frac {z^6}{2}+\frac {11\,z^4}{16}-\frac {5\,z^2}{32}+\frac {25}{256},z,k\right )}^2\,x\,18353520+{\mathrm {root}\left (z^8-\frac {z^6}{2}+\frac {11\,z^4}{16}-\frac {5\,z^2}{32}+\frac {25}{256},z,k\right )}^3\,x\,30736704\right )\,13718}{48828125}\right )\,\mathrm {root}\left (z^8-\frac {z^6}{2}+\frac {11\,z^4}{16}-\frac {5\,z^2}{32}+\frac {25}{256},z,k\right ) \] Input:
int(-(x*(5*3^(1/2)*x + 3*3^(1/2)*x^3 - 10*3^(1/2)*x^5 - 12*x^2 + 20*x^4 - 2))/(2*3^(1/2)*x^5 - 12*3^(1/2)*x^3 - 4*3^(1/2)*x + 18*x^2 + 8*x^4 - 3*x^6 + 5*x^8 + 1),x)
Output:
symsum(log(-(13718*(16*root(z^8 - z^6/2 + (11*z^4)/16 - (5*z^2)/32 + 25/25 6, z, k)^4 - 4*root(z^8 - z^6/2 + (11*z^4)/16 - (5*z^2)/32 + 25/256, z, k) ^2 + 5)*(8559440*x + 156497*3^(1/2)*root(z^8 - z^6/2 + (11*z^4)/16 - (5*z^ 2)/32 + 25/256, z, k) - 2573211*root(z^8 - z^6/2 + (11*z^4)/16 - (5*z^2)/3 2 + 25/256, z, k)*x - 1767270*3^(1/2) + 5079720*3^(1/2)*root(z^8 - z^6/2 + (11*z^4)/16 - (5*z^2)/32 + 25/256, z, k)^2 - 7464368*3^(1/2)*root(z^8 - z ^6/2 + (11*z^4)/16 - (5*z^2)/32 + 25/256, z, k)^3 - 18353520*root(z^8 - z^ 6/2 + (11*z^4)/16 - (5*z^2)/32 + 25/256, z, k)^2*x + 30736704*root(z^8 - z ^6/2 + (11*z^4)/16 - (5*z^2)/32 + 25/256, z, k)^3*x))/48828125)*root(z^8 - z^6/2 + (11*z^4)/16 - (5*z^2)/32 + 25/256, z, k), k, 1, 8)
\[ \int \frac {x \left (2-5 \sqrt {3} x+12 x^2-3 \sqrt {3} x^3-20 x^4+10 \sqrt {3} x^5\right )}{1-4 \sqrt {3} x+18 x^2-12 \sqrt {3} x^3+8 x^4+2 \sqrt {3} x^5-3 x^6+5 x^8} \, dx=50 \sqrt {3}\, \left (\int \frac {x^{14}}{25 x^{16}-30 x^{14}+89 x^{12}+120 x^{10}+110 x^{8}-102 x^{6}+52 x^{4}-12 x^{2}+1}d x \right )-45 \sqrt {3}\, \left (\int \frac {x^{12}}{25 x^{16}-30 x^{14}+89 x^{12}+120 x^{10}+110 x^{8}-102 x^{6}+52 x^{4}-12 x^{2}+1}d x \right )+104 \sqrt {3}\, \left (\int \frac {x^{10}}{25 x^{16}-30 x^{14}+89 x^{12}+120 x^{10}+110 x^{8}-102 x^{6}+52 x^{4}-12 x^{2}+1}d x \right )-93 \sqrt {3}\, \left (\int \frac {x^{8}}{25 x^{16}-30 x^{14}+89 x^{12}+120 x^{10}+110 x^{8}-102 x^{6}+52 x^{4}-12 x^{2}+1}d x \right )-24 \sqrt {3}\, \left (\int \frac {x^{6}}{25 x^{16}-30 x^{14}+89 x^{12}+120 x^{10}+110 x^{8}-102 x^{6}+52 x^{4}-12 x^{2}+1}d x \right )-21 \sqrt {3}\, \left (\int \frac {x^{4}}{25 x^{16}-30 x^{14}+89 x^{12}+120 x^{10}+110 x^{8}-102 x^{6}+52 x^{4}-12 x^{2}+1}d x \right )+3 \sqrt {3}\, \left (\int \frac {x^{2}}{25 x^{16}-30 x^{14}+89 x^{12}+120 x^{10}+110 x^{8}-102 x^{6}+52 x^{4}-12 x^{2}+1}d x \right )-100 \left (\int \frac {x^{13}}{25 x^{16}-30 x^{14}+89 x^{12}+120 x^{10}+110 x^{8}-102 x^{6}+52 x^{4}-12 x^{2}+1}d x \right )+60 \left (\int \frac {x^{11}}{25 x^{16}-30 x^{14}+89 x^{12}+120 x^{10}+110 x^{8}-102 x^{6}+52 x^{4}-12 x^{2}+1}d x \right )+192 \left (\int \frac {x^{9}}{25 x^{16}-30 x^{14}+89 x^{12}+120 x^{10}+110 x^{8}-102 x^{6}+52 x^{4}-12 x^{2}+1}d x \right )-228 \left (\int \frac {x^{7}}{25 x^{16}-30 x^{14}+89 x^{12}+120 x^{10}+110 x^{8}-102 x^{6}+52 x^{4}-12 x^{2}+1}d x \right )-4 \left (\int \frac {x^{5}}{25 x^{16}-30 x^{14}+89 x^{12}+120 x^{10}+110 x^{8}-102 x^{6}+52 x^{4}-12 x^{2}+1}d x \right )-12 \left (\int \frac {x^{3}}{25 x^{16}-30 x^{14}+89 x^{12}+120 x^{10}+110 x^{8}-102 x^{6}+52 x^{4}-12 x^{2}+1}d x \right )+2 \left (\int \frac {x}{25 x^{16}-30 x^{14}+89 x^{12}+120 x^{10}+110 x^{8}-102 x^{6}+52 x^{4}-12 x^{2}+1}d x \right ) \] Input:
int(x*(2-5*3^(1/2)*x+12*x^2-3*3^(1/2)*x^3-20*x^4+10*3^(1/2)*x^5)/(1-4*3^(1 /2)*x+18*x^2-12*3^(1/2)*x^3+8*x^4+2*3^(1/2)*x^5-3*x^6+5*x^8),x)
Output:
50*sqrt(3)*int(x**14/(25*x**16 - 30*x**14 + 89*x**12 + 120*x**10 + 110*x** 8 - 102*x**6 + 52*x**4 - 12*x**2 + 1),x) - 45*sqrt(3)*int(x**12/(25*x**16 - 30*x**14 + 89*x**12 + 120*x**10 + 110*x**8 - 102*x**6 + 52*x**4 - 12*x** 2 + 1),x) + 104*sqrt(3)*int(x**10/(25*x**16 - 30*x**14 + 89*x**12 + 120*x* *10 + 110*x**8 - 102*x**6 + 52*x**4 - 12*x**2 + 1),x) - 93*sqrt(3)*int(x** 8/(25*x**16 - 30*x**14 + 89*x**12 + 120*x**10 + 110*x**8 - 102*x**6 + 52*x **4 - 12*x**2 + 1),x) - 24*sqrt(3)*int(x**6/(25*x**16 - 30*x**14 + 89*x**1 2 + 120*x**10 + 110*x**8 - 102*x**6 + 52*x**4 - 12*x**2 + 1),x) - 21*sqrt( 3)*int(x**4/(25*x**16 - 30*x**14 + 89*x**12 + 120*x**10 + 110*x**8 - 102*x **6 + 52*x**4 - 12*x**2 + 1),x) + 3*sqrt(3)*int(x**2/(25*x**16 - 30*x**14 + 89*x**12 + 120*x**10 + 110*x**8 - 102*x**6 + 52*x**4 - 12*x**2 + 1),x) - 100*int(x**13/(25*x**16 - 30*x**14 + 89*x**12 + 120*x**10 + 110*x**8 - 10 2*x**6 + 52*x**4 - 12*x**2 + 1),x) + 60*int(x**11/(25*x**16 - 30*x**14 + 8 9*x**12 + 120*x**10 + 110*x**8 - 102*x**6 + 52*x**4 - 12*x**2 + 1),x) + 19 2*int(x**9/(25*x**16 - 30*x**14 + 89*x**12 + 120*x**10 + 110*x**8 - 102*x* *6 + 52*x**4 - 12*x**2 + 1),x) - 228*int(x**7/(25*x**16 - 30*x**14 + 89*x* *12 + 120*x**10 + 110*x**8 - 102*x**6 + 52*x**4 - 12*x**2 + 1),x) - 4*int( x**5/(25*x**16 - 30*x**14 + 89*x**12 + 120*x**10 + 110*x**8 - 102*x**6 + 5 2*x**4 - 12*x**2 + 1),x) - 12*int(x**3/(25*x**16 - 30*x**14 + 89*x**12 + 1 20*x**10 + 110*x**8 - 102*x**6 + 52*x**4 - 12*x**2 + 1),x) + 2*int(x/(2...