Integrand size = 117, antiderivative size = 139 \[ \int \frac {264 \sqrt {6} x-528 \sqrt {3} x^3+24 \sqrt {2} x^5-96 x^7+60 \sqrt {2} x^9-24 x^{11}}{99+6 \sqrt {3} x^4-6 \sqrt {6} x^6+\left (-4+3 \sqrt {3}\right ) x^8+8 \sqrt {2} x^{10}-12 x^{12}+4 \sqrt {2} x^{14}-x^{16}} \, dx=2^{3/4} \sqrt [4]{25-\sqrt {141}} \arctan \left (\frac {\sqrt [4]{2 \left (25+\sqrt {141}\right )} x^2}{\sqrt {33}}-\frac {\sqrt [4]{\frac {1}{2} \left (25+\sqrt {141}\right )} x^4}{\sqrt {33}}\right )+2^{3/4} \sqrt [4]{25+\sqrt {141}} \text {arctanh}\left (\frac {\sqrt [4]{2 \left (25-\sqrt {141}\right )} x^2}{\sqrt {33}}-\frac {\sqrt [4]{\frac {1}{2} \left (25-\sqrt {141}\right )} x^4}{\sqrt {33}}\right ) \] Output:
-2^(3/4)*(25-141^(1/2))^(1/4)*arctan(-1/33*(50+2*141^(1/2))^(1/4)*x^2*33^( 1/2)+1/33*(25/2+1/2*141^(1/2))^(1/4)*x^4*33^(1/2))-2^(3/4)*(25+141^(1/2))^ (1/4)*arctanh(-1/33*(50-2*141^(1/2))^(1/4)*x^2*33^(1/2)+1/33*(25/2-1/2*141 ^(1/2))^(1/4)*x^4*33^(1/2))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 2.89 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.81 \[ \int \frac {264 \sqrt {6} x-528 \sqrt {3} x^3+24 \sqrt {2} x^5-96 x^7+60 \sqrt {2} x^9-24 x^{11}}{99+6 \sqrt {3} x^4-6 \sqrt {6} x^6+\left (-4+3 \sqrt {3}\right ) x^8+8 \sqrt {2} x^{10}-12 x^{12}+4 \sqrt {2} x^{14}-x^{16}} \, dx=3 \text {RootSum}\left [99+6 \sqrt {3} \text {$\#$1}^4-6 \sqrt {6} \text {$\#$1}^6-4 \text {$\#$1}^8+3 \sqrt {3} \text {$\#$1}^8+8 \sqrt {2} \text {$\#$1}^{10}-12 \text {$\#$1}^{12}+4 \sqrt {2} \text {$\#$1}^{14}-\text {$\#$1}^{16}\&,\frac {-22 \sqrt {6} \log (x-\text {$\#$1})+44 \sqrt {3} \log (x-\text {$\#$1}) \text {$\#$1}^2-2 \sqrt {2} \log (x-\text {$\#$1}) \text {$\#$1}^4+8 \log (x-\text {$\#$1}) \text {$\#$1}^6-5 \sqrt {2} \log (x-\text {$\#$1}) \text {$\#$1}^8+2 \log (x-\text {$\#$1}) \text {$\#$1}^{10}}{-6 \sqrt {3} \text {$\#$1}^2+9 \sqrt {6} \text {$\#$1}^4+8 \text {$\#$1}^6-6 \sqrt {3} \text {$\#$1}^6-20 \sqrt {2} \text {$\#$1}^8+36 \text {$\#$1}^{10}-14 \sqrt {2} \text {$\#$1}^{12}+4 \text {$\#$1}^{14}}\&\right ] \] Input:
Integrate[(264*Sqrt[6]*x - 528*Sqrt[3]*x^3 + 24*Sqrt[2]*x^5 - 96*x^7 + 60* Sqrt[2]*x^9 - 24*x^11)/(99 + 6*Sqrt[3]*x^4 - 6*Sqrt[6]*x^6 + (-4 + 3*Sqrt[ 3])*x^8 + 8*Sqrt[2]*x^10 - 12*x^12 + 4*Sqrt[2]*x^14 - x^16),x]
Output:
3*RootSum[99 + 6*Sqrt[3]*#1^4 - 6*Sqrt[6]*#1^6 - 4*#1^8 + 3*Sqrt[3]*#1^8 + 8*Sqrt[2]*#1^10 - 12*#1^12 + 4*Sqrt[2]*#1^14 - #1^16 & , (-22*Sqrt[6]*Log [x - #1] + 44*Sqrt[3]*Log[x - #1]*#1^2 - 2*Sqrt[2]*Log[x - #1]*#1^4 + 8*Lo g[x - #1]*#1^6 - 5*Sqrt[2]*Log[x - #1]*#1^8 + 2*Log[x - #1]*#1^10)/(-6*Sqr t[3]*#1^2 + 9*Sqrt[6]*#1^4 + 8*#1^6 - 6*Sqrt[3]*#1^6 - 20*Sqrt[2]*#1^8 + 3 6*#1^10 - 14*Sqrt[2]*#1^12 + 4*#1^14) & ]
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.
| \(\displaystyle \int \frac {-24 x^{11}+60 \sqrt {2} x^9-96 x^7+24 \sqrt {2} x^5-528 \sqrt {3} x^3+264 \sqrt {6} x}{-x^{16}+4 \sqrt {2} x^{14}-12 x^{12}+8 \sqrt {2} x^{10}+\left (3 \sqrt {3}-4\right ) x^8-6 \sqrt {6} x^6+6 \sqrt {3} x^4+99} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {12 x \left (-2 x^{10}+5 \sqrt {2} x^8-8 x^6+2 \sqrt {2} x^4-44 \sqrt {3} x^2+22 \sqrt {6}\right )}{-x^{16}+4 \sqrt {2} x^{14}-12 x^{12}+8 \sqrt {2} x^{10}+\left (3 \sqrt {3}-4\right ) x^8-6 \sqrt {6} x^6+6 \sqrt {3} x^4+99}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 12 \int \frac {x \left (-2 x^{10}+5 \sqrt {2} x^8-8 x^6+2 \sqrt {2} x^4-44 \sqrt {3} x^2+22 \sqrt {6}\right )}{-x^{16}+4 \sqrt {2} x^{14}-12 x^{12}+8 \sqrt {2} x^{10}-\left (4-3 \sqrt {3}\right ) x^8-6 \sqrt {6} x^6+6 \sqrt {3} x^4+99}dx\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle 6 \int \frac {-2 x^{10}+5 \sqrt {2} x^8-8 x^6+2 \sqrt {2} x^4-44 \sqrt {3} x^2+22 \sqrt {6}}{-x^{16}+4 \sqrt {2} x^{14}-12 x^{12}+8 \sqrt {2} x^{10}-\left (4-3 \sqrt {3}\right ) x^8-6 \sqrt {6} x^6+6 \sqrt {3} x^4+99}dx^2\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 6 \int \left (\frac {2 x^{10}}{x^{16}-4 \sqrt {2} x^{14}+12 x^{12}-8 \sqrt {2} x^{10}+4 \left (1-\frac {3 \sqrt {3}}{4}\right ) x^8+6 \sqrt {6} x^6-6 \sqrt {3} x^4-99}+\frac {5 \sqrt {2} x^8}{-x^{16}+4 \sqrt {2} x^{14}-12 x^{12}+8 \sqrt {2} x^{10}-4 \left (1-\frac {3 \sqrt {3}}{4}\right ) x^8-6 \sqrt {6} x^6+6 \sqrt {3} x^4+99}+\frac {8 x^6}{x^{16}-4 \sqrt {2} x^{14}+12 x^{12}-8 \sqrt {2} x^{10}+4 \left (1-\frac {3 \sqrt {3}}{4}\right ) x^8+6 \sqrt {6} x^6-6 \sqrt {3} x^4-99}+\frac {2 \sqrt {2} x^4}{-x^{16}+4 \sqrt {2} x^{14}-12 x^{12}+8 \sqrt {2} x^{10}-4 \left (1-\frac {3 \sqrt {3}}{4}\right ) x^8-6 \sqrt {6} x^6+6 \sqrt {3} x^4+99}+\frac {44 \sqrt {3} x^2}{x^{16}-4 \sqrt {2} x^{14}+12 x^{12}-8 \sqrt {2} x^{10}+4 \left (1-\frac {3 \sqrt {3}}{4}\right ) x^8+6 \sqrt {6} x^6-6 \sqrt {3} x^4-99}+\frac {22 \sqrt {6}}{-x^{16}+4 \sqrt {2} x^{14}-12 x^{12}+8 \sqrt {2} x^{10}-4 \left (1-\frac {3 \sqrt {3}}{4}\right ) x^8-6 \sqrt {6} x^6+6 \sqrt {3} x^4+99}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 6 \left (22 \sqrt {6} \int \frac {1}{-x^{16}+4 \sqrt {2} x^{14}-12 x^{12}+8 \sqrt {2} x^{10}-4 \left (1-\frac {3 \sqrt {3}}{4}\right ) x^8-6 \sqrt {6} x^6+6 \sqrt {3} x^4+99}dx^2+2 \sqrt {2} \int \frac {x^4}{-x^{16}+4 \sqrt {2} x^{14}-12 x^{12}+8 \sqrt {2} x^{10}-4 \left (1-\frac {3 \sqrt {3}}{4}\right ) x^8-6 \sqrt {6} x^6+6 \sqrt {3} x^4+99}dx^2+5 \sqrt {2} \int \frac {x^8}{-x^{16}+4 \sqrt {2} x^{14}-12 x^{12}+8 \sqrt {2} x^{10}-4 \left (1-\frac {3 \sqrt {3}}{4}\right ) x^8-6 \sqrt {6} x^6+6 \sqrt {3} x^4+99}dx^2+44 \sqrt {3} \int \frac {x^2}{x^{16}-4 \sqrt {2} x^{14}+12 x^{12}-8 \sqrt {2} x^{10}+4 \left (1-\frac {3 \sqrt {3}}{4}\right ) x^8+6 \sqrt {6} x^6-6 \sqrt {3} x^4-99}dx^2+8 \int \frac {x^6}{x^{16}-4 \sqrt {2} x^{14}+12 x^{12}-8 \sqrt {2} x^{10}+4 \left (1-\frac {3 \sqrt {3}}{4}\right ) x^8+6 \sqrt {6} x^6-6 \sqrt {3} x^4-99}dx^2+2 \int \frac {x^{10}}{x^{16}-4 \sqrt {2} x^{14}+12 x^{12}-8 \sqrt {2} x^{10}+4 \left (1-\frac {3 \sqrt {3}}{4}\right ) x^8+6 \sqrt {6} x^6-6 \sqrt {3} x^4-99}dx^2\right )\) |
Input:
Int[(264*Sqrt[6]*x - 528*Sqrt[3]*x^3 + 24*Sqrt[2]*x^5 - 96*x^7 + 60*Sqrt[2 ]*x^9 - 24*x^11)/(99 + 6*Sqrt[3]*x^4 - 6*Sqrt[6]*x^6 + (-4 + 3*Sqrt[3])*x^ 8 + 8*Sqrt[2]*x^10 - 12*x^12 + 4*Sqrt[2]*x^14 - x^16),x]
Output:
$Aborted
Timed out.
hanged
Input:
int((264*6^(1/2)*x-528*3^(1/2)*x^3+24*2^(1/2)*x^5-96*x^7+60*2^(1/2)*x^9-24 *x^11)/(99+6*3^(1/2)*x^4-6*6^(1/2)*x^6+(-4+3*3^(1/2))*x^8+8*2^(1/2)*x^10-1 2*x^12+4*2^(1/2)*x^14-x^16),x)
Output:
int((264*6^(1/2)*x-528*3^(1/2)*x^3+24*2^(1/2)*x^5-96*x^7+60*2^(1/2)*x^9-24 *x^11)/(99+6*3^(1/2)*x^4-6*6^(1/2)*x^6+(-4+3*3^(1/2))*x^8+8*2^(1/2)*x^10-1 2*x^12+4*2^(1/2)*x^14-x^16),x)
Timed out. \[ \int \frac {264 \sqrt {6} x-528 \sqrt {3} x^3+24 \sqrt {2} x^5-96 x^7+60 \sqrt {2} x^9-24 x^{11}}{99+6 \sqrt {3} x^4-6 \sqrt {6} x^6+\left (-4+3 \sqrt {3}\right ) x^8+8 \sqrt {2} x^{10}-12 x^{12}+4 \sqrt {2} x^{14}-x^{16}} \, dx=\text {Timed out} \] Input:
integrate((264*6^(1/2)*x-528*3^(1/2)*x^3+24*2^(1/2)*x^5-96*x^7+60*2^(1/2)* x^9-24*x^11)/(99+6*3^(1/2)*x^4-6*x^6*6^(1/2)+(-4+3*3^(1/2))*x^8+8*2^(1/2)* x^10-12*x^12+4*2^(1/2)*x^14-x^16),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {264 \sqrt {6} x-528 \sqrt {3} x^3+24 \sqrt {2} x^5-96 x^7+60 \sqrt {2} x^9-24 x^{11}}{99+6 \sqrt {3} x^4-6 \sqrt {6} x^6+\left (-4+3 \sqrt {3}\right ) x^8+8 \sqrt {2} x^{10}-12 x^{12}+4 \sqrt {2} x^{14}-x^{16}} \, dx=\text {Timed out} \] Input:
integrate((264*6**(1/2)*x-528*3**(1/2)*x**3+24*2**(1/2)*x**5-96*x**7+60*2* *(1/2)*x**9-24*x**11)/(99+6*3**(1/2)*x**4-6*x**6*6**(1/2)+(-4+3*3**(1/2))* x**8+8*2**(1/2)*x**10-12*x**12+4*2**(1/2)*x**14-x**16),x)
Output:
Timed out
\[ \int \frac {264 \sqrt {6} x-528 \sqrt {3} x^3+24 \sqrt {2} x^5-96 x^7+60 \sqrt {2} x^9-24 x^{11}}{99+6 \sqrt {3} x^4-6 \sqrt {6} x^6+\left (-4+3 \sqrt {3}\right ) x^8+8 \sqrt {2} x^{10}-12 x^{12}+4 \sqrt {2} x^{14}-x^{16}} \, dx=\int { \frac {12 \, {\left (2 \, x^{11} - 5 \, \sqrt {2} x^{9} + 8 \, x^{7} - 2 \, \sqrt {2} x^{5} + 44 \, \sqrt {3} x^{3} - 22 \, \sqrt {6} x\right )}}{x^{16} - 4 \, \sqrt {2} x^{14} + 12 \, x^{12} - 8 \, \sqrt {2} x^{10} - x^{8} {\left (3 \, \sqrt {3} - 4\right )} + 6 \, \sqrt {6} x^{6} - 6 \, \sqrt {3} x^{4} - 99} \,d x } \] Input:
integrate((264*6^(1/2)*x-528*3^(1/2)*x^3+24*2^(1/2)*x^5-96*x^7+60*2^(1/2)* x^9-24*x^11)/(99+6*3^(1/2)*x^4-6*x^6*6^(1/2)+(-4+3*3^(1/2))*x^8+8*2^(1/2)* x^10-12*x^12+4*2^(1/2)*x^14-x^16),x, algorithm="maxima")
Output:
12*integrate((2*x^11 - 5*sqrt(2)*x^9 + 8*x^7 - 2*sqrt(2)*x^5 + 44*sqrt(3)* x^3 - 22*sqrt(6)*x)/(x^16 - 4*sqrt(2)*x^14 + 12*x^12 - 8*sqrt(2)*x^10 - x^ 8*(3*sqrt(3) - 4) + 6*sqrt(6)*x^6 - 6*sqrt(3)*x^4 - 99), x)
\[ \int \frac {264 \sqrt {6} x-528 \sqrt {3} x^3+24 \sqrt {2} x^5-96 x^7+60 \sqrt {2} x^9-24 x^{11}}{99+6 \sqrt {3} x^4-6 \sqrt {6} x^6+\left (-4+3 \sqrt {3}\right ) x^8+8 \sqrt {2} x^{10}-12 x^{12}+4 \sqrt {2} x^{14}-x^{16}} \, dx=\int { \frac {12 \, {\left (2 \, x^{11} - 5 \, \sqrt {2} x^{9} + 8 \, x^{7} - 2 \, \sqrt {2} x^{5} + 44 \, \sqrt {3} x^{3} - 22 \, \sqrt {6} x\right )}}{x^{16} - 4 \, \sqrt {2} x^{14} + 12 \, x^{12} - 8 \, \sqrt {2} x^{10} - x^{8} {\left (3 \, \sqrt {3} - 4\right )} + 6 \, \sqrt {6} x^{6} - 6 \, \sqrt {3} x^{4} - 99} \,d x } \] Input:
integrate((264*6^(1/2)*x-528*3^(1/2)*x^3+24*2^(1/2)*x^5-96*x^7+60*2^(1/2)* x^9-24*x^11)/(99+6*3^(1/2)*x^4-6*x^6*6^(1/2)+(-4+3*3^(1/2))*x^8+8*2^(1/2)* x^10-12*x^12+4*2^(1/2)*x^14-x^16),x, algorithm="giac")
Output:
undef
Timed out. \[ \int \frac {264 \sqrt {6} x-528 \sqrt {3} x^3+24 \sqrt {2} x^5-96 x^7+60 \sqrt {2} x^9-24 x^{11}}{99+6 \sqrt {3} x^4-6 \sqrt {6} x^6+\left (-4+3 \sqrt {3}\right ) x^8+8 \sqrt {2} x^{10}-12 x^{12}+4 \sqrt {2} x^{14}-x^{16}} \, dx=\text {Hanged} \] Input:
int((264*6^(1/2)*x - 528*3^(1/2)*x^3 + 24*2^(1/2)*x^5 + 60*2^(1/2)*x^9 - 9 6*x^7 - 24*x^11)/(x^8*(3*3^(1/2) - 4) + 6*3^(1/2)*x^4 + 8*2^(1/2)*x^10 - 6 *6^(1/2)*x^6 + 4*2^(1/2)*x^14 - 12*x^12 - x^16 + 99),x)
Output:
\text{Hanged}
\[ \int \frac {264 \sqrt {6} x-528 \sqrt {3} x^3+24 \sqrt {2} x^5-96 x^7+60 \sqrt {2} x^9-24 x^{11}}{99+6 \sqrt {3} x^4-6 \sqrt {6} x^6+\left (-4+3 \sqrt {3}\right ) x^8+8 \sqrt {2} x^{10}-12 x^{12}+4 \sqrt {2} x^{14}-x^{16}} \, dx=\int \frac {264 \sqrt {6}\, x -528 \sqrt {3}\, x^{3}+24 \sqrt {2}\, x^{5}-96 x^{7}+60 \sqrt {2}\, x^{9}-24 x^{11}}{99+6 \sqrt {3}\, x^{4}-6 x^{6} \sqrt {6}+\left (-4+3 \sqrt {3}\right ) x^{8}+8 \sqrt {2}\, x^{10}-12 x^{12}+4 \sqrt {2}\, x^{14}-x^{16}}d x \] Input:
int((264*6^(1/2)*x-528*3^(1/2)*x^3+24*2^(1/2)*x^5-96*x^7+60*2^(1/2)*x^9-24 *x^11)/(99+6*3^(1/2)*x^4-6*x^6*6^(1/2)+(-4+3*3^(1/2))*x^8+8*2^(1/2)*x^10-1 2*x^12+4*2^(1/2)*x^14-x^16),x)
Output:
int((264*6^(1/2)*x-528*3^(1/2)*x^3+24*2^(1/2)*x^5-96*x^7+60*2^(1/2)*x^9-24 *x^11)/(99+6*3^(1/2)*x^4-6*x^6*6^(1/2)+(-4+3*3^(1/2))*x^8+8*2^(1/2)*x^10-1 2*x^12+4*2^(1/2)*x^14-x^16),x)