Integrand size = 77, antiderivative size = 82 \[ \int \frac {40 \left (-12+8 \sqrt {15}\right ) x-5040 x^3+4200 \sqrt {15} x^5}{4+\left (560-360 \sqrt {15}\right ) x^2+5 \left (296-80 \sqrt {15}\right ) x^4-2100 \left (2+\sqrt {15}\right ) x^6+11025 x^8} \, dx=\sqrt [4]{\frac {3}{5}} \arctan \left (\sqrt {-\frac {1}{2}+\frac {2}{\sqrt {15}}}+\frac {7}{2} \sqrt [4]{\frac {5}{3}} x^2\right )+\sqrt [4]{\frac {3}{5}} \text {arctanh}\left (\sqrt {\frac {1}{2}+\frac {2}{\sqrt {15}}}-\frac {1}{2} 3^{3/4} \sqrt [4]{5} x^2\right ) \] Output:
1/5*3^(1/4)*5^(3/4)*arctan(1/30*(-450+120*15^(1/2))^(1/2)+7/6*5^(1/4)*3^(3 /4)*x^2)-1/5*3^(1/4)*5^(3/4)*arctanh(-1/30*(450+120*15^(1/2))^(1/2)+1/2*5^ (1/4)*3^(3/4)*x^2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.18 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.09 \[ \int \frac {40 \left (-12+8 \sqrt {15}\right ) x-5040 x^3+4200 \sqrt {15} x^5}{4+\left (560-360 \sqrt {15}\right ) x^2+5 \left (296-80 \sqrt {15}\right ) x^4-2100 \left (2+\sqrt {15}\right ) x^6+11025 x^8} \, dx=-\text {RootSum}\left [4+560 \text {$\#$1}^2-360 \sqrt {15} \text {$\#$1}^2+1480 \text {$\#$1}^4-400 \sqrt {15} \text {$\#$1}^4-4200 \text {$\#$1}^6-2100 \sqrt {15} \text {$\#$1}^6+11025 \text {$\#$1}^8\&,\frac {-12 \log (x-\text {$\#$1})+8 \sqrt {15} \log (x-\text {$\#$1})-126 \log (x-\text {$\#$1}) \text {$\#$1}^2+105 \sqrt {15} \log (x-\text {$\#$1}) \text {$\#$1}^4}{-28+18 \sqrt {15}-148 \text {$\#$1}^2+40 \sqrt {15} \text {$\#$1}^2+630 \text {$\#$1}^4+315 \sqrt {15} \text {$\#$1}^4-2205 \text {$\#$1}^6}\&\right ] \] Input:
Integrate[(40*(-12 + 8*Sqrt[15])*x - 5040*x^3 + 4200*Sqrt[15]*x^5)/(4 + (5 60 - 360*Sqrt[15])*x^2 + 5*(296 - 80*Sqrt[15])*x^4 - 2100*(2 + Sqrt[15])*x ^6 + 11025*x^8),x]
Output:
-RootSum[4 + 560*#1^2 - 360*Sqrt[15]*#1^2 + 1480*#1^4 - 400*Sqrt[15]*#1^4 - 4200*#1^6 - 2100*Sqrt[15]*#1^6 + 11025*#1^8 & , (-12*Log[x - #1] + 8*Sqr t[15]*Log[x - #1] - 126*Log[x - #1]*#1^2 + 105*Sqrt[15]*Log[x - #1]*#1^4)/ (-28 + 18*Sqrt[15] - 148*#1^2 + 40*Sqrt[15]*#1^2 + 630*#1^4 + 315*Sqrt[15] *#1^4 - 2205*#1^6) & ]
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.
| \(\displaystyle \int \frac {4200 \sqrt {15} x^5-5040 x^3+40 \left (8 \sqrt {15}-12\right ) x}{11025 x^8-2100 \left (2+\sqrt {15}\right ) x^6+5 \left (296-80 \sqrt {15}\right ) x^4+\left (560-360 \sqrt {15}\right ) x^2+4} \, dx\) |
| \(\Big \downarrow \) 2028 |
| \(\displaystyle \int \frac {x \left (4200 \sqrt {15} x^4-5040 x^2+40 \left (8 \sqrt {15}-12\right )\right )}{11025 x^8-2100 \left (2+\sqrt {15}\right ) x^6+5 \left (296-80 \sqrt {15}\right ) x^4+\left (560-360 \sqrt {15}\right ) x^2+4}dx\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle \frac {1}{2} \int -\frac {40 \left (-105 \sqrt {15} x^4+126 x^2+4 \left (3-2 \sqrt {15}\right )\right )}{11025 x^8-2100 \left (2+\sqrt {15}\right ) x^6+40 \left (37-10 \sqrt {15}\right ) x^4+40 \left (14-9 \sqrt {15}\right ) x^2+4}dx^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -20 \int \frac {-105 \sqrt {15} x^4+126 x^2+4 \left (3-2 \sqrt {15}\right )}{11025 x^8-2100 \left (2+\sqrt {15}\right ) x^6+40 \left (37-10 \sqrt {15}\right ) x^4+40 \left (14-9 \sqrt {15}\right ) x^2+4}dx^2\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -20 \int \left (\frac {105 \sqrt {15} x^4}{-11025 x^8+2100 \left (2+\sqrt {15}\right ) x^6-40 \left (37-10 \sqrt {15}\right ) x^4-40 \left (14-9 \sqrt {15}\right ) x^2-4}+\frac {126 x^2}{11025 x^8-2100 \left (2+\sqrt {15}\right ) x^6+40 \left (37-10 \sqrt {15}\right ) x^4+40 \left (14-9 \sqrt {15}\right ) x^2+4}+\frac {4 \left (3-2 \sqrt {15}\right )}{11025 x^8-2100 \left (2+\sqrt {15}\right ) x^6+40 \left (37-10 \sqrt {15}\right ) x^4+40 \left (14-9 \sqrt {15}\right ) x^2+4}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -20 \left (105 \sqrt {15} \int \frac {x^4}{-11025 x^8+2100 \left (2+\sqrt {15}\right ) x^6-40 \left (37-10 \sqrt {15}\right ) x^4-40 \left (14-9 \sqrt {15}\right ) x^2-4}dx^2+4 \left (3-2 \sqrt {15}\right ) \int \frac {1}{11025 x^8-2100 \left (2+\sqrt {15}\right ) x^6+40 \left (37-10 \sqrt {15}\right ) x^4+40 \left (14-9 \sqrt {15}\right ) x^2+4}dx^2+126 \int \frac {x^2}{11025 x^8-2100 \left (2+\sqrt {15}\right ) x^6+40 \left (37-10 \sqrt {15}\right ) x^4+40 \left (14-9 \sqrt {15}\right ) x^2+4}dx^2\right )\) |
Input:
Int[(40*(-12 + 8*Sqrt[15])*x - 5040*x^3 + 4200*Sqrt[15]*x^5)/(4 + (560 - 3 60*Sqrt[15])*x^2 + 5*(296 - 80*Sqrt[15])*x^4 - 2100*(2 + Sqrt[15])*x^6 + 1 1025*x^8),x]
Output:
$Aborted
Time = 0.43 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.73
| method | result | size |
| default | \(-\frac {15^{\frac {1}{4}} \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (90 x^{2}-6 \sqrt {15}-30\right ) \sqrt {5}\, 15^{\frac {3}{4}}}{900}\right )}{5}+\frac {15^{\frac {1}{4}} \sqrt {5}\, \arctan \left (\frac {\left (490 x^{2}-14 \sqrt {15}+70\right ) \sqrt {5}\, 15^{\frac {3}{4}}}{2100}\right )}{5}\) | \(60\) |
Input:
int((40*(-12+8*15^(1/2))*x-5040*x^3+4200*15^(1/2)*x^5)/(4+(560-360*15^(1/2 ))*x^2+5*(296-80*15^(1/2))*x^4-2100*(2+15^(1/2))*x^6+11025*x^8),x,method=_ RETURNVERBOSE)
Output:
-1/5*15^(1/4)*5^(1/2)*arctanh(1/900*(90*x^2-6*15^(1/2)-30)*5^(1/2)*15^(3/4 ))+1/5*15^(1/4)*5^(1/2)*arctan(1/2100*(490*x^2-14*15^(1/2)+70)*5^(1/2)*15^ (3/4))
Time = 0.09 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.39 \[ \int \frac {40 \left (-12+8 \sqrt {15}\right ) x-5040 x^3+4200 \sqrt {15} x^5}{4+\left (560-360 \sqrt {15}\right ) x^2+5 \left (296-80 \sqrt {15}\right ) x^4-2100 \left (2+\sqrt {15}\right ) x^6+11025 x^8} \, dx=\left (\frac {3}{5}\right )^{\frac {1}{4}} \arctan \left (\frac {5}{6} \, \sqrt {15} \left (\frac {3}{5}\right )^{\frac {1}{4}} x^{2} + \frac {5}{6} \, \left (\frac {3}{5}\right )^{\frac {3}{4}} {\left (2 \, x^{2} + 1\right )} - \frac {1}{2} \, \left (\frac {3}{5}\right )^{\frac {1}{4}}\right ) + \frac {1}{2} \, \left (\frac {3}{5}\right )^{\frac {1}{4}} \log \left (63 \, x^{2} - \sqrt {15} {\left (10 \, \left (\frac {3}{5}\right )^{\frac {3}{4}} + 5 \, \sqrt {\frac {3}{5}} + 3\right )} - 6 \, \sqrt {\frac {3}{5}} - 12 \, \left (\frac {3}{5}\right )^{\frac {1}{4}} - 6\right ) - \frac {1}{2} \, \left (\frac {3}{5}\right )^{\frac {1}{4}} \log \left (63 \, x^{2} + \sqrt {15} {\left (10 \, \left (\frac {3}{5}\right )^{\frac {3}{4}} - 5 \, \sqrt {\frac {3}{5}} - 3\right )} - 6 \, \sqrt {\frac {3}{5}} + 12 \, \left (\frac {3}{5}\right )^{\frac {1}{4}} - 6\right ) \] Input:
integrate((40*(-12+8*15^(1/2))*x-5040*x^3+4200*15^(1/2)*x^5)/(4+(560-360*1 5^(1/2))*x^2+5*(296-80*15^(1/2))*x^4-2100*(2+15^(1/2))*x^6+11025*x^8),x, a lgorithm="fricas")
Output:
(3/5)^(1/4)*arctan(5/6*sqrt(15)*(3/5)^(1/4)*x^2 + 5/6*(3/5)^(3/4)*(2*x^2 + 1) - 1/2*(3/5)^(1/4)) + 1/2*(3/5)^(1/4)*log(63*x^2 - sqrt(15)*(10*(3/5)^( 3/4) + 5*sqrt(3/5) + 3) - 6*sqrt(3/5) - 12*(3/5)^(1/4) - 6) - 1/2*(3/5)^(1 /4)*log(63*x^2 + sqrt(15)*(10*(3/5)^(3/4) - 5*sqrt(3/5) - 3) - 6*sqrt(3/5) + 12*(3/5)^(1/4) - 6)
Timed out. \[ \int \frac {40 \left (-12+8 \sqrt {15}\right ) x-5040 x^3+4200 \sqrt {15} x^5}{4+\left (560-360 \sqrt {15}\right ) x^2+5 \left (296-80 \sqrt {15}\right ) x^4-2100 \left (2+\sqrt {15}\right ) x^6+11025 x^8} \, dx=\text {Timed out} \] Input:
integrate((40*(-12+8*15**(1/2))*x-5040*x**3+4200*15**(1/2)*x**5)/(4+(560-3 60*15**(1/2))*x**2+5*(296-80*15**(1/2))*x**4-2100*(2+15**(1/2))*x**6+11025 *x**8),x)
Output:
Timed out
\[ \int \frac {40 \left (-12+8 \sqrt {15}\right ) x-5040 x^3+4200 \sqrt {15} x^5}{4+\left (560-360 \sqrt {15}\right ) x^2+5 \left (296-80 \sqrt {15}\right ) x^4-2100 \left (2+\sqrt {15}\right ) x^6+11025 x^8} \, dx=\int { \frac {40 \, {\left (105 \, \sqrt {15} x^{5} - 126 \, x^{3} + 4 \, x {\left (2 \, \sqrt {15} - 3\right )}\right )}}{11025 \, x^{8} - 2100 \, x^{6} {\left (\sqrt {15} + 2\right )} - 40 \, x^{4} {\left (10 \, \sqrt {15} - 37\right )} - 40 \, x^{2} {\left (9 \, \sqrt {15} - 14\right )} + 4} \,d x } \] Input:
integrate((40*(-12+8*15^(1/2))*x-5040*x^3+4200*15^(1/2)*x^5)/(4+(560-360*1 5^(1/2))*x^2+5*(296-80*15^(1/2))*x^4-2100*(2+15^(1/2))*x^6+11025*x^8),x, a lgorithm="maxima")
Output:
40*integrate((105*sqrt(15)*x^5 - 126*x^3 + 4*x*(2*sqrt(15) - 3))/(11025*x^ 8 - 2100*x^6*(sqrt(15) + 2) - 40*x^4*(10*sqrt(15) - 37) - 40*x^2*(9*sqrt(1 5) - 14) + 4), x)
Timed out. \[ \int \frac {40 \left (-12+8 \sqrt {15}\right ) x-5040 x^3+4200 \sqrt {15} x^5}{4+\left (560-360 \sqrt {15}\right ) x^2+5 \left (296-80 \sqrt {15}\right ) x^4-2100 \left (2+\sqrt {15}\right ) x^6+11025 x^8} \, dx=\text {Timed out} \] Input:
integrate((40*(-12+8*15^(1/2))*x-5040*x^3+4200*15^(1/2)*x^5)/(4+(560-360*1 5^(1/2))*x^2+5*(296-80*15^(1/2))*x^4-2100*(2+15^(1/2))*x^6+11025*x^8),x, a lgorithm="giac")
Output:
Timed out
Time = 11.62 (sec) , antiderivative size = 761, normalized size of antiderivative = 9.28 \[ \int \frac {40 \left (-12+8 \sqrt {15}\right ) x-5040 x^3+4200 \sqrt {15} x^5}{4+\left (560-360 \sqrt {15}\right ) x^2+5 \left (296-80 \sqrt {15}\right ) x^4-2100 \left (2+\sqrt {15}\right ) x^6+11025 x^8} \, dx=\text {Too large to display} \] Input:
int(-(4200*15^(1/2)*x^5 - 5040*x^3 + 40*x*(8*15^(1/2) - 12))/(5*x^4*(80*15 ^(1/2) - 296) + x^2*(360*15^(1/2) - 560) - 11025*x^8 + 2100*x^6*(15^(1/2) + 2) - 4),x)
Output:
symsum(log(1272777503760*15^(1/2)*root(134752761600*15^(1/2)*z^4 - 7379679 45840*z^4 - 5053228560*15^(1/2) + 27673797969, z, k) - 12652154396784*root (134752761600*15^(1/2)*z^4 - 737967945840*z^4 - 5053228560*15^(1/2) + 2767 3797969, z, k) + 47072975040*15^(1/2) - 1245012050400*15^(1/2)*root(134752 761600*15^(1/2)*z^4 - 737967945840*z^4 - 5053228560*15^(1/2) + 27673797969 , z, k)^2 + 1261817605440*15^(1/2)*root(134752761600*15^(1/2)*z^4 - 737967 945840*z^4 - 5053228560*15^(1/2) + 27673797969, z, k)^3 - 16342805423680*1 5^(1/2)*root(134752761600*15^(1/2)*z^4 - 737967945840*z^4 - 5053228560*15^ (1/2) + 27673797969, z, k)^4 - 24703130457600*15^(1/2)*root(134752761600*1 5^(1/2)*z^4 - 737967945840*z^4 - 5053228560*15^(1/2) + 27673797969, z, k)^ 5 + 6416990566400*15^(1/2)*root(134752761600*15^(1/2)*z^4 - 737967945840*z ^4 - 5053228560*15^(1/2) + 27673797969, z, k)^6 - 2966711800400910*root(13 4752761600*15^(1/2)*z^4 - 737967945840*z^4 - 5053228560*15^(1/2) + 2767379 7969, z, k)*x^2 - 282772703535840*15^(1/2)*x^2 + 39784538365926*root(13475 2761600*15^(1/2)*z^4 - 737967945840*z^4 - 5053228560*15^(1/2) + 2767379796 9, z, k)^2 - 147474447170400*root(134752761600*15^(1/2)*z^4 - 737967945840 *z^4 - 5053228560*15^(1/2) + 27673797969, z, k)^3 + 259431497859840*root(1 34752761600*15^(1/2)*z^4 - 737967945840*z^4 - 5053228560*15^(1/2) + 276737 97969, z, k)^4 - 399257716838400*root(134752761600*15^(1/2)*z^4 - 73796794 5840*z^4 - 5053228560*15^(1/2) + 27673797969, z, k)^5 + 843958447014400...
\[ \int \frac {40 \left (-12+8 \sqrt {15}\right ) x-5040 x^3+4200 \sqrt {15} x^5}{4+\left (560-360 \sqrt {15}\right ) x^2+5 \left (296-80 \sqrt {15}\right ) x^4-2100 \left (2+\sqrt {15}\right ) x^6+11025 x^8} \, dx =\text {Too large to display} \] Input:
int((40*(-12+8*15^(1/2))*x-5040*x^3+4200*15^(1/2)*x^5)/(4+(560-360*15^(1/2 ))*x^2+5*(296-80*15^(1/2))*x^4-2100*(2+15^(1/2))*x^6+11025*x^8),x)
Output:
40*(1157625*sqrt(15)*int(x**13/(121550625*x**16 - 92610000*x**14 - 1587600 0*x**12 - 25284000*x**10 - 27505400*x**8 - 2696000*x**6 - 1618560*x**4 + 4 480*x**2 + 16),x) - 441000*sqrt(15)*int(x**11/(121550625*x**16 - 92610000* x**14 - 15876000*x**12 - 25284000*x**10 - 27505400*x**8 - 2696000*x**6 - 1 618560*x**4 + 4480*x**2 + 16),x) - 21000*sqrt(15)*int(x**9/(121550625*x**1 6 - 92610000*x**14 - 15876000*x**12 - 25284000*x**10 - 27505400*x**8 - 269 6000*x**6 - 1618560*x**4 + 4480*x**2 + 16),x) - 50400*sqrt(15)*int(x**7/(1 21550625*x**16 - 92610000*x**14 - 15876000*x**12 - 25284000*x**10 - 275054 00*x**8 - 2696000*x**6 - 1618560*x**4 + 4480*x**2 + 16),x) - 37900*sqrt(15 )*int(x**5/(121550625*x**16 - 92610000*x**14 - 15876000*x**12 - 25284000*x **10 - 27505400*x**8 - 2696000*x**6 - 1618560*x**4 + 4480*x**2 + 16),x) + 160*sqrt(15)*int(x**3/(121550625*x**16 - 92610000*x**14 - 15876000*x**12 - 25284000*x**10 - 27505400*x**8 - 2696000*x**6 - 1618560*x**4 + 4480*x**2 + 16),x) + 32*sqrt(15)*int(x/(121550625*x**16 - 92610000*x**14 - 15876000* x**12 - 25284000*x**10 - 27505400*x**8 - 2696000*x**6 - 1618560*x**4 + 448 0*x**2 + 16),x) + 1918350*int(x**11/(121550625*x**16 - 92610000*x**14 - 15 876000*x**12 - 25284000*x**10 - 27505400*x**8 - 2696000*x**6 - 1618560*x** 4 + 4480*x**2 + 16),x) + 1026900*int(x**9/(121550625*x**16 - 92610000*x**1 4 - 15876000*x**12 - 25284000*x**10 - 27505400*x**8 - 2696000*x**6 - 16185 60*x**4 + 4480*x**2 + 16),x) + 682920*int(x**7/(121550625*x**16 - 92610...