Integrand size = 93, antiderivative size = 191 \[ \int \frac {\sqrt {5} \left (-24-2 \sqrt {21}\right )+\sqrt {5} \left (-105+15 \sqrt {21}\right ) x^2}{-16+\left (-48 \sqrt {5}+4 \sqrt {105}\right ) x+\left (50+60 \sqrt {21}\right ) x^2+\left (-210 \sqrt {5}-30 \sqrt {105}\right ) x^3+525 x^4} \, dx=-\frac {1}{2} \sqrt {5} \text {RootSum}\left [16+48 \sqrt {5} \text {$\#$1}-4 \sqrt {105} \text {$\#$1}-50 \text {$\#$1}^2-60 \sqrt {21} \text {$\#$1}^2+210 \sqrt {5} \text {$\#$1}^3+30 \sqrt {105} \text {$\#$1}^3-525 \text {$\#$1}^4\&,\frac {-24 \log (x-\text {$\#$1})-2 \sqrt {21} \log (x-\text {$\#$1})-105 \log (x-\text {$\#$1}) \text {$\#$1}^2+15 \sqrt {21} \log (x-\text {$\#$1}) \text {$\#$1}^2}{24 \sqrt {5}-2 \sqrt {105}-50 \text {$\#$1}-60 \sqrt {21} \text {$\#$1}+315 \sqrt {5} \text {$\#$1}^2+45 \sqrt {105} \text {$\#$1}^2-1050 \text {$\#$1}^3}\&\right ] \] Output:
-1/2*5^(1/2)*RootSum(_Z1 -> 16+48*5^(1/2)*_Z1-4*105^(1/2)*_Z1-50*_Z1^2-60* 21^(1/2)*_Z1^2+210*5^(1/2)*_Z1^3+30*105^(1/2)*_Z1^3-525*_Z1^4,_Z1 -> (-24* ln(x-_Z1)-2*21^(1/2)*ln(x-_Z1)-105*ln(x-_Z1)*_Z1^2+15*21^(1/2)*ln(x-_Z1)*_ Z1^2)/(24*5^(1/2)-2*105^(1/2)-50*_Z1-60*21^(1/2)*_Z1+315*5^(1/2)*_Z1^2+45* 105^(1/2)*_Z1^2-1050*_Z1^3))
Time = 0.17 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {5} \left (-24-2 \sqrt {21}\right )+\sqrt {5} \left (-105+15 \sqrt {21}\right ) x^2}{-16+\left (-48 \sqrt {5}+4 \sqrt {105}\right ) x+\left (50+60 \sqrt {21}\right ) x^2+\left (-210 \sqrt {5}-30 \sqrt {105}\right ) x^3+525 x^4} \, dx=-\frac {1}{2} \sqrt {5} \text {RootSum}\left [16+48 \sqrt {5} \text {$\#$1}-4 \sqrt {105} \text {$\#$1}-50 \text {$\#$1}^2-60 \sqrt {21} \text {$\#$1}^2+210 \sqrt {5} \text {$\#$1}^3+30 \sqrt {105} \text {$\#$1}^3-525 \text {$\#$1}^4\&,\frac {-24 \log (x-\text {$\#$1})-2 \sqrt {21} \log (x-\text {$\#$1})-105 \log (x-\text {$\#$1}) \text {$\#$1}^2+15 \sqrt {21} \log (x-\text {$\#$1}) \text {$\#$1}^2}{24 \sqrt {5}-2 \sqrt {105}-50 \text {$\#$1}-60 \sqrt {21} \text {$\#$1}+315 \sqrt {5} \text {$\#$1}^2+45 \sqrt {105} \text {$\#$1}^2-1050 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[(Sqrt[5]*(-24 - 2*Sqrt[21]) + Sqrt[5]*(-105 + 15*Sqrt[21])*x^2)/ (-16 + (-48*Sqrt[5] + 4*Sqrt[105])*x + (50 + 60*Sqrt[21])*x^2 + (-210*Sqrt [5] - 30*Sqrt[105])*x^3 + 525*x^4),x]
Output:
-1/2*(Sqrt[5]*RootSum[16 + 48*Sqrt[5]*#1 - 4*Sqrt[105]*#1 - 50*#1^2 - 60*S qrt[21]*#1^2 + 210*Sqrt[5]*#1^3 + 30*Sqrt[105]*#1^3 - 525*#1^4 & , (-24*Lo g[x - #1] - 2*Sqrt[21]*Log[x - #1] - 105*Log[x - #1]*#1^2 + 15*Sqrt[21]*Lo g[x - #1]*#1^2)/(24*Sqrt[5] - 2*Sqrt[105] - 50*#1 - 60*Sqrt[21]*#1 + 315*S qrt[5]*#1^2 + 45*Sqrt[105]*#1^2 - 1050*#1^3) & ])
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.
| \(\displaystyle \int \frac {\sqrt {5} \left (15 \sqrt {21}-105\right ) x^2+\sqrt {5} \left (-24-2 \sqrt {21}\right )}{525 x^4+\left (-210 \sqrt {5}-30 \sqrt {105}\right ) x^3+\left (50+60 \sqrt {21}\right ) x^2+\left (4 \sqrt {105}-48 \sqrt {5}\right ) x-16} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-\sqrt {5} \left (15 \sqrt {21}-105\right ) x^2-\sqrt {5} \left (-24-2 \sqrt {21}\right )}{-525 x^4+30 \sqrt {35} \left (\sqrt {3}+\sqrt {7}\right ) x^3-10 \left (5+6 \sqrt {21}\right ) x^2+4 \sqrt {15} \left (4 \sqrt {3}-\sqrt {7}\right ) x+16}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-\sqrt {5} \left (15 \sqrt {21}-105\right ) x^2-\sqrt {5} \left (-24-2 \sqrt {21}\right )}{-525 x^4+30 \sqrt {35} \left (\sqrt {3}+\sqrt {7}\right ) x^3-10 \left (5+6 \sqrt {21}\right ) x^2+4 \sqrt {5} \left (12-\sqrt {21}\right ) x+16}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {15 \sqrt {5} \left (7-\sqrt {21}\right ) x^2}{-525 x^4+30 \sqrt {35} \left (\sqrt {3}+\sqrt {7}\right ) x^3-10 \left (5+6 \sqrt {21}\right ) x^2+4 \sqrt {5} \left (12-\sqrt {21}\right ) x+16}+\frac {2 \sqrt {5} \left (12+\sqrt {21}\right )}{-525 x^4+30 \sqrt {35} \left (\sqrt {3}+\sqrt {7}\right ) x^3-10 \left (5+6 \sqrt {21}\right ) x^2+4 \sqrt {5} \left (12-\sqrt {21}\right ) x+16}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \sqrt {15} \left (4 \sqrt {3}+\sqrt {7}\right ) \int \frac {1}{-525 x^4+30 \sqrt {35} \left (\sqrt {3}+\sqrt {7}\right ) x^3-10 \left (5+6 \sqrt {21}\right ) x^2+4 \sqrt {5} \left (12-\sqrt {21}\right ) x+16}dx-15 \sqrt {35} \left (\sqrt {3}-\sqrt {7}\right ) \int \frac {x^2}{-525 x^4+30 \sqrt {35} \left (\sqrt {3}+\sqrt {7}\right ) x^3-10 \left (5+6 \sqrt {21}\right ) x^2+4 \sqrt {5} \left (12-\sqrt {21}\right ) x+16}dx\) |
Input:
Int[(Sqrt[5]*(-24 - 2*Sqrt[21]) + Sqrt[5]*(-105 + 15*Sqrt[21])*x^2)/(-16 + (-48*Sqrt[5] + 4*Sqrt[105])*x + (50 + 60*Sqrt[21])*x^2 + (-210*Sqrt[5] - 30*Sqrt[105])*x^3 + 525*x^4),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs. \(2(14)=28\).
Time = 0.60 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.27
| method | result | size |
| default | \(\sqrt {5}\, \left (\frac {\sqrt {21}\, \sqrt {7}\, \arctan \left (\frac {\left (-2 \sqrt {5}\, \sqrt {21}+70 x \right ) \sqrt {7}}{70}\right )}{35}+\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (-6 \sqrt {5}+30 x \right ) \sqrt {3}}{30}\right )}{5}\right )\) | \(52\) |
| risch | \(\frac {\sqrt {15}\, \ln \left (\sqrt {5}\, \sqrt {15}-3 \sqrt {5}+15 x \right )}{10}-\frac {\sqrt {15}\, \ln \left (-\sqrt {5}\, \sqrt {15}-3 \sqrt {5}+15 x \right )}{10}+\frac {\sqrt {15}\, \arctan \left (\frac {\left (-3 \sqrt {5}\, \sqrt {21}+105 x \right ) \sqrt {5}\, \sqrt {21}\, \sqrt {15}}{1575}\right )}{5}\) | \(76\) |
Input:
int((5^(1/2)*(-24-2*21^(1/2))+5^(1/2)*(-105+15*21^(1/2))*x^2)/(-16+(-48*5^ (1/2)+4*105^(1/2))*x+(50+60*21^(1/2))*x^2+(-210*5^(1/2)-30*105^(1/2))*x^3+ 525*x^4),x,method=_RETURNVERBOSE)
Output:
5^(1/2)*(1/35*21^(1/2)*7^(1/2)*arctan(1/70*(-2*5^(1/2)*21^(1/2)+70*x)*7^(1 /2))+1/5*3^(1/2)*arctanh(1/30*(-6*5^(1/2)+30*x)*3^(1/2)))
Timed out. \[ \int \frac {\sqrt {5} \left (-24-2 \sqrt {21}\right )+\sqrt {5} \left (-105+15 \sqrt {21}\right ) x^2}{-16+\left (-48 \sqrt {5}+4 \sqrt {105}\right ) x+\left (50+60 \sqrt {21}\right ) x^2+\left (-210 \sqrt {5}-30 \sqrt {105}\right ) x^3+525 x^4} \, dx=\text {Timed out} \] Input:
integrate((5^(1/2)*(-24-2*21^(1/2))+5^(1/2)*(-105+15*21^(1/2))*x^2)/(-16+( -48*5^(1/2)+4*105^(1/2))*x+(50+60*21^(1/2))*x^2+(-210*5^(1/2)-30*105^(1/2) )*x^3+525*x^4),x, algorithm="fricas")
Output:
Timed out
Exception generated. \[ \int \frac {\sqrt {5} \left (-24-2 \sqrt {21}\right )+\sqrt {5} \left (-105+15 \sqrt {21}\right ) x^2}{-16+\left (-48 \sqrt {5}+4 \sqrt {105}\right ) x+\left (50+60 \sqrt {21}\right ) x^2+\left (-210 \sqrt {5}-30 \sqrt {105}\right ) x^3+525 x^4} \, dx=\text {Exception raised: PolynomialError} \] Input:
integrate((5**(1/2)*(-24-2*21**(1/2))+5**(1/2)*(-105+15*21**(1/2))*x**2)/( -16+(-48*5**(1/2)+4*105**(1/2))*x+(50+60*21**(1/2))*x**2+(-210*5**(1/2)-30 *105**(1/2))*x**3+525*x**4),x)
Output:
Exception raised: PolynomialError >> 1/(-11312*_t**4 + 1680*sqrt(21)*_t**4 - 3360*_t**3 + 288*sqrt(21)*_t**3 - 600*sqrt(21)*_t**2 + 2568*_t**2 - 72* sqrt(21)*_t + 360*_t - 207 + 45*sqrt(21)) contains an element of the set o f generators.
\[ \int \frac {\sqrt {5} \left (-24-2 \sqrt {21}\right )+\sqrt {5} \left (-105+15 \sqrt {21}\right ) x^2}{-16+\left (-48 \sqrt {5}+4 \sqrt {105}\right ) x+\left (50+60 \sqrt {21}\right ) x^2+\left (-210 \sqrt {5}-30 \sqrt {105}\right ) x^3+525 x^4} \, dx=\int { \frac {15 \, \sqrt {5} x^{2} {\left (\sqrt {21} - 7\right )} - 2 \, \sqrt {5} {\left (\sqrt {21} + 12\right )}}{525 \, x^{4} - 30 \, x^{3} {\left (\sqrt {105} + 7 \, \sqrt {5}\right )} + 10 \, x^{2} {\left (6 \, \sqrt {21} + 5\right )} + 4 \, x {\left (\sqrt {105} - 12 \, \sqrt {5}\right )} - 16} \,d x } \] Input:
integrate((5^(1/2)*(-24-2*21^(1/2))+5^(1/2)*(-105+15*21^(1/2))*x^2)/(-16+( -48*5^(1/2)+4*105^(1/2))*x+(50+60*21^(1/2))*x^2+(-210*5^(1/2)-30*105^(1/2) )*x^3+525*x^4),x, algorithm="maxima")
Output:
integrate((15*sqrt(5)*x^2*(sqrt(21) - 7) - 2*sqrt(5)*(sqrt(21) + 12))/(525 *x^4 - 30*x^3*(sqrt(105) + 7*sqrt(5)) + 10*x^2*(6*sqrt(21) + 5) + 4*x*(sqr t(105) - 12*sqrt(5)) - 16), x)
Default grade assigned because unable to parse optimal
Time = 0.17 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.35 \[ \int \frac {\sqrt {5} \left (-24-2 \sqrt {21}\right )+\sqrt {5} \left (-105+15 \sqrt {21}\right ) x^2}{-16+\left (-48 \sqrt {5}+4 \sqrt {105}\right ) x+\left (50+60 \sqrt {21}\right ) x^2+\left (-210 \sqrt {5}-30 \sqrt {105}\right ) x^3+525 x^4} \, dx=\frac {1}{35} \, \sqrt {105} \sqrt {7} \arctan \left (\frac {1}{35} \, \sqrt {7} {\left (35 \, x - \sqrt {105}\right )}\right ) - \frac {1}{10} \, \sqrt {5} \sqrt {3} \log \left (\frac {{\left | 3000 \, x - 600 \, \sqrt {5} - 1000 \, \sqrt {3} \right |}}{{\left | 3000 \, x - 600 \, \sqrt {5} + 1000 \, \sqrt {3} \right |}}\right ) \] Input:
integrate((5^(1/2)*(-24-2*21^(1/2))+5^(1/2)*(-105+15*21^(1/2))*x^2)/(-16+( -48*5^(1/2)+4*105^(1/2))*x+(50+60*21^(1/2))*x^2+(-210*5^(1/2)-30*105^(1/2) )*x^3+525*x^4),x, algorithm="giac")
Output:
1/35*sqrt(105)*sqrt(7)*arctan(1/35*sqrt(7)*(35*x - sqrt(105))) - 1/10*sqrt (5)*sqrt(3)*log(abs(3000*x - 600*sqrt(5) - 1000*sqrt(3))/abs(3000*x - 600* sqrt(5) + 1000*sqrt(3)))
Time = 13.03 (sec) , antiderivative size = 1659, normalized size of antiderivative = 8.69 \[ \int \frac {\sqrt {5} \left (-24-2 \sqrt {21}\right )+\sqrt {5} \left (-105+15 \sqrt {21}\right ) x^2}{-16+\left (-48 \sqrt {5}+4 \sqrt {105}\right ) x+\left (50+60 \sqrt {21}\right ) x^2+\left (-210 \sqrt {5}-30 \sqrt {105}\right ) x^3+525 x^4} \, dx=\text {Too large to display} \] Input:
int((5^(1/2)*(2*21^(1/2) + 24) - 5^(1/2)*x^2*(15*21^(1/2) - 105))/(x^3*(21 0*5^(1/2) + 30*105^(1/2)) - x^2*(60*21^(1/2) + 50) + x*(48*5^(1/2) - 4*105 ^(1/2)) - 525*x^4 + 16),x)
Output:
symsum(log((4232*5^(1/2)*root(4126651200*5^(1/2)*21^(1/2)*105^(1/2)*z^4 + 11534040000*5^(1/2)*105^(1/2)*z^4 - 42151200000*21^(1/2)*z^4 - 63105572600 0*z^4 - 368373600*5^(1/2)*21^(1/2)*105^(1/2)*z^2 + 768474000*5^(1/2)*105^( 1/2)*z^2 - 3842370000*21^(1/2)*z^2 + 38679228000*z^2 + 1508220000*5^(1/2)* 105^(1/2)*z - 7541100000*21^(1/2)*z - 20468700*5^(1/2)*21^(1/2)*105^(1/2) - 103950000*5^(1/2)*105^(1/2) + 170572500*21^(1/2) + 6598753875, z, k))/64 3125 - (304*x)/6125 + (5192*105^(1/2)*root(4126651200*5^(1/2)*21^(1/2)*105 ^(1/2)*z^4 + 11534040000*5^(1/2)*105^(1/2)*z^4 - 42151200000*21^(1/2)*z^4 - 631055726000*z^4 - 368373600*5^(1/2)*21^(1/2)*105^(1/2)*z^2 + 768474000* 5^(1/2)*105^(1/2)*z^2 - 3842370000*21^(1/2)*z^2 + 38679228000*z^2 + 150822 0000*5^(1/2)*105^(1/2)*z - 7541100000*21^(1/2)*z - 20468700*5^(1/2)*21^(1/ 2)*105^(1/2) - 103950000*5^(1/2)*105^(1/2) + 170572500*21^(1/2) + 65987538 75, z, k))/1929375 - (11824*root(4126651200*5^(1/2)*21^(1/2)*105^(1/2)*z^4 + 11534040000*5^(1/2)*105^(1/2)*z^4 - 42151200000*21^(1/2)*z^4 - 63105572 6000*z^4 - 368373600*5^(1/2)*21^(1/2)*105^(1/2)*z^2 + 768474000*5^(1/2)*10 5^(1/2)*z^2 - 3842370000*21^(1/2)*z^2 + 38679228000*z^2 + 1508220000*5^(1/ 2)*105^(1/2)*z - 7541100000*21^(1/2)*z - 20468700*5^(1/2)*21^(1/2)*105^(1/ 2) - 103950000*5^(1/2)*105^(1/2) + 170572500*21^(1/2) + 6598753875, z, k)* x)/77175 + (304*21^(1/2)*x)/25725 + (908*5^(1/2))/643125 - (428*105^(1/2)) /1929375 - (28624*5^(1/2)*root(4126651200*5^(1/2)*21^(1/2)*105^(1/2)*z^...
Time = 191.01 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.34 \[ \int \frac {\sqrt {5} \left (-24-2 \sqrt {21}\right )+\sqrt {5} \left (-105+15 \sqrt {21}\right ) x^2}{-16+\left (-48 \sqrt {5}+4 \sqrt {105}\right ) x+\left (50+60 \sqrt {21}\right ) x^2+\left (-210 \sqrt {5}-30 \sqrt {105}\right ) x^3+525 x^4} \, dx=\frac {\sqrt {15}\, \left (-4 \mathit {atan} \left (\frac {\sqrt {105}-35 x}{5 \sqrt {7}}\right )-2 \mathit {atanh} \left (\frac {15 x}{3 \sqrt {5}-5 \sqrt {3}}\right )-2 \,\mathrm {log}\left (\sqrt {15}\, x -\sqrt {5}-\sqrt {3}\right )+\mathrm {log}\left (2 \sqrt {15}+15 x^{2}-8\right )\right )}{20} \] Input:
int((5^(1/2)*(-24-2*21^(1/2))+5^(1/2)*(-105+15*21^(1/2))*x^2)/(-16+(-48*5^ (1/2)+4*105^(1/2))*x+(50+60*21^(1/2))*x^2+(-210*5^(1/2)-30*105^(1/2))*x^3+ 525*x^4),x)
Output:
(sqrt(15)*( - 4*atan((sqrt(105) - 35*x)/(5*sqrt(7))) - 2*atanh((15*x)/(3*s qrt(5) - 5*sqrt(3))) - 2*log(sqrt(15)*x - sqrt(5) - sqrt(3)) + log(2*sqrt( 15) + 15*x**2 - 8)))/20