Integrand size = 116, antiderivative size = 204 \[ \int \frac {\left (-20 \sqrt {3}+3 \sqrt {5}\right ) x+36 \sqrt {5} x^3-75 \sqrt {3} x^5}{8 \sqrt {3}+4 \sqrt {3} \left (70-90 \sqrt {15}\right ) x^2+4 \sqrt {3} \left (140-25 \sqrt {15}\right ) x^4+4 \sqrt {3} \left (-150-150 \sqrt {15}\right ) x^6+1800 \sqrt {3} x^8} \, dx=-\frac {1}{80} \text {RootSum}\left [2+70 \text {$\#$1}^2-90 \sqrt {15} \text {$\#$1}^2+140 \text {$\#$1}^4-25 \sqrt {15} \text {$\#$1}^4-150 \text {$\#$1}^6-150 \sqrt {15} \text {$\#$1}^6+450 \text {$\#$1}^8\&,\frac {20 \sqrt {3} \log (x-\text {$\#$1})-3 \sqrt {5} \log (x-\text {$\#$1})-36 \sqrt {5} \log (x-\text {$\#$1}) \text {$\#$1}^2+75 \sqrt {3} \log (x-\text {$\#$1}) \text {$\#$1}^4}{7 \sqrt {3}-27 \sqrt {5}+28 \sqrt {3} \text {$\#$1}^2-15 \sqrt {5} \text {$\#$1}^2-45 \sqrt {3} \text {$\#$1}^4-135 \sqrt {5} \text {$\#$1}^4+180 \sqrt {3} \text {$\#$1}^6}\&\right ] \] Output:
-1/80*RootSum(_Z1 -> 2+70*_Z1^2-90*15^(1/2)*_Z1^2+140*_Z1^4-25*15^(1/2)*_Z 1^4-150*_Z1^6-150*15^(1/2)*_Z1^6+450*_Z1^8,_Z1 -> (20*3^(1/2)*ln(x-_Z1)-3* 5^(1/2)*ln(x-_Z1)-36*5^(1/2)*ln(x-_Z1)*_Z1^2+75*3^(1/2)*ln(x-_Z1)*_Z1^4)/( 7*3^(1/2)-27*5^(1/2)+28*3^(1/2)*_Z1^2-15*5^(1/2)*_Z1^2-45*3^(1/2)*_Z1^4-13 5*5^(1/2)*_Z1^4+180*3^(1/2)*_Z1^6))
Time = 0.18 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.57 \[ \int \frac {\left (-20 \sqrt {3}+3 \sqrt {5}\right ) x+36 \sqrt {5} x^3-75 \sqrt {3} x^5}{8 \sqrt {3}+4 \sqrt {3} \left (70-90 \sqrt {15}\right ) x^2+4 \sqrt {3} \left (140-25 \sqrt {15}\right ) x^4+4 \sqrt {3} \left (-150-150 \sqrt {15}\right ) x^6+1800 \sqrt {3} x^8} \, dx=-\frac {\left (20 \sqrt {3}-3 \sqrt {5}-36 \sqrt {5} x^2+75 \sqrt {3} x^4\right ) \text {RootSum}\left [2 \sqrt {3}+70 \sqrt {3} \text {$\#$1}^2-270 \sqrt {5} \text {$\#$1}^2+140 \sqrt {3} \text {$\#$1}^4-75 \sqrt {5} \text {$\#$1}^4-150 \sqrt {3} \text {$\#$1}^6-450 \sqrt {5} \text {$\#$1}^6+450 \sqrt {3} \text {$\#$1}^8\&,\frac {-75 \log (x-\text {$\#$1})+23 \sqrt {15} \log (x-\text {$\#$1})-180 \log (x-\text {$\#$1}) \text {$\#$1}^2+36 \sqrt {15} \log (x-\text {$\#$1}) \text {$\#$1}^2-225 \log (x-\text {$\#$1}) \text {$\#$1}^4+75 \sqrt {15} \log (x-\text {$\#$1}) \text {$\#$1}^4}{7 \sqrt {3}-27 \sqrt {5}+28 \sqrt {3} \text {$\#$1}^2-15 \sqrt {5} \text {$\#$1}^2-45 \sqrt {3} \text {$\#$1}^4-135 \sqrt {5} \text {$\#$1}^4+180 \sqrt {3} \text {$\#$1}^6}\&\right ]}{80 \left (-75+23 \sqrt {15}+36 \left (-5+\sqrt {15}\right ) x^2+75 \left (-3+\sqrt {15}\right ) x^4\right )} \] Input:
Integrate[((-20*Sqrt[3] + 3*Sqrt[5])*x + 36*Sqrt[5]*x^3 - 75*Sqrt[3]*x^5)/ (8*Sqrt[3] + 4*Sqrt[3]*(70 - 90*Sqrt[15])*x^2 + 4*Sqrt[3]*(140 - 25*Sqrt[1 5])*x^4 + 4*Sqrt[3]*(-150 - 150*Sqrt[15])*x^6 + 1800*Sqrt[3]*x^8),x]
Output:
-1/80*((20*Sqrt[3] - 3*Sqrt[5] - 36*Sqrt[5]*x^2 + 75*Sqrt[3]*x^4)*RootSum[ 2*Sqrt[3] + 70*Sqrt[3]*#1^2 - 270*Sqrt[5]*#1^2 + 140*Sqrt[3]*#1^4 - 75*Sqr t[5]*#1^4 - 150*Sqrt[3]*#1^6 - 450*Sqrt[5]*#1^6 + 450*Sqrt[3]*#1^8 & , (-7 5*Log[x - #1] + 23*Sqrt[15]*Log[x - #1] - 180*Log[x - #1]*#1^2 + 36*Sqrt[1 5]*Log[x - #1]*#1^2 - 225*Log[x - #1]*#1^4 + 75*Sqrt[15]*Log[x - #1]*#1^4) /(7*Sqrt[3] - 27*Sqrt[5] + 28*Sqrt[3]*#1^2 - 15*Sqrt[5]*#1^2 - 45*Sqrt[3]* #1^4 - 135*Sqrt[5]*#1^4 + 180*Sqrt[3]*#1^6) & ])/(-75 + 23*Sqrt[15] + 36*( -5 + Sqrt[15])*x^2 + 75*(-3 + Sqrt[15])*x^4)
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.
| \(\displaystyle \int \frac {-75 \sqrt {3} x^5+36 \sqrt {5} x^3+\left (3 \sqrt {5}-20 \sqrt {3}\right ) x}{1800 \sqrt {3} x^8+4 \sqrt {3} \left (-150-150 \sqrt {15}\right ) x^6+4 \sqrt {3} \left (140-25 \sqrt {15}\right ) x^4+4 \sqrt {3} \left (70-90 \sqrt {15}\right ) x^2+8 \sqrt {3}} \, dx\) |
| \(\Big \downarrow \) 2028 |
| \(\displaystyle \int \frac {x \left (-75 \sqrt {3} x^4+36 \sqrt {5} x^2+3 \sqrt {5}-20 \sqrt {3}\right )}{1800 \sqrt {3} x^8+4 \sqrt {3} \left (-150-150 \sqrt {15}\right ) x^6+4 \sqrt {3} \left (140-25 \sqrt {15}\right ) x^4+4 \sqrt {3} \left (70-90 \sqrt {15}\right ) x^2+8 \sqrt {3}}dx\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle \frac {1}{2} \int -\frac {75 \sqrt {3} x^4-36 \sqrt {5} x^2-3 \sqrt {5}+20 \sqrt {3}}{4 \left (450 \sqrt {3} x^8-150 \sqrt {3} \left (1+\sqrt {15}\right ) x^6+5 \sqrt {3} \left (28-5 \sqrt {15}\right ) x^4+10 \sqrt {3} \left (7-9 \sqrt {15}\right ) x^2+2 \sqrt {3}\right )}dx^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{8} \int \frac {75 \sqrt {3} x^4-36 \sqrt {5} x^2-3 \sqrt {5}+20 \sqrt {3}}{450 \sqrt {3} x^8-150 \sqrt {3} \left (1+\sqrt {15}\right ) x^6+5 \sqrt {3} \left (28-5 \sqrt {15}\right ) x^4+10 \sqrt {3} \left (7-9 \sqrt {15}\right ) x^2+2 \sqrt {3}}dx^2\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{8} \int \left (\frac {75 x^4}{450 x^8-150 \left (1+\sqrt {15}\right ) x^6+5 \left (28-5 \sqrt {15}\right ) x^4+10 \left (7-9 \sqrt {15}\right ) x^2+2}+\frac {12 \sqrt {15} x^2}{-450 x^8+150 \left (1+\sqrt {15}\right ) x^6-5 \left (28-5 \sqrt {15}\right ) x^4-10 \left (7-9 \sqrt {15}\right ) x^2-2}+\frac {\sqrt {15} \left (1-4 \sqrt {\frac {5}{3}}\right )}{-450 x^8+150 \left (1+\sqrt {15}\right ) x^6-5 \left (28-5 \sqrt {15}\right ) x^4-10 \left (7-9 \sqrt {15}\right ) x^2-2}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{8} \left (\left (20-\sqrt {15}\right ) \int \frac {1}{-450 x^8+150 \left (1+\sqrt {15}\right ) x^6-5 \left (28-5 \sqrt {15}\right ) x^4-10 \left (7-9 \sqrt {15}\right ) x^2-2}dx^2-12 \sqrt {15} \int \frac {x^2}{-450 x^8+150 \left (1+\sqrt {15}\right ) x^6-5 \left (28-5 \sqrt {15}\right ) x^4-10 \left (7-9 \sqrt {15}\right ) x^2-2}dx^2-75 \int \frac {x^4}{450 x^8-150 \left (1+\sqrt {15}\right ) x^6+5 \left (28-5 \sqrt {15}\right ) x^4+10 \left (7-9 \sqrt {15}\right ) x^2+2}dx^2\right )\) |
Input:
Int[((-20*Sqrt[3] + 3*Sqrt[5])*x + 36*Sqrt[5]*x^3 - 75*Sqrt[3]*x^5)/(8*Sqr t[3] + 4*Sqrt[3]*(70 - 90*Sqrt[15])*x^2 + 4*Sqrt[3]*(140 - 25*Sqrt[15])*x^ 4 + 4*Sqrt[3]*(-150 - 150*Sqrt[15])*x^6 + 1800*Sqrt[3]*x^8),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(70\) vs. \(2(11)=22\).
Time = 0.78 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.35
| method | result | size |
| default | \(\frac {\sqrt {3}\, \left (\frac {\sqrt {3}\, \sqrt {5}\, 15^{\frac {3}{4}} \operatorname {arctanh}\left (\frac {\left (20 x^{2}-2 \sqrt {15}-10\right ) \sqrt {5}\, 15^{\frac {3}{4}}}{300}\right )}{300}-\frac {\sqrt {3}\, \sqrt {5}\, 15^{\frac {3}{4}} \arctan \left (\frac {\left (90 x^{2}-6 \sqrt {15}+30\right ) \sqrt {5}\, 15^{\frac {3}{4}}}{900}\right )}{300}\right )}{12}\) | \(71\) |
Input:
int(((-20*3^(1/2)+3*5^(1/2))*x+36*5^(1/2)*x^3-75*3^(1/2)*x^5)/(8*3^(1/2)+4 *3^(1/2)*(70-90*15^(1/2))*x^2+4*3^(1/2)*(140-25*15^(1/2))*x^4+4*3^(1/2)*(- 150-150*15^(1/2))*x^6+1800*3^(1/2)*x^8),x,method=_RETURNVERBOSE)
Output:
1/12*3^(1/2)*(1/300*3^(1/2)*5^(1/2)*15^(3/4)*arctanh(1/300*(20*x^2-2*15^(1 /2)-10)*5^(1/2)*15^(3/4))-1/300*3^(1/2)*5^(1/2)*15^(3/4)*arctan(1/900*(90* x^2-6*15^(1/2)+30)*5^(1/2)*15^(3/4)))
Timed out. \[ \int \frac {\left (-20 \sqrt {3}+3 \sqrt {5}\right ) x+36 \sqrt {5} x^3-75 \sqrt {3} x^5}{8 \sqrt {3}+4 \sqrt {3} \left (70-90 \sqrt {15}\right ) x^2+4 \sqrt {3} \left (140-25 \sqrt {15}\right ) x^4+4 \sqrt {3} \left (-150-150 \sqrt {15}\right ) x^6+1800 \sqrt {3} x^8} \, dx=\text {Timed out} \] Input:
integrate(((-20*3^(1/2)+3*5^(1/2))*x+36*5^(1/2)*x^3-75*3^(1/2)*x^5)/(8*3^( 1/2)+4*3^(1/2)*(70-90*15^(1/2))*x^2+4*3^(1/2)*(140-25*15^(1/2))*x^4+4*3^(1 /2)*(-150-150*15^(1/2))*x^6+1800*3^(1/2)*x^8),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {\left (-20 \sqrt {3}+3 \sqrt {5}\right ) x+36 \sqrt {5} x^3-75 \sqrt {3} x^5}{8 \sqrt {3}+4 \sqrt {3} \left (70-90 \sqrt {15}\right ) x^2+4 \sqrt {3} \left (140-25 \sqrt {15}\right ) x^4+4 \sqrt {3} \left (-150-150 \sqrt {15}\right ) x^6+1800 \sqrt {3} x^8} \, dx=\text {Timed out} \] Input:
integrate(((-20*3**(1/2)+3*5**(1/2))*x+36*5**(1/2)*x**3-75*3**(1/2)*x**5)/ (8*3**(1/2)+4*3**(1/2)*(70-90*15**(1/2))*x**2+4*3**(1/2)*(140-25*15**(1/2) )*x**4+4*3**(1/2)*(-150-150*15**(1/2))*x**6+1800*3**(1/2)*x**8),x)
Output:
Timed out
\[ \int \frac {\left (-20 \sqrt {3}+3 \sqrt {5}\right ) x+36 \sqrt {5} x^3-75 \sqrt {3} x^5}{8 \sqrt {3}+4 \sqrt {3} \left (70-90 \sqrt {15}\right ) x^2+4 \sqrt {3} \left (140-25 \sqrt {15}\right ) x^4+4 \sqrt {3} \left (-150-150 \sqrt {15}\right ) x^6+1800 \sqrt {3} x^8} \, dx=\int { -\frac {75 \, \sqrt {3} x^{5} - 36 \, \sqrt {5} x^{3} - x {\left (3 \, \sqrt {5} - 20 \, \sqrt {3}\right )}}{4 \, {\left (450 \, \sqrt {3} x^{8} - 150 \, \sqrt {3} x^{6} {\left (\sqrt {15} + 1\right )} - 5 \, \sqrt {3} x^{4} {\left (5 \, \sqrt {15} - 28\right )} - 10 \, \sqrt {3} x^{2} {\left (9 \, \sqrt {15} - 7\right )} + 2 \, \sqrt {3}\right )}} \,d x } \] Input:
integrate(((-20*3^(1/2)+3*5^(1/2))*x+36*5^(1/2)*x^3-75*3^(1/2)*x^5)/(8*3^( 1/2)+4*3^(1/2)*(70-90*15^(1/2))*x^2+4*3^(1/2)*(140-25*15^(1/2))*x^4+4*3^(1 /2)*(-150-150*15^(1/2))*x^6+1800*3^(1/2)*x^8),x, algorithm="maxima")
Output:
-1/4*integrate((75*sqrt(3)*x^5 - 36*sqrt(5)*x^3 - x*(3*sqrt(5) - 20*sqrt(3 )))/(450*sqrt(3)*x^8 - 150*sqrt(3)*x^6*(sqrt(15) + 1) - 5*sqrt(3)*x^4*(5*s qrt(15) - 28) - 10*sqrt(3)*x^2*(9*sqrt(15) - 7) + 2*sqrt(3)), x)
\[ \int \frac {\left (-20 \sqrt {3}+3 \sqrt {5}\right ) x+36 \sqrt {5} x^3-75 \sqrt {3} x^5}{8 \sqrt {3}+4 \sqrt {3} \left (70-90 \sqrt {15}\right ) x^2+4 \sqrt {3} \left (140-25 \sqrt {15}\right ) x^4+4 \sqrt {3} \left (-150-150 \sqrt {15}\right ) x^6+1800 \sqrt {3} x^8} \, dx=\int { -\frac {75 \, \sqrt {3} x^{5} - 36 \, \sqrt {5} x^{3} - x {\left (3 \, \sqrt {5} - 20 \, \sqrt {3}\right )}}{4 \, {\left (450 \, \sqrt {3} x^{8} - 150 \, \sqrt {3} x^{6} {\left (\sqrt {15} + 1\right )} - 5 \, \sqrt {3} x^{4} {\left (5 \, \sqrt {15} - 28\right )} - 10 \, \sqrt {3} x^{2} {\left (9 \, \sqrt {15} - 7\right )} + 2 \, \sqrt {3}\right )}} \,d x } \] Input:
integrate(((-20*3^(1/2)+3*5^(1/2))*x+36*5^(1/2)*x^3-75*3^(1/2)*x^5)/(8*3^( 1/2)+4*3^(1/2)*(70-90*15^(1/2))*x^2+4*3^(1/2)*(140-25*15^(1/2))*x^4+4*3^(1 /2)*(-150-150*15^(1/2))*x^6+1800*3^(1/2)*x^8),x, algorithm="giac")
Output:
sage0*x
Time = 14.82 (sec) , antiderivative size = 761, normalized size of antiderivative = 3.73 \[ \int \frac {\left (-20 \sqrt {3}+3 \sqrt {5}\right ) x+36 \sqrt {5} x^3-75 \sqrt {3} x^5}{8 \sqrt {3}+4 \sqrt {3} \left (70-90 \sqrt {15}\right ) x^2+4 \sqrt {3} \left (140-25 \sqrt {15}\right ) x^4+4 \sqrt {3} \left (-150-150 \sqrt {15}\right ) x^6+1800 \sqrt {3} x^8} \, dx=\text {Too large to display} \] Input:
int((75*3^(1/2)*x^5 - 36*5^(1/2)*x^3 + x*(20*3^(1/2) - 3*5^(1/2)))/(4*3^(1 /2)*x^2*(90*15^(1/2) - 70) - 1800*3^(1/2)*x^8 - 8*3^(1/2) + 4*3^(1/2)*x^4* (25*15^(1/2) - 140) + 4*3^(1/2)*x^6*(150*15^(1/2) + 150)),x)
Output:
symsum(log((519438023*15^(1/2)*root(164974559232000000*15^(1/2)*z^4 - 1750 227232358400000*z^4 - 419552000*15^(1/2) + 4451058025, z, k))/637729200000 000000 - (13405639*root(164974559232000000*15^(1/2)*z^4 - 1750227232358400 000*z^4 - 419552000*15^(1/2) + 4451058025, z, k))/25509168000000000 + (336 71*15^(1/2))/170061120000000000 + (123517*15^(1/2)*root(164974559232000000 *15^(1/2)*z^4 - 1750227232358400000*z^4 - 419552000*15^(1/2) + 4451058025, z, k)^2)/29893556250000 + (97241057*15^(1/2)*root(164974559232000000*15^( 1/2)*z^4 - 1750227232358400000*z^4 - 419552000*15^(1/2) + 4451058025, z, k )^3)/2491129687500 - (395459293*15^(1/2)*root(164974559232000000*15^(1/2)* z^4 - 1750227232358400000*z^4 - 419552000*15^(1/2) + 4451058025, z, k)^4)/ 3736694531250 + (15882735488*15^(1/2)*root(164974559232000000*15^(1/2)*z^4 - 1750227232358400000*z^4 - 419552000*15^(1/2) + 4451058025, z, k)^5)/415 18828125 + (164204969984*15^(1/2)*root(164974559232000000*15^(1/2)*z^4 - 1 750227232358400000*z^4 - 419552000*15^(1/2) + 4451058025, z, k)^6)/2491129 6875 - (185540082571*root(164974559232000000*15^(1/2)*z^4 - 17502272323584 00000*z^4 - 419552000*15^(1/2) + 4451058025, z, k)*x^2)/306110016000000000 - (5307890617*15^(1/2)*x^2)/8162933760000000000 + (114841366009*root(1649 74559232000000*15^(1/2)*z^4 - 1750227232358400000*z^4 - 419552000*15^(1/2) + 4451058025, z, k)^2)/191318760000000000 + (6880723*root(164974559232000 000*15^(1/2)*z^4 - 1750227232358400000*z^4 - 419552000*15^(1/2) + 44510...
\[ \int \frac {\left (-20 \sqrt {3}+3 \sqrt {5}\right ) x+36 \sqrt {5} x^3-75 \sqrt {3} x^5}{8 \sqrt {3}+4 \sqrt {3} \left (70-90 \sqrt {15}\right ) x^2+4 \sqrt {3} \left (140-25 \sqrt {15}\right ) x^4+4 \sqrt {3} \left (-150-150 \sqrt {15}\right ) x^6+1800 \sqrt {3} x^8} \, dx =\text {Too large to display} \] Input:
int(((-20*3^(1/2)+3*5^(1/2))*x+36*5^(1/2)*x^3-75*3^(1/2)*x^5)/(8*3^(1/2)+4 *3^(1/2)*(70-90*15^(1/2))*x^2+4*3^(1/2)*(140-25*15^(1/2))*x^4+4*3^(1/2)*(- 150-150*15^(1/2))*x^6+1800*3^(1/2)*x^8),x)
Output:
( - 5850*sqrt(15)*int(x**11/(202500*x**16 - 135000*x**14 - 189000*x**12 - 91500*x**10 - 413975*x**8 - 48500*x**6 - 116040*x**4 + 280*x**2 + 4),x) - 3225*sqrt(15)*int(x**9/(202500*x**16 - 135000*x**14 - 189000*x**12 - 91500 *x**10 - 413975*x**8 - 48500*x**6 - 116040*x**4 + 280*x**2 + 4),x) - 8220* sqrt(15)*int(x**7/(202500*x**16 - 135000*x**14 - 189000*x**12 - 91500*x**1 0 - 413975*x**8 - 48500*x**6 - 116040*x**4 + 280*x**2 + 4),x) + 480*sqrt(1 5)*int(x**5/(202500*x**16 - 135000*x**14 - 189000*x**12 - 91500*x**10 - 41 3975*x**8 - 48500*x**6 - 116040*x**4 + 280*x**2 + 4),x) - 1706*sqrt(15)*in t(x**3/(202500*x**16 - 135000*x**14 - 189000*x**12 - 91500*x**10 - 413975* x**8 - 48500*x**6 - 116040*x**4 + 280*x**2 + 4),x) + 2*sqrt(15)*int(x/(202 500*x**16 - 135000*x**14 - 189000*x**12 - 91500*x**10 - 413975*x**8 - 4850 0*x**6 - 116040*x**4 + 280*x**2 + 4),x) - 33750*int(x**13/(202500*x**16 - 135000*x**14 - 189000*x**12 - 91500*x**10 - 413975*x**8 - 48500*x**6 - 116 040*x**4 + 280*x**2 + 4),x) + 11250*int(x**11/(202500*x**16 - 135000*x**14 - 189000*x**12 - 91500*x**10 - 413975*x**8 - 48500*x**6 - 116040*x**4 + 2 80*x**2 + 4),x) + 7500*int(x**9/(202500*x**16 - 135000*x**14 - 189000*x**1 2 - 91500*x**10 - 413975*x**8 - 48500*x**6 - 116040*x**4 + 280*x**2 + 4),x ) + 4500*int(x**7/(202500*x**16 - 135000*x**14 - 189000*x**12 - 91500*x**1 0 - 413975*x**8 - 48500*x**6 - 116040*x**4 + 280*x**2 + 4),x) + 13625*int( x**5/(202500*x**16 - 135000*x**14 - 189000*x**12 - 91500*x**10 - 413975...