Integrand size = 26, antiderivative size = 248 \[ \int \frac {x^2}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {(-1)^{2/3} \arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{27\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {(-1)^{2/3} \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{81 \sqrt [3]{2} \sqrt [6]{3} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{81\ 2^{5/6} \sqrt [6]{3} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}} \]
1/162*(-1)^(2/3)*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/(24-18*(-3)^(2/3)*2^(1/ 3))^(1/2))*2^(1/6)*3^(5/6)/(1+(-1)^(1/3))^2/(4-3*(-3)^(2/3)*2^(1/3))^(1/2) -1/486*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/(-12+9*2^(1/3)*3^(2/3))^(1/2) )*2^(1/6)*3^(5/6)/(-4+3*2^(1/3)*3^(2/3))^(1/2)+1/486*(-1)^(2/3)*arctan((3* (-2)^(2/3)*3^(1/3)+2*x)/(24+18*(-2)^(1/3)*3^(2/3))^(1/2))*2^(2/3)*3^(5/6)/ (8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.24 \[ \int \frac {x^2}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {1}{6} \text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}}{36+162 \text {$\#$1}+12 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ] \]
RootSum[216 + 108*#1^2 + 324*#1^3 + 18*#1^4 + #1^6 & , (Log[x - #1]*#1)/(3 6 + 162*#1 + 12*#1^2 + #1^4) & ]/6
Time = 0.65 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2466, 2009}
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.
| \(\displaystyle \int \frac {x^2}{x^6+18 x^4+324 x^3+108 x^2+216} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 1259712 \int \left (\frac {(-1)^{2/3}}{68024448 \sqrt [3]{2} 3^{2/3} \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}+\frac {1}{68024448 \sqrt [3]{2} 3^{2/3} \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}-\frac {(-1)^{2/3}}{22674816 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 1259712 \left (\frac {(-1)^{2/3} \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{102036672\ 2^{5/6} \sqrt [6]{3} \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac {(-1)^{2/3} \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{34012224\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{102036672\ 2^{5/6} \sqrt [6]{3} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}}\right )\) |
1259712*(((-1)^(2/3)*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2 )^(1/3)*3^(2/3))]])/(102036672*2^(5/6)*3^(1/6)*Sqrt[4 + 3*(-2)^(1/3)*3^(2/ 3)]) + ((-1)^(2/3)*ArcTan[(2^(1/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(34012224*2^(5/6)*3^(1/6)*(1 + (-1)^(1/3))^2*Sqr t[4 - 3*(-3)^(2/3)*2^(1/3)]) - ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/S qrt[3*(-4 + 3*2^(1/3)*3^(2/3))]]/(102036672*2^(5/6)*3^(1/6)*Sqrt[-4 + 3*2^ (1/3)*3^(2/3)]))
3.2.46.3.1 Defintions of rubi rules used
Int[(u_.)*(Q6_)^(p_), x_Symbol] :> With[{a = Coeff[Q6, x, 0], b = Coeff[Q6, x, 2], c = Coeff[Q6, x, 3], d = Coeff[Q6, x, 4], e = Coeff[Q6, x, 6]}, Sim p[1/(3^(3*p)*a^(2*p)) Int[ExpandIntegrand[u*(3*a + 3*Rt[a, 3]^2*Rt[c, 3]* x + b*x^2)^p*(3*a - 3*(-1)^(1/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p*(3*a + 3* (-1)^(2/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p, x], x], x] /; EqQ[b^2 - 3*a*d, 0] && EqQ[b^3 - 27*a^2*e, 0]] /; ILtQ[p, 0] && PolyQ[Q6, x, 6] && EqQ[Coef f[Q6, x, 1], 0] && EqQ[Coeff[Q6, x, 5], 0] && RationalFunctionQ[u, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.23
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(56\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(56\) |
1/6*sum(_R^2/(_R^5+12*_R^3+162*_R^2+36*_R)*ln(x-_R),_R=RootOf(_Z^6+18*_Z^4 +324*_Z^3+108*_Z^2+216))
Leaf count of result is larger than twice the leaf count of optimal. 1277 vs. \(2 (162) = 324\).
Time = 0.93 (sec) , antiderivative size = 1277, normalized size of antiderivative = 5.15 \[ \int \frac {x^2}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\text {Too large to display} \]
1/324*sqrt(1/633)*sqrt(6*18^(2/3) + 8*18^(1/3) + 81)*log(1/211*sqrt(1/633) *(3*(6*18^(2/3) + 8*18^(1/3) + 81)^2 - 3741*18^(2/3) - 4988*18^(1/3) - 248 67)*sqrt(6*18^(2/3) + 8*18^(1/3) + 81) - 1/422*(6*18^(2/3) + 8*18^(1/3) + 81)^2 + 2*x + 729/211*18^(2/3) + 972/211*18^(1/3) + 8289/422) - 1/324*sqrt (1/633)*sqrt(6*18^(2/3) + 8*18^(1/3) + 81)*log(-1/211*sqrt(1/633)*(3*(6*18 ^(2/3) + 8*18^(1/3) + 81)^2 - 3741*18^(2/3) - 4988*18^(1/3) - 24867)*sqrt( 6*18^(2/3) + 8*18^(1/3) + 81) - 1/422*(6*18^(2/3) + 8*18^(1/3) + 81)^2 + 2 *x + 729/211*18^(2/3) + 972/211*18^(1/3) + 8289/422) - 1/136728*sqrt(1266) *sqrt(-2/3*18^(2/3) + sqrt(-1/27*(6*18^(2/3) + 8*18^(1/3) + 81)^2 + 36*18^ (2/3) + 48*18^(1/3) + 371) - 8/9*18^(1/3) + 18)*log(2*(6*18^(2/3) + 8*18^( 1/3) + 81)^2 + 18*sqrt(-1/27*(6*18^(2/3) + 8*18^(1/3) + 81)^2 + 36*18^(2/3 ) + 48*18^(1/3) + 371)*(6*18^(2/3) + 8*18^(1/3) + 81) + 1/211*(6*sqrt(1266 )*(6*18^(2/3) + 8*18^(1/3) + 81)^2 + 9*sqrt(-1/27*(6*18^(2/3) + 8*18^(1/3) + 81)^2 + 36*18^(2/3) + 48*18^(1/3) + 371)*(6*sqrt(1266)*(6*18^(2/3) + 8* 18^(1/3) + 81) - 211*sqrt(1266)) - 1247*sqrt(1266)*(6*18^(2/3) + 8*18^(1/3 ) + 81) + 51273*sqrt(1266))*sqrt(-2/3*18^(2/3) + sqrt(-1/27*(6*18^(2/3) + 8*18^(1/3) + 81)^2 + 36*18^(2/3) + 48*18^(1/3) + 371) - 8/9*18^(1/3) + 18) + 3376*x - 2916*18^(2/3) - 3888*18^(1/3) - 16578) + 1/136728*sqrt(1266)*s qrt(-2/3*18^(2/3) + sqrt(-1/27*(6*18^(2/3) + 8*18^(1/3) + 81)^2 + 36*18^(2 /3) + 48*18^(1/3) + 371) - 8/9*18^(1/3) + 18)*log(2*(6*18^(2/3) + 8*18^...
Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.19 \[ \int \frac {x^2}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\operatorname {RootSum} {\left (732274264442769408 t^{6} - 2677850419968 t^{4} + 2834352 t^{2} - 1, \left ( t \mapsto t \log {\left (10170475895038464 t^{5} - 5231726283456 t^{4} - 31809932496 t^{3} + 19131876 t^{2} + 19683 t + x - \frac {27}{2} \right )} \right )\right )} \]
RootSum(732274264442769408*_t**6 - 2677850419968*_t**4 + 2834352*_t**2 - 1 , Lambda(_t, _t*log(10170475895038464*_t**5 - 5231726283456*_t**4 - 318099 32496*_t**3 + 19131876*_t**2 + 19683*_t + x - 27/2)))
\[ \int \frac {x^2}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {x^{2}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \]
\[ \int \frac {x^2}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {x^{2}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \]
Time = 9.53 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\sum _{k=1}^6\ln \left (-\frac {216\,\left (32134205039616\,x-1836660096\,{\mathrm {root}\left (z^6-2834352\,z^4+2677850419968\,z^2-732274264442769408,z,k\right )}^2-1889568\,{\mathrm {root}\left (z^6-2834352\,z^4+2677850419968\,z^2-732274264442769408,z,k\right )}^3+972\,{\mathrm {root}\left (z^6-2834352\,z^4+2677850419968\,z^2-732274264442769408,z,k\right )}^4+{\mathrm {root}\left (z^6-2834352\,z^4+2677850419968\,z^2-732274264442769408,z,k\right )}^5+132239526912\,x\,\mathrm {root}\left (z^6-2834352\,z^4+2677850419968\,z^2-732274264442769408,z,k\right )+204073344\,x\,{\mathrm {root}\left (z^6-2834352\,z^4+2677850419968\,z^2-732274264442769408,z,k\right )}^2+139968\,x\,{\mathrm {root}\left (z^6-2834352\,z^4+2677850419968\,z^2-732274264442769408,z,k\right )}^3+36\,x\,{\mathrm {root}\left (z^6-2834352\,z^4+2677850419968\,z^2-732274264442769408,z,k\right )}^4+863230245120\,\mathrm {root}\left (z^6-2834352\,z^4+2677850419968\,z^2-732274264442769408,z,k\right )+781932322630656\right )}{{\mathrm {root}\left (z^6-2834352\,z^4+2677850419968\,z^2-732274264442769408,z,k\right )}^5}\right )\,\mathrm {root}\left (z^6-\frac {z^4}{273456}+\frac {z^2}{258356853504}-\frac {1}{732274264442769408},z,k\right ) \]
symsum(log(-(216*(32134205039616*x - 1836660096*root(z^6 - 2834352*z^4 + 2 677850419968*z^2 - 732274264442769408, z, k)^2 - 1889568*root(z^6 - 283435 2*z^4 + 2677850419968*z^2 - 732274264442769408, z, k)^3 + 972*root(z^6 - 2 834352*z^4 + 2677850419968*z^2 - 732274264442769408, z, k)^4 + root(z^6 - 2834352*z^4 + 2677850419968*z^2 - 732274264442769408, z, k)^5 + 1322395269 12*x*root(z^6 - 2834352*z^4 + 2677850419968*z^2 - 732274264442769408, z, k ) + 204073344*x*root(z^6 - 2834352*z^4 + 2677850419968*z^2 - 7322742644427 69408, z, k)^2 + 139968*x*root(z^6 - 2834352*z^4 + 2677850419968*z^2 - 732 274264442769408, z, k)^3 + 36*x*root(z^6 - 2834352*z^4 + 2677850419968*z^2 - 732274264442769408, z, k)^4 + 863230245120*root(z^6 - 2834352*z^4 + 267 7850419968*z^2 - 732274264442769408, z, k) + 781932322630656))/root(z^6 - 2834352*z^4 + 2677850419968*z^2 - 732274264442769408, z, k)^5)*root(z^6 - z^4/273456 + z^2/258356853504 - 1/732274264442769408, z, k), k, 1, 6)