Integrand size = 26, antiderivative size = 395 \[ \int \frac {x^5}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=-\frac {\sqrt [3]{-2} \left (1+\sqrt [3]{-2} 3^{2/3}\right ) \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{3^{5/6} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\sqrt [6]{\frac {3}{2}} \left (1-(-3)^{2/3} \sqrt [3]{2}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{\left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\left (1-\sqrt [3]{2} 3^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{\sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac {1}{216} \left (36+2^{2/3} \sqrt [3]{3} \left (1+i \sqrt {3}\right )\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )+\frac {1}{108} \left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )+\frac {1}{108} \left (18-2^{2/3} \sqrt [3]{3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right ) \]
1/108*(18-(-2)^(2/3)*3^(1/3))*ln(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)+1/108*(18-2 ^(2/3)*3^(1/3))*ln(6+3*2^(2/3)*3^(1/3)*x+x^2)+1/216*ln(6-3*(-3)^(1/3)*2^(2 /3)*x+x^2)*(36+2^(2/3)*3^(1/3)*(1+I*3^(1/2)))+1/2*3^(1/6)*2^(5/6)*(1-(-3)^ (2/3)*2^(1/3))*arctan(2^(1/6)*(3*(-3)^(1/3)-2^(1/3)*x)/(12-9*(-3)^(2/3)*2^ (1/3))^(1/2))/(1+(-1)^(1/3))^2/(4-3*(-3)^(2/3)*2^(1/3))^(1/2)-1/6*(1-2^(1/ 3)*3^(2/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/(-12+9*2^(1/3)*3^(2/3))^ (1/2))*2^(5/6)*3^(1/6)/(-4+3*2^(1/3)*3^(2/3))^(1/2)-1/3*(-2)^(1/3)*(1+(-2) ^(1/3)*3^(2/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/(24+18*(-2)^(1/3)*3^(2/3 ))^(1/2))*3^(1/6)/(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 = 61, normalized size of antiderivative = 0.15 \[ \int \frac {x^5}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {1}{6} \text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}^4}{36+162 \text {$\#$1}+12 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ] \]
RootSum[216 + 108*#1^2 + 324*#1^3 + 18*#1^4 + #1^6 & , (Log[x - #1]*#1^4)/ (36 + 162*#1 + 12*#1^2 + #1^4) & ]/6
Time = 1.36 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.02, 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.
| \(\displaystyle \int \frac {x^5}{x^6+18 x^4+324 x^3+108 x^2+216} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 1259712 \int \left (\frac {(-1)^{2/3} \left (\left (1-3 (-3)^{2/3} \sqrt [3]{2}\right ) x+3 \sqrt [3]{-3} 2^{2/3}\right )}{3779136 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}+\frac {(-1)^{2/3} \left (3 (-2)^{2/3} \sqrt [3]{3}-\left (1+3 \sqrt [3]{-2} 3^{2/3}\right ) x\right )}{11337408 \sqrt [3]{2} 3^{2/3} \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}+\frac {\left (18-2^{2/3} \sqrt [3]{3}\right ) x+6 \sqrt [3]{2} 3^{2/3}}{68024448 \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 1259712 \left (\frac {(-1)^{2/3} \left ((-1)^{2/3}-\sqrt [3]{2} 3^{2/3}\right ) \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{1259712 \sqrt [6]{2} 3^{5/6} \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac {\left (2 (-3)^{2/3}-2^{2/3}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{419904\ 6^{5/6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\left (1-\sqrt [3]{2} 3^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{1259712 \sqrt [6]{2} 3^{5/6} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}}+\frac {\left (36+2^{2/3} \sqrt [3]{3}+i 2^{2/3} 3^{5/6}\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{272097792}+\frac {\left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{136048896}+\frac {\left (18-2^{2/3} \sqrt [3]{3}\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{136048896}\right )\) |
1259712*(((-1)^(2/3)*((-1)^(2/3) - 2^(1/3)*3^(2/3))*ArcTan[(3*(-2)^(2/3)*3 ^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(1259712*2^(1/6)*3^(5/6 )*Sqrt[4 + 3*(-2)^(1/3)*3^(2/3)]) - ((2*(-3)^(2/3) - 2^(2/3))*ArcTan[(2^(1 /6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(4199 04*6^(5/6)*(1 + (-1)^(1/3))^2*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) - ((1 - 2^(1 /3)*3^(2/3))*ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1 /3)*3^(2/3))]])/(1259712*2^(1/6)*3^(5/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)]) + ( (36 + 2^(2/3)*3^(1/3) + I*2^(2/3)*3^(5/6))*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2])/272097792 + ((18 - (-2)^(2/3)*3^(1/3))*Log[6 + 3*(-2)^(2/3)*3^(1/3 )*x + x^2])/136048896 + ((18 - 2^(2/3)*3^(1/3))*Log[6 + 3*2^(2/3)*3^(1/3)* x + x^2])/136048896)
3.2.43.3.1 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 = 0.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.14
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\textit {\_R}^{5} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(56\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\textit {\_R}^{5} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(56\) |
1/6*sum(_R^5/(_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))
Timed out. \[ \int \frac {x^5}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\text {Timed out} \]
Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.18 \[ \int \frac {x^5}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\operatorname {RootSum} {\left (72662865048 t^{6} - 72662865048 t^{5} + 24163559388 t^{4} - 2646786132 t^{3} - 6626610 t^{2} - 4374 t - 1, \left ( t \mapsto t \log {\left (- \frac {89236417131047376 t^{5}}{833243797} + \frac {89301949532998128 t^{4}}{833243797} - \frac {29740560281805852 t^{3}}{833243797} + \frac {192466080408420 t^{2}}{49014341} + \frac {5867255361684 t}{833243797} + x + \frac {5365044886}{2499731391} \right )} \right )\right )} \]
RootSum(72662865048*_t**6 - 72662865048*_t**5 + 24163559388*_t**4 - 264678 6132*_t**3 - 6626610*_t**2 - 4374*_t - 1, Lambda(_t, _t*log(-8923641713104 7376*_t**5/833243797 + 89301949532998128*_t**4/833243797 - 297405602818058 52*_t**3/833243797 + 192466080408420*_t**2/49014341 + 5867255361684*_t/833 243797 + x + 5365044886/2499731391)))
\[ \int \frac {x^5}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {x^{5}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \]
\[ \int \frac {x^5}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {x^{5}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \]
Time = 0.38 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.08 \[ \int \frac {x^5}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\sum _{k=1}^6\ln \left (\frac {362797056\,\left (19236852\,x\,\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )-19131876\,x-6482268\,x\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^2+742851\,x\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^3-4130\,x\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^4+x\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^5-154944576\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^2+17047422\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^3+27054\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^4+9\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^5+465542316\,\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )-465542316\right )}{{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^5}\right )\,\mathrm {root}\left (z^6-z^5+\frac {421\,z^4}{1266}-\frac {100853\,z^3}{2768742}-\frac {505\,z^2}{5537484}-\frac {z}{16612452}-\frac {1}{72662865048},z,k\right ) \]
symsum(log((362797056*(19236852*x*root(z^6 + 4374*z^5 + 6626610*z^4 + 2646 786132*z^3 - 24163559388*z^2 + 72662865048*z - 72662865048, z, k) - 191318 76*x - 6482268*x*root(z^6 + 4374*z^5 + 6626610*z^4 + 2646786132*z^3 - 2416 3559388*z^2 + 72662865048*z - 72662865048, z, k)^2 + 742851*x*root(z^6 + 4 374*z^5 + 6626610*z^4 + 2646786132*z^3 - 24163559388*z^2 + 72662865048*z - 72662865048, z, k)^3 - 4130*x*root(z^6 + 4374*z^5 + 6626610*z^4 + 2646786 132*z^3 - 24163559388*z^2 + 72662865048*z - 72662865048, z, k)^4 + x*root( z^6 + 4374*z^5 + 6626610*z^4 + 2646786132*z^3 - 24163559388*z^2 + 72662865 048*z - 72662865048, z, k)^5 - 154944576*root(z^6 + 4374*z^5 + 6626610*z^4 + 2646786132*z^3 - 24163559388*z^2 + 72662865048*z - 72662865048, z, k)^2 + 17047422*root(z^6 + 4374*z^5 + 6626610*z^4 + 2646786132*z^3 - 241635593 88*z^2 + 72662865048*z - 72662865048, z, k)^3 + 27054*root(z^6 + 4374*z^5 + 6626610*z^4 + 2646786132*z^3 - 24163559388*z^2 + 72662865048*z - 7266286 5048, z, k)^4 + 9*root(z^6 + 4374*z^5 + 6626610*z^4 + 2646786132*z^3 - 241 63559388*z^2 + 72662865048*z - 72662865048, z, k)^5 + 465542316*root(z^6 + 4374*z^5 + 6626610*z^4 + 2646786132*z^3 - 24163559388*z^2 + 72662865048*z - 72662865048, z, k) - 465542316))/root(z^6 + 4374*z^5 + 6626610*z^4 + 26 46786132*z^3 - 24163559388*z^2 + 72662865048*z - 72662865048, z, k)^5)*roo t(z^6 - z^5 + (421*z^4)/1266 - (100853*z^3)/2768742 - (505*z^2)/5537484 - z/16612452 - 1/72662865048, z, k), k, 1, 6)