Integrand size = 26, antiderivative size = 361 \[ \int \frac {x^3}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=-\frac {\arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{6 \sqrt [6]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\sqrt [3]{-1} \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{9\ 2^{2/3} 3^{5/6} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{18 \sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}-\frac {(-1)^{2/3} \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{36 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {(-1)^{2/3} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{108 \sqrt [3]{2} 3^{2/3}}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{108 \sqrt [3]{2} 3^{2/3}} \] Output:
-1/36*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/(24-18*(-3)^(2/3)*2^(1/3))^(1/2))* 2^(5/6)*3^(1/6)/(1+(-1)^(1/3))^2/(4-3*(-3)^(2/3)*2^(1/3))^(1/2)+1/54*(-1)^ (1/3)*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/(24+18*(-2)^(1/3)*3^(2/3))^(1/2))* 2^(1/3)*3^(1/6)/(8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))^(1/2)+1/108*arct anh(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)-1/216*(-1)^(2/3)*ln(6-3*(-3)^(1/3)*2^( 2/3)*x+x^2)*2^(2/3)*3^(1/3)/(1+(-1)^(1/3))^2+1/648*(-1)^(2/3)*ln(6+3*(-2)^ (2/3)*3^(1/3)*x+x^2)*2^(2/3)*3^(1/3)+1/648*ln(6+3*2^(2/3)*3^(1/3)*x+x^2)*2 ^(2/3)*3^(1/3)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.17 \[ \int \frac {x^3}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {1}{6} \text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}^2}{36+162 \text {$\#$1}+12 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ] \] Input:
Integrate[x^3/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6),x]
Output:
RootSum[216 + 108*#1^2 + 324*#1^3 + 18*#1^4 + #1^6 & , (Log[x - #1]*#1^2)/ (36 + 162*#1 + 12*#1^2 + #1^4) & ]/6
Time = 1.43 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.98, 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.
| \(\displaystyle \int \frac {x^3}{x^6+18 x^4+324 x^3+108 x^2+216} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 1259712 \int \left (-\frac {(-1)^{2/3} x}{22674816 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}+\frac {(-1)^{2/3} x}{68024448 \sqrt [3]{2} 3^{2/3} \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}+\frac {x}{68024448 \sqrt [3]{2} 3^{2/3} \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 1259712 \left (\frac {\sqrt [3]{-1} \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{22674816 \sqrt [6]{2} 3^{5/6} \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac {\arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{7558272 \sqrt [6]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{22674816 \sqrt [6]{2} 3^{5/6} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}}-\frac {(-1)^{2/3} \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{45349632 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {(-1)^{2/3} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{136048896 \sqrt [3]{2} 3^{2/3}}+\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{136048896 \sqrt [3]{2} 3^{2/3}}\right )\) |
Input:
Int[x^3/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6),x]
Output:
1259712*(((-1)^(1/3)*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2 )^(1/3)*3^(2/3))]])/(22674816*2^(1/6)*3^(5/6)*Sqrt[4 + 3*(-2)^(1/3)*3^(2/3 )]) - ArcTan[(2^(1/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3) *2^(1/3))]]/(7558272*2^(1/6)*3^(5/6)*(1 + (-1)^(1/3))^2*Sqrt[4 - 3*(-3)^(2 /3)*2^(1/3)]) + ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2 ^(1/3)*3^(2/3))]]/(22674816*2^(1/6)*3^(5/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)]) - ((-1)^(2/3)*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2])/(45349632*2^(1/3)*3^( 2/3)*(1 + (-1)^(1/3))^2) + ((-1)^(2/3)*Log[6 + 3*(-2)^(2/3)*3^(1/3)*x + x^ 2])/(136048896*2^(1/3)*3^(2/3)) + Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2]/(1360 48896*2^(1/3)*3^(2/3)))
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 = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.16
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(56\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(56\) |
Input:
int(x^3/(x^6+18*x^4+324*x^3+108*x^2+216),x,method=_RETURNVERBOSE)
Output:
1/6*sum(_R^3/(_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. \[ \int \frac {x^3}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\text {Timed out} \] Input:
integrate(x^3/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="fricas")
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.17 \[ \int \frac {x^3}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\operatorname {RootSum} {\left (3390158631679488 t^{6} - 74384733888 t^{4} - 1332145440 t^{3} - 1417176 t^{2} - 1, \left ( t \mapsto t \log {\left (- \frac {8482372214243328 t^{5}}{415817} + \frac {2216055910930560 t^{4}}{415817} - \frac {2062546612992 t^{3}}{415817} - \frac {57027208896 t^{2}}{415817} - \frac {416583756 t}{415817} + x - \frac {89938}{415817} \right )} \right )\right )} \] Input:
integrate(x**3/(x**6+18*x**4+324*x**3+108*x**2+216),x)
Output:
RootSum(3390158631679488*_t**6 - 74384733888*_t**4 - 1332145440*_t**3 - 14 17176*_t**2 - 1, Lambda(_t, _t*log(-8482372214243328*_t**5/415817 + 221605 5910930560*_t**4/415817 - 2062546612992*_t**3/415817 - 57027208896*_t**2/4 15817 - 416583756*_t/415817 + x - 89938/415817)))
\[ \int \frac {x^3}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {x^{3}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \] Input:
integrate(x^3/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="maxima")
Output:
integrate(x^3/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)
\[ \int \frac {x^3}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {x^{3}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \] Input:
integrate(x^3/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="giac")
Output:
integrate(x^3/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)
Time = 22.47 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.76 \[ \int \frac {x^3}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\sum _{k=1}^6\ln \left (-\frac {23328\,\left (297538935552\,x-7992872640\,x\,\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )+52488\,x\,{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^3+2904\,x\,{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^4+x\,{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^5-153055008\,{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^2-2764368\,{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^3-1620\,{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^4-3673320192\,\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )+7240114098432\right )}{{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^5}\right )\,\mathrm {root}\left (z^6-\frac {z^4}{45576}-\frac {235\,z^3}{598048272}-\frac {z^2}{2392193088}-\frac {1}{3390158631679488},z,k\right ) \] Input:
int(x^3/(108*x^2 + 324*x^3 + 18*x^4 + x^6 + 216),x)
Output:
symsum(log(-(23328*(297538935552*x - 7992872640*x*root(z^6 + 1417176*z^4 + 1332145440*z^3 + 74384733888*z^2 - 3390158631679488, z, k) + 52488*x*root (z^6 + 1417176*z^4 + 1332145440*z^3 + 74384733888*z^2 - 3390158631679488, z, k)^3 + 2904*x*root(z^6 + 1417176*z^4 + 1332145440*z^3 + 74384733888*z^2 - 3390158631679488, z, k)^4 + x*root(z^6 + 1417176*z^4 + 1332145440*z^3 + 74384733888*z^2 - 3390158631679488, z, k)^5 - 153055008*root(z^6 + 141717 6*z^4 + 1332145440*z^3 + 74384733888*z^2 - 3390158631679488, z, k)^2 - 276 4368*root(z^6 + 1417176*z^4 + 1332145440*z^3 + 74384733888*z^2 - 339015863 1679488, z, k)^3 - 1620*root(z^6 + 1417176*z^4 + 1332145440*z^3 + 74384733 888*z^2 - 3390158631679488, z, k)^4 - 3673320192*root(z^6 + 1417176*z^4 + 1332145440*z^3 + 74384733888*z^2 - 3390158631679488, z, k) + 7240114098432 ))/root(z^6 + 1417176*z^4 + 1332145440*z^3 + 74384733888*z^2 - 33901586316 79488, z, k)^5)*root(z^6 - z^4/45576 - (235*z^3)/598048272 - z^2/239219308 8 - 1/3390158631679488, z, k), k, 1, 6)
\[ \int \frac {x^3}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int \frac {x^{3}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}d x \] Input:
int(x^3/(x^6+18*x^4+324*x^3+108*x^2+216),x)
Output:
int(x**3/(x**6 + 18*x**4 + 324*x**3 + 108*x**2 + 216),x)