Integrand size = 30, antiderivative size = 455 \[ \int \frac {(2+3 x)^2}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {\left (18 \sqrt [3]{-6}+25 (-2)^{2/3}+6\ 3^{2/3}\right ) \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{486 \sqrt [6]{3} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac {(-1)^{2/3} \left (25+3 (-3)^{2/3} \sqrt [3]{2}+9 \sqrt [3]{-3} 2^{2/3}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{81\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\left (25-9\ 2^{2/3} \sqrt [3]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{243\ 2^{5/6} \sqrt [6]{3} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\left (-\frac {1}{6}\right )^{2/3} \left (\sqrt [3]{-3}+3 \sqrt [3]{2}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2}-\frac {\left ((-6)^{2/3}+\sqrt [3]{-2}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{324 \sqrt [3]{3}}+\frac {\left (1-\sqrt [3]{2} 3^{2/3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{162\ 2^{2/3} \sqrt [3]{3}} \] Output:
1/1458*(18*(-6)^(1/3)+25*(-2)^(2/3)+6*3^(2/3))*arctan((3*(-2)^(2/3)*3^(1/3 )+2*x)/(24+18*(-2)^(1/3)*3^(2/3))^(1/2))*3^(5/6)/(8+9*I*2^(1/3)*3^(1/6)+3* 2^(1/3)*3^(2/3))^(1/2)+1/486*(-1)^(2/3)*(25+3*(-3)^(2/3)*2^(1/3)+9*(-3)^(1 /3)*2^(2/3))*arctan(2^(1/6)*(3*(-3)^(1/3)-2^(1/3)*x)/(12-9*(-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 )-1/1458*(25-9*2^(2/3)*3^(1/3)+3*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^(1/6)*3^(5/6)/(-4+3*2^(1/3) *3^(2/3))^(1/2)+1/324*(-1)^(2/3)*6^(1/3)*((-3)^(1/3)+3*2^(1/3))*ln(6-3*(-3 )^(1/3)*2^(2/3)*x+x^2)/(1+(-1)^(1/3))^2-1/972*((-6)^(2/3)+(-2)^(1/3))*ln(6 +3*(-2)^(2/3)*3^(1/3)*x+x^2)*3^(2/3)+1/972*(1-2^(1/3)*3^(2/3))*ln(6+3*2^(2 /3)*3^(1/3)*x+x^2)*2^(1/3)*3^(2/3)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.19 \[ \int \frac {(2+3 x)^2}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {81}{2} \text {RootSum}\left [125128+64608 \text {$\#$1}-39612 \text {$\#$1}^2+7292 \text {$\#$1}^3+222 \text {$\#$1}^4-12 \text {$\#$1}^5+\text {$\#$1}^6\&,\frac {\log (2+3 x-\text {$\#$1}) \text {$\#$1}^2}{10768-13204 \text {$\#$1}+3646 \text {$\#$1}^2+148 \text {$\#$1}^3-10 \text {$\#$1}^4+\text {$\#$1}^5}\&\right ] \] Input:
Integrate[(2 + 3*x)^2/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6),x]
Output:
(81*RootSum[125128 + 64608*#1 - 39612*#1^2 + 7292*#1^3 + 222*#1^4 - 12*#1^ 5 + #1^6 & , (Log[2 + 3*x - #1]*#1^2)/(10768 - 13204*#1 + 3646*#1^2 + 148* #1^3 - 10*#1^4 + #1^5) & ])/2
Time = 2.10 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, 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.
| \(\displaystyle \int \frac {(3 x+2)^2}{x^6+18 x^4+324 x^3+108 x^2+216} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 1259712 \int \left (-\frac {(-1)^{2/3} \left (-\left (\left (6+\sqrt [3]{-3} 2^{2/3}\right ) x\right )+18 \sqrt [3]{-3} 2^{2/3}+6 (-3)^{2/3} \sqrt [3]{2}+25\right )}{68024448 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}+\frac {(-1)^{2/3} \left (-\left (\left (6-(-2)^{2/3} \sqrt [3]{3}\right ) x\right )-6 \sqrt [3]{-2} 3^{2/3}-18 (-2)^{2/3} \sqrt [3]{3}+25\right )}{204073344 \sqrt [3]{2} 3^{2/3} \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}+\frac {2 \sqrt [3]{2} 3^{2/3} \left (1-\sqrt [3]{2} 3^{2/3}\right ) x-36 \sqrt [3]{2} 3^{2/3}+25\ 2^{2/3} \sqrt [3]{3}+36}{1224440064 \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 1259712 \left (\frac {(-1)^{2/3} \left (25-9 (-2)^{2/3} \sqrt [3]{3}-3 \sqrt [3]{-2} 3^{2/3}\right ) \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{306110016\ 2^{5/6} \sqrt [6]{3} \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac {(-1)^{2/3} \left (25+3 (-3)^{2/3} \sqrt [3]{2}+9 \sqrt [3]{-3} 2^{2/3}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{102036672\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\left (27\ 2^{5/6} \sqrt [6]{3}-25 \sqrt [6]{2} 3^{5/6}-9 \sqrt {6}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{1836660096 \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}}+\frac {\left (-\frac {1}{6}\right )^{2/3} \left (\sqrt [3]{-3}+3 \sqrt [3]{2}\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{68024448 \left (1+\sqrt [3]{-1}\right )^2}+\frac {(-1)^{2/3} \left ((-2)^{2/3}-2\ 3^{2/3}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{408146688 \sqrt [3]{6}}+\frac {\left (1-\sqrt [3]{2} 3^{2/3}\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{204073344\ 2^{2/3} \sqrt [3]{3}}\right )\) |
Input:
Int[(2 + 3*x)^2/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6),x]
Output:
1259712*(((-1)^(2/3)*(25 - 9*(-2)^(2/3)*3^(1/3) - 3*(-2)^(1/3)*3^(2/3))*Ar cTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(30 6110016*2^(5/6)*3^(1/6)*Sqrt[4 + 3*(-2)^(1/3)*3^(2/3)]) + ((-1)^(2/3)*(25 + 3*(-3)^(2/3)*2^(1/3) + 9*(-3)^(1/3)*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))]])/(102036672*2^(5/6)*3 ^(1/6)*(1 + (-1)^(1/3))^2*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) + ((27*2^(5/6)*3 ^(1/6) - 25*2^(1/6)*3^(5/6) - 9*Sqrt[6])*ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^( 1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]])/(1836660096*Sqrt[-4 + 3*2^(1/3 )*3^(2/3)]) + ((-1/6)^(2/3)*((-3)^(1/3) + 3*2^(1/3))*Log[6 - 3*(-3)^(1/3)* 2^(2/3)*x + x^2])/(68024448*(1 + (-1)^(1/3))^2) + ((-1)^(2/3)*((-2)^(2/3) - 2*3^(2/3))*Log[6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2])/(408146688*6^(1/3)) + ((1 - 2^(1/3)*3^(2/3))*Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2])/(204073344*2^(2 /3)*3^(1/3)))
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 = 0.14 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.14
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (9 \textit {\_R}^{2}+12 \textit {\_R} +4\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(63\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (9 \textit {\_R}^{2}+12 \textit {\_R} +4\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(63\) |
Input:
int((2+3*x)^2/(x^6+18*x^4+324*x^3+108*x^2+216),x,method=_RETURNVERBOSE)
Output:
1/6*sum((9*_R^2+12*_R+4)/(_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. \[ \int \frac {(2+3 x)^2}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\text {Timed out} \] Input:
integrate((2+3*x)^2/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="fricas" )
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.14 \[ \int \frac {(2+3 x)^2}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\operatorname {RootSum} {\left (533827938778778898432 t^{6} - 40127588543220480 t^{4} + 2296802223171072 t^{3} + 32984196026544 t^{2} + 90717389856 t - 244640881, \left ( t \mapsto t \log {\left (\frac {6135155667731323161149373799929216 t^{5}}{307381025025457458399935} + \frac {69510029598991783751411734577568 t^{4}}{307381025025457458399935} - \frac {167454421024100513865438958272 t^{3}}{23644694232727496799995} + \frac {47147410389051283066608263328 t^{2}}{307381025025457458399935} + \frac {994701165304699703409532677 t}{614762050050914916799870} + x - \frac {4878034002593517908678517}{2459048200203659667199480} \right )} \right )\right )} \] Input:
integrate((2+3*x)**2/(x**6+18*x**4+324*x**3+108*x**2+216),x)
Output:
RootSum(533827938778778898432*_t**6 - 40127588543220480*_t**4 + 2296802223 171072*_t**3 + 32984196026544*_t**2 + 90717389856*_t - 244640881, Lambda(_ t, _t*log(6135155667731323161149373799929216*_t**5/30738102502545745839993 5 + 69510029598991783751411734577568*_t**4/307381025025457458399935 - 1674 54421024100513865438958272*_t**3/23644694232727496799995 + 471474103890512 83066608263328*_t**2/307381025025457458399935 + 99470116530469970340953267 7*_t/614762050050914916799870 + x - 4878034002593517908678517/245904820020 3659667199480)))
\[ \int \frac {(2+3 x)^2}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{2}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \] Input:
integrate((2+3*x)^2/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="maxima" )
Output:
integrate((3*x + 2)^2/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)
\[ \int \frac {(2+3 x)^2}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{2}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \] Input:
integrate((2+3*x)^2/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="giac")
Output:
integrate((3*x + 2)^2/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)
Time = 22.10 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^2}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx =\text {Too large to display} \] Input:
int((3*x + 2)^2/(108*x^2 + 324*x^3 + 18*x^4 + x^6 + 216),x)
Output:
symsum(log(588343450752*root(z^6 - (185*z^4)/2461104 + (11579*z^3)/2691217 224 + (431011*z^2)/6975635044608 + (1296259*z)/7627856921278848 - 24464088 1/533827938778778898432, z, k)^3 - 654156*x - 228683908*root(z^6 - (185*z^ 4)/2461104 + (11579*z^3)/2691217224 + (431011*z^2)/6975635044608 + (129625 9*z)/7627856921278848 - 244640881/533827938778778898432, z, k)*x - 2987691 6096*root(z^6 - (185*z^4)/2461104 + (11579*z^3)/2691217224 + (431011*z^2)/ 6975635044608 + (1296259*z)/7627856921278848 - 244640881/53382793877877889 8432, z, k)^2*x - 1695645415296*root(z^6 - (185*z^4)/2461104 + (11579*z^3) /2691217224 + (431011*z^2)/6975635044608 + (1296259*z)/7627856921278848 - 244640881/533827938778778898432, z, k)^3*x - 37159307062272*root(z^6 - (18 5*z^4)/2461104 + (11579*z^3)/2691217224 + (431011*z^2)/6975635044608 + (12 96259*z)/7627856921278848 - 244640881/533827938778778898432, z, k)^4*x - 2 89207845356544*root(z^6 - (185*z^4)/2461104 + (11579*z^3)/2691217224 + (43 1011*z^2)/6975635044608 + (1296259*z)/7627856921278848 - 244640881/5338279 38778778898432, z, k)^5*x - 3605925600*root(z^6 - (185*z^4)/2461104 + (115 79*z^3)/2691217224 + (431011*z^2)/6975635044608 + (1296259*z)/762785692127 8848 - 244640881/533827938778778898432, z, k)^2 - 170444700*root(z^6 - (18 5*z^4)/2461104 + (11579*z^3)/2691217224 + (431011*z^2)/6975635044608 + (12 96259*z)/7627856921278848 - 244640881/533827938778778898432, z, k) - 61171 801157376*root(z^6 - (185*z^4)/2461104 + (11579*z^3)/2691217224 + (4310...
\[ \int \frac {(2+3 x)^2}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=9 \left (\int \frac {x^{2}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}d x \right )+12 \left (\int \frac {x}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}d x \right )+4 \left (\int \frac {1}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}d x \right ) \] Input:
int((2+3*x)^2/(x^6+18*x^4+324*x^3+108*x^2+216),x)
Output:
9*int(x**2/(x**6 + 18*x**4 + 324*x**3 + 108*x**2 + 216),x) + 12*int(x/(x** 6 + 18*x**4 + 324*x**3 + 108*x**2 + 216),x) + 4*int(1/(x**6 + 18*x**4 + 32 4*x**3 + 108*x**2 + 216),x)