Integrand size = 31, antiderivative size = 237 \[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=19 x-\frac {17 x^3}{3}+x^5+\frac {25 x \left (3-x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {3}{16} \sqrt {\frac {3}{2} \left (-8669+5011 \sqrt {3}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {3}{16} \sqrt {\frac {3}{2} \left (-8669+5011 \sqrt {3}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {3}{32} \sqrt {\frac {3}{2} \left (8669+5011 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )-\frac {3}{32} \sqrt {\frac {3}{2} \left (8669+5011 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right ) \]
19*x-17/3*x^3+x^5+25/8*x*(-x^2+3)/(x^4+2*x^2+3)+3/32*arctan((-2*x+(-2+2*3^ (1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-52014+30066*3^(1/2))^(1/2)-3/32*arcta n((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-52014+30066*3^(1/2))^( 1/2)+3/64*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(52014+30066*3^(1/2))^(1/ 2)-3/64*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(52014+30066*3^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.56 \[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=19 x-\frac {17 x^3}{3}+x^5-\frac {25 x \left (-3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {9 \left (90 i+31 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{16 \sqrt {2-2 i \sqrt {2}}}+\frac {9 \left (-90 i+31 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{16 \sqrt {2+2 i \sqrt {2}}} \]
19*x - (17*x^3)/3 + x^5 - (25*x*(-3 + x^2))/(8*(3 + 2*x^2 + x^4)) + (9*(90 *I + 31*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/(16*Sqrt[2 - (2*I)*Sqrt[2] ]) + (9*(-90*I + 31*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(16*Sqrt[2 + ( 2*I)*Sqrt[2]])
Time = 0.53 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2197, 27, 2205, 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.
| \(\displaystyle \int \frac {x^6 \left (5 x^6+3 x^4+x^2+4\right )}{\left (x^4+2 x^2+3\right )^2} \, dx\) |
| \(\Big \downarrow \) 2197 |
| \(\displaystyle \frac {1}{48} \int -\frac {6 \left (-40 x^8+56 x^6-175 x^2+75\right )}{x^4+2 x^2+3}dx+\frac {25 x \left (3-x^2\right )}{8 \left (x^4+2 x^2+3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {25 x \left (3-x^2\right )}{8 \left (x^4+2 x^2+3\right )}-\frac {1}{8} \int \frac {-40 x^8+56 x^6-175 x^2+75}{x^4+2 x^2+3}dx\) |
| \(\Big \downarrow \) 2205 |
| \(\displaystyle \frac {25 x \left (3-x^2\right )}{8 \left (x^4+2 x^2+3\right )}-\frac {1}{8} \int \left (-40 x^4+136 x^2+\frac {9 \left (59-31 x^2\right )}{x^4+2 x^2+3}-152\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{8} \left (\frac {3}{2} \sqrt {\frac {3}{2} \left (5011 \sqrt {3}-8669\right )} \arctan \left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {3}{2} \sqrt {\frac {3}{2} \left (5011 \sqrt {3}-8669\right )} \arctan \left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+8 x^5-\frac {136 x^3}{3}+\frac {3}{4} \sqrt {\frac {3}{2} \left (8669+5011 \sqrt {3}\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {3}{4} \sqrt {\frac {3}{2} \left (8669+5011 \sqrt {3}\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+152 x\right )+\frac {25 x \left (3-x^2\right )}{8 \left (x^4+2 x^2+3\right )}\) |
(25*x*(3 - x^2))/(8*(3 + 2*x^2 + x^4)) + (152*x - (136*x^3)/3 + 8*x^5 + (3 *Sqrt[(3*(-8669 + 5011*Sqrt[3]))/2]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/ Sqrt[2*(1 + Sqrt[3])]])/2 - (3*Sqrt[(3*(-8669 + 5011*Sqrt[3]))/2]*ArcTan[( Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/2 + (3*Sqrt[(3*(8669 + 5011*Sqrt[3]))/2]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/4 - (3 *Sqrt[(3*(8669 + 5011*Sqrt[3]))/2]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/4)/8
3.2.10.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{Qx = PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x], d = Coeff[Pol ynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Polynomial Remainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4) ^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c*x ^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*Qx + b^2*d*(2*p + 3) - 2* a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; Fre eQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[Pq, x^2], 1] && NeQ[b^2 - 4 *a*c, 0] && LtQ[p, -1] && IGtQ[m/2, 0]
Int[(Px_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandInte grand[Px/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x^ 2] && Expon[Px, x^2] > 1
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.30
| method | result | size |
| risch | \(x^{5}-\frac {17 x^{3}}{3}+19 x +\frac {-\frac {25}{8} x^{3}+\frac {75}{8} x}{x^{4}+2 x^{2}+3}+\frac {9 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{2}+3\right )}{\sum }\frac {\left (31 \textit {\_R}^{2}-59\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}\right )}{32}\) | \(72\) |
| default | \(x^{5}-\frac {17 x^{3}}{3}+19 x +\frac {-\frac {25}{8} x^{3}+\frac {75}{8} x}{x^{4}+2 x^{2}+3}+\frac {3 \left (76 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+135 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}-x \sqrt {-2+2 \sqrt {3}}\right )}{64}+\frac {3 \left (-118 \sqrt {3}+\frac {\left (76 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+135 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{16 \sqrt {2+2 \sqrt {3}}}+\frac {3 \left (-76 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-135 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}+x \sqrt {-2+2 \sqrt {3}}\right )}{64}+\frac {3 \left (-118 \sqrt {3}-\frac {\left (-76 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-135 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{16 \sqrt {2+2 \sqrt {3}}}\) | \(288\) |
x^5-17/3*x^3+19*x+(-25/8*x^3+75/8*x)/(x^4+2*x^2+3)+9/32*sum((31*_R^2-59)/( _R^3+_R)*ln(x-_R),_R=RootOf(_Z^4+2*_Z^2+3))
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.89 \[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=\frac {96 \, x^{9} - 352 \, x^{7} + 1024 \, x^{5} + 1716 \, x^{3} - 3 \, {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {8073 i \, \sqrt {2} + 234063} \log \left (\sqrt {8073 i \, \sqrt {2} + 234063} {\left (76 i \, \sqrt {2} + 59\right )} + 45099 \, x\right ) + 3 \, {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {8073 i \, \sqrt {2} + 234063} \log \left (\sqrt {8073 i \, \sqrt {2} + 234063} {\left (-76 i \, \sqrt {2} - 59\right )} + 45099 \, x\right ) + 3 \, {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {-8073 i \, \sqrt {2} + 234063} \log \left ({\left (76 i \, \sqrt {2} - 59\right )} \sqrt {-8073 i \, \sqrt {2} + 234063} + 45099 \, x\right ) - 3 \, {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {-8073 i \, \sqrt {2} + 234063} \log \left ({\left (-76 i \, \sqrt {2} + 59\right )} \sqrt {-8073 i \, \sqrt {2} + 234063} + 45099 \, x\right ) + 6372 \, x}{96 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \]
1/96*(96*x^9 - 352*x^7 + 1024*x^5 + 1716*x^3 - 3*(x^4 + 2*x^2 + 3)*sqrt(80 73*I*sqrt(2) + 234063)*log(sqrt(8073*I*sqrt(2) + 234063)*(76*I*sqrt(2) + 5 9) + 45099*x) + 3*(x^4 + 2*x^2 + 3)*sqrt(8073*I*sqrt(2) + 234063)*log(sqrt (8073*I*sqrt(2) + 234063)*(-76*I*sqrt(2) - 59) + 45099*x) + 3*(x^4 + 2*x^2 + 3)*sqrt(-8073*I*sqrt(2) + 234063)*log((76*I*sqrt(2) - 59)*sqrt(-8073*I* sqrt(2) + 234063) + 45099*x) - 3*(x^4 + 2*x^2 + 3)*sqrt(-8073*I*sqrt(2) + 234063)*log((-76*I*sqrt(2) + 59)*sqrt(-8073*I*sqrt(2) + 234063) + 45099*x) + 6372*x)/(x^4 + 2*x^2 + 3)
Leaf count of result is larger than twice the leaf count of optimal. 1205 vs. \(2 (199) = 398\).
Time = 0.74 (sec) , antiderivative size = 1205, normalized size of antiderivative = 5.08 \[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=\text {Too large to display} \]
x**5 - 17*x**3/3 + 19*x + (-25*x**3 + 75*x)/(8*x**4 + 16*x**2 + 24) - 3*sq rt(26007/2048 + 15033*sqrt(3)/2048)*log(x**2 + x*(-304*sqrt(2)*sqrt(8669 + 5011*sqrt(3))/299 - 433349*sqrt(6)*sqrt(8669 + 5011*sqrt(3))/1498289 + 15 2*sqrt(3)*sqrt(8669 + 5011*sqrt(3))*sqrt(43440359*sqrt(3) + 75240962)/1498 289) - 2882918249387*sqrt(2)*sqrt(43440359*sqrt(3) + 75240962)/22448699275 21 - 993398584*sqrt(6)*sqrt(43440359*sqrt(3) + 75240962)/1343965233 + 4993 6376949404567/2244869927521 + 17261871038090*sqrt(3)/1343965233) + 3*sqrt( 26007/2048 + 15033*sqrt(3)/2048)*log(x**2 + x*(-152*sqrt(3)*sqrt(8669 + 50 11*sqrt(3))*sqrt(43440359*sqrt(3) + 75240962)/1498289 + 433349*sqrt(6)*sqr t(8669 + 5011*sqrt(3))/1498289 + 304*sqrt(2)*sqrt(8669 + 5011*sqrt(3))/299 ) - 2882918249387*sqrt(2)*sqrt(43440359*sqrt(3) + 75240962)/2244869927521 - 993398584*sqrt(6)*sqrt(43440359*sqrt(3) + 75240962)/1343965233 + 4993637 6949404567/2244869927521 + 17261871038090*sqrt(3)/1343965233) - 2*sqrt(-27 *sqrt(2)*sqrt(43440359*sqrt(3) + 75240962)/1024 + 234063/2048 + 405891*sqr t(3)/2048)*atan(2996578*sqrt(3)*x/(17641*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(4344 0359*sqrt(3) + 75240962) + 8669 + 15033*sqrt(3)) + 152*sqrt(43440359*sqrt( 3) + 75240962)*sqrt(-2*sqrt(2)*sqrt(43440359*sqrt(3) + 75240962) + 8669 + 15033*sqrt(3))) - 1523344*sqrt(6)*sqrt(8669 + 5011*sqrt(3))/(17641*sqrt(2) *sqrt(-2*sqrt(2)*sqrt(43440359*sqrt(3) + 75240962) + 8669 + 15033*sqrt(3)) + 152*sqrt(43440359*sqrt(3) + 75240962)*sqrt(-2*sqrt(2)*sqrt(43440359*...
\[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=\int { \frac {{\left (5 \, x^{6} + 3 \, x^{4} + x^{2} + 4\right )} x^{6}}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}} \,d x } \]
x^5 - 17/3*x^3 + 19*x - 25/8*(x^3 - 3*x)/(x^4 + 2*x^2 + 3) + 9/8*integrate ((31*x^2 - 59)/(x^4 + 2*x^2 + 3), x)
Leaf count of result is larger than twice the leaf count of optimal. 576 vs. \(2 (166) = 332\).
Time = 0.63 (sec) , antiderivative size = 576, normalized size of antiderivative = 2.43 \[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=x^{5} - \frac {17}{3} \, x^{3} - \frac {1}{2304} \, \sqrt {2} {\left (31 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 558 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 558 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 31 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2124 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} - 2124 \cdot 3^{\frac {1}{4}} \sqrt {-6 \, \sqrt {3} + 18}\right )} \arctan \left (\frac {3^{\frac {3}{4}} {\left (x + 3^{\frac {1}{4}} \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}\right )}}{3 \, \sqrt {\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}}\right ) - \frac {1}{2304} \, \sqrt {2} {\left (31 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 558 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 558 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 31 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2124 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} - 2124 \cdot 3^{\frac {1}{4}} \sqrt {-6 \, \sqrt {3} + 18}\right )} \arctan \left (\frac {3^{\frac {3}{4}} {\left (x - 3^{\frac {1}{4}} \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}\right )}}{3 \, \sqrt {\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}}\right ) - \frac {1}{4608} \, \sqrt {2} {\left (558 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 31 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 31 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 558 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} + 2124 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} + 2124 \cdot 3^{\frac {1}{4}} \sqrt {6 \, \sqrt {3} + 18}\right )} \log \left (x^{2} + 2 \cdot 3^{\frac {1}{4}} x \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}} + \sqrt {3}\right ) + \frac {1}{4608} \, \sqrt {2} {\left (558 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 31 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 31 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 558 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} + 2124 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} + 2124 \cdot 3^{\frac {1}{4}} \sqrt {6 \, \sqrt {3} + 18}\right )} \log \left (x^{2} - 2 \cdot 3^{\frac {1}{4}} x \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}} + \sqrt {3}\right ) + 19 \, x - \frac {25 \, {\left (x^{3} - 3 \, x\right )}}{8 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \]
x^5 - 17/3*x^3 - 1/2304*sqrt(2)*(31*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 558*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 558*3^(3/4)*(s qrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 31*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) + 2 124*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) - 2124*3^(1/4)*sqrt(-6*sqrt(3) + 18))*arctan(1/3*3^(3/4)*(x + 3^(1/4)*sqrt(-1/6*sqrt(3) + 1/2))/sqrt(1/6*sq rt(3) + 1/2)) - 1/2304*sqrt(2)*(31*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 558*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 558*3^(3/4)*(sq rt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 31*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) + 21 24*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) - 2124*3^(1/4)*sqrt(-6*sqrt(3) + 1 8))*arctan(1/3*3^(3/4)*(x - 3^(1/4)*sqrt(-1/6*sqrt(3) + 1/2))/sqrt(1/6*sqr t(3) + 1/2)) - 1/4608*sqrt(2)*(558*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*s qrt(3) + 18) - 31*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 31*3^(3/4)*(6* sqrt(3) + 18)^(3/2) + 558*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) + 212 4*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) + 2124*3^(1/4)*sqrt(6*sqrt(3) + 18 ))*log(x^2 + 2*3^(1/4)*x*sqrt(-1/6*sqrt(3) + 1/2) + sqrt(3)) + 1/4608*sqrt (2)*(558*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 31*3^(3/4)* sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 31*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 558* 3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) + 2124*3^(1/4)*sqrt(2)*sqrt(-6* sqrt(3) + 18) + 2124*3^(1/4)*sqrt(6*sqrt(3) + 18))*log(x^2 - 2*3^(1/4)*x*s qrt(-1/6*sqrt(3) + 1/2) + sqrt(3)) + 19*x - 25/8*(x^3 - 3*x)/(x^4 + 2*x...
Time = 8.62 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.69 \[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=19\,x+\frac {\frac {75\,x}{8}-\frac {25\,x^3}{8}}{x^4+2\,x^2+3}-\frac {17\,x^3}{3}+x^5-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {26007-\sqrt {2}\,897{}\mathrm {i}}\,24219{}\mathrm {i}}{64\,\left (-\frac {1380483}{16}+\frac {\sqrt {2}\,4286763{}\mathrm {i}}{128}\right )}-\frac {24219\,\sqrt {2}\,x\,\sqrt {26007-\sqrt {2}\,897{}\mathrm {i}}}{128\,\left (-\frac {1380483}{16}+\frac {\sqrt {2}\,4286763{}\mathrm {i}}{128}\right )}\right )\,\sqrt {26007-\sqrt {2}\,897{}\mathrm {i}}\,3{}\mathrm {i}}{16}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {26007+\sqrt {2}\,897{}\mathrm {i}}\,24219{}\mathrm {i}}{64\,\left (\frac {1380483}{16}+\frac {\sqrt {2}\,4286763{}\mathrm {i}}{128}\right )}+\frac {24219\,\sqrt {2}\,x\,\sqrt {26007+\sqrt {2}\,897{}\mathrm {i}}}{128\,\left (\frac {1380483}{16}+\frac {\sqrt {2}\,4286763{}\mathrm {i}}{128}\right )}\right )\,\sqrt {26007+\sqrt {2}\,897{}\mathrm {i}}\,3{}\mathrm {i}}{16} \]
19*x + ((75*x)/8 - (25*x^3)/8)/(2*x^2 + x^4 + 3) - (atan((x*(26007 - 2^(1/ 2)*897i)^(1/2)*24219i)/(64*((2^(1/2)*4286763i)/128 - 1380483/16)) - (24219 *2^(1/2)*x*(26007 - 2^(1/2)*897i)^(1/2))/(128*((2^(1/2)*4286763i)/128 - 13 80483/16)))*(26007 - 2^(1/2)*897i)^(1/2)*3i)/16 + (atan((x*(2^(1/2)*897i + 26007)^(1/2)*24219i)/(64*((2^(1/2)*4286763i)/128 + 1380483/16)) + (24219* 2^(1/2)*x*(2^(1/2)*897i + 26007)^(1/2))/(128*((2^(1/2)*4286763i)/128 + 138 0483/16)))*(2^(1/2)*897i + 26007)^(1/2)*3i)/16 - (17*x^3)/3 + x^5