\(\int \frac {4-7 x^2+x^4}{(1+2 x^2)^2 (2+5 x^2+3 x^4)^{5/2}} \, dx\) [223]

Optimal result

Integrand size = 36, antiderivative size = 290 \[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right )^2 \left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=-\frac {x \left (49+53 x^2\right )}{\left (1+2 x^2\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}+\frac {152 x}{\left (1+2 x^2\right ) \sqrt {2+5 x^2+3 x^4}}-\frac {967 x \left (2+3 x^2\right )}{2 \sqrt {2+5 x^2+3 x^4}}+\frac {x \left (3533+2901 x^2\right )}{2 \sqrt {2+5 x^2+3 x^4}}+\frac {967 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{\sqrt {2} \sqrt {2+5 x^2+3 x^4}}+\frac {3075 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {2+5 x^2+3 x^4}}-\frac {1656 \sqrt {3} \left (1+x^2\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{\sqrt {\frac {1+x^2}{2+3 x^2}} \sqrt {2+5 x^2+3 x^4}} \] Output:

-x*(53*x^2+49)/(2*x^2+1)/(3*x^4+5*x^2+2)^(3/2)+152*x/(2*x^2+1)/(3*x^4+5*x^ 
2+2)^(1/2)-967/2*x*(3*x^2+2)/(3*x^4+5*x^2+2)^(1/2)+1/2*x*(2901*x^2+3533)/( 
3*x^4+5*x^2+2)^(1/2)+967/2*2^(1/2)*(x^2+1)*((3*x^2+2)/(x^2+1))^(1/2)*Ellip 
ticE(x/(x^2+1)^(1/2),1/2*I*2^(1/2))/(3*x^4+5*x^2+2)^(1/2)+3075/2*2^(1/2)*( 
x^2+1)*((3*x^2+2)/(x^2+1))^(1/2)*InverseJacobiAM(arctan(x),1/2*I*2^(1/2))/ 
(3*x^4+5*x^2+2)^(1/2)-1656*(x^2+1)*EllipticPi(x*6^(1/2)/(6*x^2+4)^(1/2),-1 
/3,1/3*3^(1/2))*3^(1/2)/((x^2+1)/(3*x^2+2))^(1/2)/(3*x^4+5*x^2+2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.50 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.92 \[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right )^2 \left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\frac {262 x+270 x^3+2665 x \left (2+5 x^2+3 x^4\right )+2157 x^3 \left (2+5 x^2+3 x^4\right )+\frac {496 x \left (2+5 x^2+3 x^4\right )^2}{1+2 x^2}+967 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \left (2+5 x^2+3 x^4\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )-533 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \left (2+5 x^2+3 x^4\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )+828 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \left (2+5 x^2+3 x^4\right ) \operatorname {EllipticPi}\left (\frac {4}{3},i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{2 \left (2+5 x^2+3 x^4\right )^{3/2}} \] Input:

Integrate[(4 - 7*x^2 + x^4)/((1 + 2*x^2)^2*(2 + 5*x^2 + 3*x^4)^(5/2)),x]
 

Output:

(262*x + 270*x^3 + 2665*x*(2 + 5*x^2 + 3*x^4) + 2157*x^3*(2 + 5*x^2 + 3*x^ 
4) + (496*x*(2 + 5*x^2 + 3*x^4)^2)/(1 + 2*x^2) + (967*I)*Sqrt[3]*Sqrt[1 + 
x^2]*Sqrt[2 + 3*x^2]*(2 + 5*x^2 + 3*x^4)*EllipticE[I*ArcSinh[Sqrt[3/2]*x], 
 2/3] - (533*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*(2 + 5*x^2 + 3*x^4)* 
EllipticF[I*ArcSinh[Sqrt[3/2]*x], 2/3] + (828*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqr 
t[2 + 3*x^2]*(2 + 5*x^2 + 3*x^4)*EllipticPi[4/3, I*ArcSinh[Sqrt[3/2]*x], 2 
/3])/(2*(2 + 5*x^2 + 3*x^4)^(3/2))
 

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 539, normalized size of antiderivative = 1.86, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2258, 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.

\(\displaystyle \int \frac {x^4-7 x^2+4}{\left (2 x^2+1\right )^2 \left (3 x^4+5 x^2+2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 2258

\(\displaystyle \int \left (\frac {111}{\left (x^2+1\right ) \sqrt {3 x^4+5 x^2+2}}-\frac {1056}{\left (2 x^2+1\right ) \sqrt {3 x^4+5 x^2+2}}+\frac {1251}{\left (3 x^2+2\right ) \sqrt {3 x^4+5 x^2+2}}+\frac {12}{\left (x^2+1\right )^2 \sqrt {3 x^4+5 x^2+2}}+\frac {124}{\left (2 x^2+1\right )^2 \sqrt {3 x^4+5 x^2+2}}+\frac {738}{\left (3 x^2+2\right )^2 \sqrt {3 x^4+5 x^2+2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1770 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {3 x^4+5 x^2+2}}+\frac {42 \sqrt {2} \left (3 x^2+2\right ) \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {\frac {3 x^2+2}{x^2+1}} \sqrt {3 x^4+5 x^2+2}}-\frac {549 \left (3 x^2+2\right ) \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {\frac {3 x^2+2}{x^2+1}} \sqrt {3 x^4+5 x^2+2}}+\frac {124 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{\sqrt {3 x^4+5 x^2+2}}-\frac {266 \sqrt {2} \left (3 x^2+2\right ) E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{\sqrt {\frac {3 x^2+2}{x^2+1}} \sqrt {3 x^4+5 x^2+2}}+\frac {1251 \left (3 x^2+2\right ) E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{\sqrt {2} \sqrt {\frac {3 x^2+2}{x^2+1}} \sqrt {3 x^4+5 x^2+2}}-\frac {1656 \sqrt {3} \left (x^2+1\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{\sqrt {\frac {x^2+1}{3 x^2+2}} \sqrt {3 x^4+5 x^2+2}}+\frac {248 \sqrt {3 x^4+5 x^2+2} x}{2 x^2+1}+\frac {3015 \left (x^2+1\right ) x}{2 \sqrt {3 x^4+5 x^2+2}}-\frac {4 \left (3 x^2+2\right ) x}{\left (x^2+1\right ) \sqrt {3 x^4+5 x^2+2}}-\frac {1253 \left (3 x^2+2\right ) x}{2 \sqrt {3 x^4+5 x^2+2}}+\frac {369 \left (x^2+1\right ) x}{\left (3 x^2+2\right ) \sqrt {3 x^4+5 x^2+2}}\)

Input:

Int[(4 - 7*x^2 + x^4)/((1 + 2*x^2)^2*(2 + 5*x^2 + 3*x^4)^(5/2)),x]
 

Output:

(3015*x*(1 + x^2))/(2*Sqrt[2 + 5*x^2 + 3*x^4]) + (369*x*(1 + x^2))/((2 + 3 
*x^2)*Sqrt[2 + 5*x^2 + 3*x^4]) - (1253*x*(2 + 3*x^2))/(2*Sqrt[2 + 5*x^2 + 
3*x^4]) - (4*x*(2 + 3*x^2))/((1 + x^2)*Sqrt[2 + 5*x^2 + 3*x^4]) + (248*x*S 
qrt[2 + 5*x^2 + 3*x^4])/(1 + 2*x^2) + (1251*(2 + 3*x^2)*EllipticE[ArcTan[x 
], -1/2])/(Sqrt[2]*Sqrt[(2 + 3*x^2)/(1 + x^2)]*Sqrt[2 + 5*x^2 + 3*x^4]) - 
(266*Sqrt[2]*(2 + 3*x^2)*EllipticE[ArcTan[x], -1/2])/(Sqrt[(2 + 3*x^2)/(1 
+ x^2)]*Sqrt[2 + 5*x^2 + 3*x^4]) + (124*Sqrt[2]*(1 + x^2)*Sqrt[(2 + 3*x^2) 
/(1 + x^2)]*EllipticE[ArcTan[x], -1/2])/Sqrt[2 + 5*x^2 + 3*x^4] - (549*(2 
+ 3*x^2)*EllipticF[ArcTan[x], -1/2])/(Sqrt[2]*Sqrt[(2 + 3*x^2)/(1 + x^2)]* 
Sqrt[2 + 5*x^2 + 3*x^4]) + (42*Sqrt[2]*(2 + 3*x^2)*EllipticF[ArcTan[x], -1 
/2])/(Sqrt[(2 + 3*x^2)/(1 + x^2)]*Sqrt[2 + 5*x^2 + 3*x^4]) + (1770*Sqrt[2] 
*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticF[ArcTan[x], -1/2])/Sqrt[2 
+ 5*x^2 + 3*x^4] - (1656*Sqrt[3]*(1 + x^2)*EllipticPi[-1/3, ArcTan[Sqrt[3/ 
2]*x], 1/3])/(Sqrt[(1 + x^2)/(2 + 3*x^2)]*Sqrt[2 + 5*x^2 + 3*x^4])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2258

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ 
(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + b*x^2 + c*x^4], Px*(d + e 
*x^2)^q*(a + b*x^2 + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e}, x] 
&& PolyQ[Px, x] && IntegerQ[p + 1/2] && IntegerQ[q]
 

Maple [A] (verified)

Time = 5.85 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.68

method	result	size
risch	\(\frac {x \left (17406 x^{8}+58911 x^{6}+72950 x^{4}+39013 x^{2}+7576\right )}{2 \left (2 x^{2}+1\right ) \left (3 x^{4}+5 x^{2}+2\right )^{\frac {3}{2}}}+\frac {651 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{2 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {967 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{2 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {1242 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{\sqrt {3 x^{4}+5 x^{2}+2}}\)	\(197\)
elliptic	\(\frac {744 x \sqrt {3 x^{4}+5 x^{2}+2}}{6 x^{2}+3}+\frac {\left (15 x^{3}+\frac {131}{9} x \right ) \sqrt {3 x^{4}+5 x^{2}+2}}{\left (x^{4}+\frac {5}{3} x^{2}+\frac {2}{3}\right )^{2}}-\frac {6 \left (-\frac {719}{4} x^{3}-\frac {2665}{12} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}-\frac {158 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {967 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{2 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {1242 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{\sqrt {3 x^{4}+5 x^{2}+2}}\)	\(223\)
default	\(\frac {\left (\frac {5}{18} x^{3}+\frac {13}{54} x \right ) \sqrt {3 x^{4}+5 x^{2}+2}}{4 \left (x^{4}+\frac {5}{3} x^{2}+\frac {2}{3}\right )^{2}}-\frac {3 \left (\frac {115}{12} x^{3}+\frac {145}{18} x \right )}{2 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {3907 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{24 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {115 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{24 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {744 x \sqrt {3 x^{4}+5 x^{2}+2}}{6 x^{2}+3}+\frac {31 \left (\frac {29}{18} x^{3}+\frac {85}{54} x \right ) \sqrt {3 x^{4}+5 x^{2}+2}}{4 \left (x^{4}+\frac {5}{3} x^{2}+\frac {2}{3}\right )^{2}}-\frac {93 \left (-\frac {329}{12} x^{3}-\frac {569}{18} x \right )}{2 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {11719 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{24 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {1242 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{\sqrt {3 x^{4}+5 x^{2}+2}}-\frac {4 \left (-\frac {11}{18} x^{3}-\frac {31}{54} x \right ) \sqrt {3 x^{4}+5 x^{2}+2}}{\left (x^{4}+\frac {5}{3} x^{2}+\frac {2}{3}\right )^{2}}+\frac {-182 x^{3}-\frac {376}{3} x}{\sqrt {3 x^{4}+5 x^{2}+2}}\)	\(404\)

Input:

int((x^4-7*x^2+4)/(2*x^2+1)^2/(3*x^4+5*x^2+2)^(5/2),x,method=_RETURNVERBOS 
E)
 

Output:

1/2*x*(17406*x^8+58911*x^6+72950*x^4+39013*x^2+7576)/(2*x^2+1)/(3*x^4+5*x^ 
2+2)^(3/2)+651/2*I*(x^2+1)^(1/2)*(6*x^2+4)^(1/2)/(3*x^4+5*x^2+2)^(1/2)*Ell 
ipticF(I*x,1/2*6^(1/2))-967/2*I*(x^2+1)^(1/2)*(6*x^2+4)^(1/2)/(3*x^4+5*x^2 
+2)^(1/2)*(EllipticF(I*x,1/2*6^(1/2))-EllipticE(I*x,1/2*6^(1/2)))+1242*I*( 
x^2+1)^(1/2)*(1+3/2*x^2)^(1/2)/(3*x^4+5*x^2+2)^(1/2)*EllipticPi(I*x,2,1/2* 
I*(-3)^(1/2)*2^(1/2))
 

Fricas [F]

\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right )^2 \left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\int { \frac {x^{4} - 7 \, x^{2} + 4}{{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {5}{2}} {\left (2 \, x^{2} + 1\right )}^{2}} \,d x } \] Input:

integrate((x^4-7*x^2+4)/(2*x^2+1)^2/(3*x^4+5*x^2+2)^(5/2),x, algorithm="fr 
icas")
 

Output:

integral(sqrt(3*x^4 + 5*x^2 + 2)*(x^4 - 7*x^2 + 4)/(108*x^16 + 648*x^14 + 
1683*x^12 + 2471*x^10 + 2243*x^8 + 1289*x^6 + 458*x^4 + 92*x^2 + 8), x)
 

Sympy [F]

\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right )^2 \left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\int \frac {x^{4} - 7 x^{2} + 4}{\left (\left (x^{2} + 1\right ) \left (3 x^{2} + 2\right )\right )^{\frac {5}{2}} \left (2 x^{2} + 1\right )^{2}}\, dx \] Input:

integrate((x**4-7*x**2+4)/(2*x**2+1)**2/(3*x**4+5*x**2+2)**(5/2),x)
 

Output:

Integral((x**4 - 7*x**2 + 4)/(((x**2 + 1)*(3*x**2 + 2))**(5/2)*(2*x**2 + 1 
)**2), x)
 

Maxima [F]

\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right )^2 \left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\int { \frac {x^{4} - 7 \, x^{2} + 4}{{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {5}{2}} {\left (2 \, x^{2} + 1\right )}^{2}} \,d x } \] Input:

integrate((x^4-7*x^2+4)/(2*x^2+1)^2/(3*x^4+5*x^2+2)^(5/2),x, algorithm="ma 
xima")
 

Output:

integrate((x^4 - 7*x^2 + 4)/((3*x^4 + 5*x^2 + 2)^(5/2)*(2*x^2 + 1)^2), x)
 

Giac [F]

\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right )^2 \left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\int { \frac {x^{4} - 7 \, x^{2} + 4}{{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {5}{2}} {\left (2 \, x^{2} + 1\right )}^{2}} \,d x } \] Input:

integrate((x^4-7*x^2+4)/(2*x^2+1)^2/(3*x^4+5*x^2+2)^(5/2),x, algorithm="gi 
ac")
 

Output:

integrate((x^4 - 7*x^2 + 4)/((3*x^4 + 5*x^2 + 2)^(5/2)*(2*x^2 + 1)^2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right )^2 \left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\int \frac {x^4-7\,x^2+4}{{\left (2\,x^2+1\right )}^2\,{\left (3\,x^4+5\,x^2+2\right )}^{5/2}} \,d x \] Input:

int((x^4 - 7*x^2 + 4)/((2*x^2 + 1)^2*(5*x^2 + 3*x^4 + 2)^(5/2)),x)
 

Output:

int((x^4 - 7*x^2 + 4)/((2*x^2 + 1)^2*(5*x^2 + 3*x^4 + 2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right )^2 \left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=4 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{108 x^{16}+648 x^{14}+1683 x^{12}+2471 x^{10}+2243 x^{8}+1289 x^{6}+458 x^{4}+92 x^{2}+8}d x \right )+\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{4}}{108 x^{16}+648 x^{14}+1683 x^{12}+2471 x^{10}+2243 x^{8}+1289 x^{6}+458 x^{4}+92 x^{2}+8}d x -7 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{108 x^{16}+648 x^{14}+1683 x^{12}+2471 x^{10}+2243 x^{8}+1289 x^{6}+458 x^{4}+92 x^{2}+8}d x \right ) \] Input:

int((x^4-7*x^2+4)/(2*x^2+1)^2/(3*x^4+5*x^2+2)^(5/2),x)
 

Output:

4*int(sqrt(3*x**4 + 5*x**2 + 2)/(108*x**16 + 648*x**14 + 1683*x**12 + 2471 
*x**10 + 2243*x**8 + 1289*x**6 + 458*x**4 + 92*x**2 + 8),x) + int((sqrt(3* 
x**4 + 5*x**2 + 2)*x**4)/(108*x**16 + 648*x**14 + 1683*x**12 + 2471*x**10 
+ 2243*x**8 + 1289*x**6 + 458*x**4 + 92*x**2 + 8),x) - 7*int((sqrt(3*x**4 
+ 5*x**2 + 2)*x**2)/(108*x**16 + 648*x**14 + 1683*x**12 + 2471*x**10 + 224 
3*x**8 + 1289*x**6 + 458*x**4 + 92*x**2 + 8),x)