\(\int \frac {(1+2 x^2)^3 (4-7 x^2+x^4)}{\sqrt {2+5 x^2+3 x^4}} \, dx\) [204]

Optimal result

Integrand size = 36, antiderivative size = 229 \[ \int \frac {\left (1+2 x^2\right )^3 \left (4-7 x^2+x^4\right )}{\sqrt {2+5 x^2+3 x^4}} \, dx=\frac {299197 x \left (2+3 x^2\right )}{76545 \sqrt {2+5 x^2+3 x^4}}-\frac {3359 x \sqrt {2+5 x^2+3 x^4}}{5103}+\frac {5602 x^3 \sqrt {2+5 x^2+3 x^4}}{2835}-\frac {1508}{567} x^5 \sqrt {2+5 x^2+3 x^4}+\frac {8}{27} x^7 \sqrt {2+5 x^2+3 x^4}-\frac {299197 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{76545 \sqrt {2+5 x^2+3 x^4}}+\frac {13565 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{5103 \sqrt {2+5 x^2+3 x^4}} \] Output:

299197/76545*x*(3*x^2+2)/(3*x^4+5*x^2+2)^(1/2)-3359/5103*x*(3*x^4+5*x^2+2) 
^(1/2)+5602/2835*x^3*(3*x^4+5*x^2+2)^(1/2)-1508/567*x^5*(3*x^4+5*x^2+2)^(1 
/2)+8/27*x^7*(3*x^4+5*x^2+2)^(1/2)-299197/76545*2^(1/2)*(x^2+1)*((3*x^2+2) 
/(x^2+1))^(1/2)*EllipticE(x/(x^2+1)^(1/2),1/2*I*2^(1/2))/(3*x^4+5*x^2+2)^( 
1/2)+13565/5103*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)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.17 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.63 \[ \int \frac {\left (1+2 x^2\right )^3 \left (4-7 x^2+x^4\right )}{\sqrt {2+5 x^2+3 x^4}} \, dx=\frac {3 x \left (-33590+16861 x^2+65985 x^4-172926 x^6-165780 x^8+22680 x^{10}\right )-299197 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )+163547 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{76545 \sqrt {2+5 x^2+3 x^4}} \] Input:

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

Output:

(3*x*(-33590 + 16861*x^2 + 65985*x^4 - 172926*x^6 - 165780*x^8 + 22680*x^1 
0) - (299197*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticE[I*ArcSinh[ 
Sqrt[3/2]*x], 2/3] + (163547*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*Elli 
pticF[I*ArcSinh[Sqrt[3/2]*x], 2/3])/(76545*Sqrt[2 + 5*x^2 + 3*x^4])
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2207, 2207, 2207, 27, 2207, 1503, 1413, 1456}

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.

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

\(\Big \downarrow \) 2207

\(\displaystyle \frac {1}{27} \int \frac {-1508 x^8-1354 x^6+189 x^4+459 x^2+108}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {8}{27} \sqrt {3 x^4+5 x^2+2} x^7\)

\(\Big \downarrow \) 2207

\(\displaystyle \frac {1}{27} \left (\frac {1}{21} \int \frac {16806 x^6+19049 x^4+9639 x^2+2268}{\sqrt {3 x^4+5 x^2+2}}dx-\frac {1508}{21} x^5 \sqrt {3 x^4+5 x^2+2}\right )+\frac {8}{27} \sqrt {3 x^4+5 x^2+2} x^7\)

\(\Big \downarrow \) 2207

\(\displaystyle \frac {1}{27} \left (\frac {1}{21} \left (\frac {1}{15} \int \frac {3 \left (-16795 x^4+14583 x^2+11340\right )}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {5602}{5} \sqrt {3 x^4+5 x^2+2} x^3\right )-\frac {1508}{21} x^5 \sqrt {3 x^4+5 x^2+2}\right )+\frac {8}{27} \sqrt {3 x^4+5 x^2+2} x^7\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{27} \left (\frac {1}{21} \left (\frac {1}{5} \int \frac {-16795 x^4+14583 x^2+11340}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {5602}{5} \sqrt {3 x^4+5 x^2+2} x^3\right )-\frac {1508}{21} x^5 \sqrt {3 x^4+5 x^2+2}\right )+\frac {8}{27} \sqrt {3 x^4+5 x^2+2} x^7\)

\(\Big \downarrow \) 2207

\(\displaystyle \frac {1}{27} \left (\frac {1}{21} \left (\frac {1}{5} \left (\frac {1}{9} \int \frac {299197 x^2+135650}{\sqrt {3 x^4+5 x^2+2}}dx-\frac {16795}{9} x \sqrt {3 x^4+5 x^2+2}\right )+\frac {5602}{5} \sqrt {3 x^4+5 x^2+2} x^3\right )-\frac {1508}{21} x^5 \sqrt {3 x^4+5 x^2+2}\right )+\frac {8}{27} \sqrt {3 x^4+5 x^2+2} x^7\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {1}{27} \left (\frac {1}{21} \left (\frac {1}{5} \left (\frac {1}{9} \left (135650 \int \frac {1}{\sqrt {3 x^4+5 x^2+2}}dx+299197 \int \frac {x^2}{\sqrt {3 x^4+5 x^2+2}}dx\right )-\frac {16795}{9} x \sqrt {3 x^4+5 x^2+2}\right )+\frac {5602}{5} \sqrt {3 x^4+5 x^2+2} x^3\right )-\frac {1508}{21} x^5 \sqrt {3 x^4+5 x^2+2}\right )+\frac {8}{27} \sqrt {3 x^4+5 x^2+2} x^7\)

\(\Big \downarrow \) 1413

\(\displaystyle \frac {1}{27} \left (\frac {1}{21} \left (\frac {1}{5} \left (\frac {1}{9} \left (299197 \int \frac {x^2}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {67825 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {3 x^4+5 x^2+2}}\right )-\frac {16795}{9} x \sqrt {3 x^4+5 x^2+2}\right )+\frac {5602}{5} \sqrt {3 x^4+5 x^2+2} x^3\right )-\frac {1508}{21} x^5 \sqrt {3 x^4+5 x^2+2}\right )+\frac {8}{27} \sqrt {3 x^4+5 x^2+2} x^7\)

\(\Big \downarrow \) 1456

\(\displaystyle \frac {1}{27} \left (\frac {1}{21} \left (\frac {1}{5} \left (\frac {1}{9} \left (\frac {67825 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {3 x^4+5 x^2+2}}+299197 \left (\frac {x \left (3 x^2+2\right )}{3 \sqrt {3 x^4+5 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{3 \sqrt {3 x^4+5 x^2+2}}\right )\right )-\frac {16795}{9} x \sqrt {3 x^4+5 x^2+2}\right )+\frac {5602}{5} \sqrt {3 x^4+5 x^2+2} x^3\right )-\frac {1508}{21} x^5 \sqrt {3 x^4+5 x^2+2}\right )+\frac {8}{27} \sqrt {3 x^4+5 x^2+2} x^7\)

Input:

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

Output:

(8*x^7*Sqrt[2 + 5*x^2 + 3*x^4])/27 + ((-1508*x^5*Sqrt[2 + 5*x^2 + 3*x^4])/ 
21 + ((5602*x^3*Sqrt[2 + 5*x^2 + 3*x^4])/5 + ((-16795*x*Sqrt[2 + 5*x^2 + 3 
*x^4])/9 + (299197*((x*(2 + 3*x^2))/(3*Sqrt[2 + 5*x^2 + 3*x^4]) - (Sqrt[2] 
*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticE[ArcTan[x], -1/2])/(3*Sqrt 
[2 + 5*x^2 + 3*x^4])) + (67825*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])/9)/5)/21)/27
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1413

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b 
^2 - 4*a*c, 2]}, Simp[(2*a + (b - q)*x^2)*(Sqrt[(2*a + (b + q)*x^2)/(2*a + 
(b - q)*x^2)]/(2*a*Rt[(b - q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]))*EllipticF 
[ArcTan[Rt[(b - q)/(2*a), 2]*x], -2*(q/(b - q))], x] /; PosQ[(b - q)/a]] /; 
 FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
 

rule 1456

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[b^2 - 4*a*c, 2]}, Simp[x*((b - q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4 
])), x] - Simp[Rt[(b - q)/(2*a), 2]*(2*a + (b - q)*x^2)*(Sqrt[(2*a + (b + q 
)*x^2)/(2*a + (b - q)*x^2)]/(2*c*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[ArcTan 
[Rt[(b - q)/(2*a), 2]*x], -2*(q/(b - q))], x] /; PosQ[(b - q)/a]] /; FreeQ[ 
{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
 

rule 1503

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[d   Int[1/Sqrt[a + b*x^2 + c*x^4] 
, x], x] + Simp[e   Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b + q) 
/a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]
 

rule 2207

Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{n = 
 Expon[Px, x^2], e = Coeff[Px, x^2, Expon[Px, x^2]]}, Simp[e*x^(2*n - 3)*(( 
a + b*x^2 + c*x^4)^(p + 1)/(c*(2*n + 4*p + 1))), x] + Simp[1/(c*(2*n + 4*p 
+ 1))   Int[(a + b*x^2 + c*x^4)^p*ExpandToSum[c*(2*n + 4*p + 1)*Px - a*e*(2 
*n - 3)*x^(2*n - 4) - b*e*(2*n + 2*p - 1)*x^(2*n - 2) - c*e*(2*n + 4*p + 1) 
*x^(2*n), x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Px, x^2] && Expon[ 
Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] &&  !LtQ[p, -1]
 

Maple [A] (verified)

Time = 15.97 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.59

method	result	size
risch	\(\frac {x \left (7560 x^{6}-67860 x^{4}+50418 x^{2}-16795\right ) \sqrt {3 x^{4}+5 x^{2}+2}}{25515}-\frac {13565 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{5103 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {299197 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{76545 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(135\)
default	\(-\frac {13565 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{5103 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {299197 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{76545 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {3359 x \sqrt {3 x^{4}+5 x^{2}+2}}{5103}+\frac {5602 x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{2835}-\frac {1508 x^{5} \sqrt {3 x^{4}+5 x^{2}+2}}{567}+\frac {8 x^{7} \sqrt {3 x^{4}+5 x^{2}+2}}{27}\)	\(175\)
elliptic	\(-\frac {13565 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{5103 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {299197 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{76545 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {3359 x \sqrt {3 x^{4}+5 x^{2}+2}}{5103}+\frac {5602 x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{2835}-\frac {1508 x^{5} \sqrt {3 x^{4}+5 x^{2}+2}}{567}+\frac {8 x^{7} \sqrt {3 x^{4}+5 x^{2}+2}}{27}\)	\(175\)

Input:

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

Output:

1/25515*x*(7560*x^6-67860*x^4+50418*x^2-16795)*(3*x^4+5*x^2+2)^(1/2)-13565 
/5103*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))+299197/76545*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)))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.36 \[ \int \frac {\left (1+2 x^2\right )^3 \left (4-7 x^2+x^4\right )}{\sqrt {2+5 x^2+3 x^4}} \, dx=-\frac {598394 \, \sqrt {3} \sqrt {-\frac {2}{3}} x E(\arcsin \left (\frac {\sqrt {-\frac {2}{3}}}{x}\right )\,|\,\frac {3}{2}) - 1208819 \, \sqrt {3} \sqrt {-\frac {2}{3}} x F(\arcsin \left (\frac {\sqrt {-\frac {2}{3}}}{x}\right )\,|\,\frac {3}{2}) - 3 \, {\left (22680 \, x^{8} - 203580 \, x^{6} + 151254 \, x^{4} - 50385 \, x^{2} + 299197\right )} \sqrt {3 \, x^{4} + 5 \, x^{2} + 2}}{229635 \, x} \] Input:

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

Output:

-1/229635*(598394*sqrt(3)*sqrt(-2/3)*x*elliptic_e(arcsin(sqrt(-2/3)/x), 3/ 
2) - 1208819*sqrt(3)*sqrt(-2/3)*x*elliptic_f(arcsin(sqrt(-2/3)/x), 3/2) - 
3*(22680*x^8 - 203580*x^6 + 151254*x^4 - 50385*x^2 + 299197)*sqrt(3*x^4 + 
5*x^2 + 2))/x
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (1+2 x^2\right )^3 \left (4-7 x^2+x^4\right )}{\sqrt {2+5 x^2+3 x^4}} \, dx=\frac {8 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{7}}{27}-\frac {1508 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{5}}{567}+\frac {5602 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{3}}{2835}-\frac {3359 \sqrt {3 x^{4}+5 x^{2}+2}\, x}{5103}+\frac {27130 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{3 x^{4}+5 x^{2}+2}d x \right )}{5103}+\frac {299197 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{3 x^{4}+5 x^{2}+2}d x \right )}{25515} \] Input:

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

Output:

(7560*sqrt(3*x**4 + 5*x**2 + 2)*x**7 - 67860*sqrt(3*x**4 + 5*x**2 + 2)*x** 
5 + 50418*sqrt(3*x**4 + 5*x**2 + 2)*x**3 - 16795*sqrt(3*x**4 + 5*x**2 + 2) 
*x + 135650*int(sqrt(3*x**4 + 5*x**2 + 2)/(3*x**4 + 5*x**2 + 2),x) + 29919 
7*int((sqrt(3*x**4 + 5*x**2 + 2)*x**2)/(3*x**4 + 5*x**2 + 2),x))/25515