\(\int \frac {(1+2 x^2) (4-7 x^2+x^4)}{(2+5 x^2+3 x^4)^{5/2}} \, dx\) [220]

Optimal result

Integrand size = 34, antiderivative size = 160 \[ \int \frac {\left (1+2 x^2\right ) \left (4-7 x^2+x^4\right )}{\left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=-\frac {x \left (113+149 x^2\right )}{9 \left (2+5 x^2+3 x^4\right )^{3/2}}+\frac {425 x}{6 \sqrt {2+5 x^2+3 x^4}}+\frac {2419 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{9 \sqrt {2} \sqrt {2+5 x^2+3 x^4}}-\frac {997 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{3 \sqrt {2} \sqrt {2+5 x^2+3 x^4}} \] Output:

-1/9*x*(149*x^2+113)/(3*x^4+5*x^2+2)^(3/2)+425/6*x/(3*x^4+5*x^2+2)^(1/2)+2 
419/18*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)-997/6*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 = 10.20 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.99 \[ \int \frac {\left (1+2 x^2\right ) \left (4-7 x^2+x^4\right )}{\left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\frac {3 x \left (4000+14927 x^2+18208 x^4+7257 x^6\right )+2419 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \left (2+5 x^2+3 x^4\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )-425 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \left (2+5 x^2+3 x^4\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{18 \left (2+5 x^2+3 x^4\right )^{3/2}} \] Input:

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

Output:

(3*x*(4000 + 14927*x^2 + 18208*x^4 + 7257*x^6) + (2419*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] - (425*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])/(18*(2 + 5*x^2 + 3*x^4)^(3/2))
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.29, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {2206, 27, 1492, 27, 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.

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

\(\Big \downarrow \) 2206

\(\displaystyle -\frac {1}{6} \int -\frac {2 \left (131-441 x^2\right )}{3 \left (3 x^4+5 x^2+2\right )^{3/2}}dx-\frac {x \left (149 x^2+113\right )}{9 \left (3 x^4+5 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \int \frac {131-441 x^2}{\left (3 x^4+5 x^2+2\right )^{3/2}}dx-\frac {x \left (149 x^2+113\right )}{9 \left (3 x^4+5 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 1492

\(\displaystyle \frac {1}{9} \left (\frac {x \left (7257 x^2+6113\right )}{2 \sqrt {3 x^4+5 x^2+2}}-\frac {1}{2} \int \frac {3 \left (2419 x^2+1994\right )}{\sqrt {3 x^4+5 x^2+2}}dx\right )-\frac {x \left (149 x^2+113\right )}{9 \left (3 x^4+5 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {x \left (7257 x^2+6113\right )}{2 \sqrt {3 x^4+5 x^2+2}}-\frac {3}{2} \int \frac {2419 x^2+1994}{\sqrt {3 x^4+5 x^2+2}}dx\right )-\frac {x \left (149 x^2+113\right )}{9 \left (3 x^4+5 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {1}{9} \left (\frac {x \left (7257 x^2+6113\right )}{2 \sqrt {3 x^4+5 x^2+2}}-\frac {3}{2} \left (1994 \int \frac {1}{\sqrt {3 x^4+5 x^2+2}}dx+2419 \int \frac {x^2}{\sqrt {3 x^4+5 x^2+2}}dx\right )\right )-\frac {x \left (149 x^2+113\right )}{9 \left (3 x^4+5 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 1413

\(\displaystyle \frac {1}{9} \left (\frac {x \left (7257 x^2+6113\right )}{2 \sqrt {3 x^4+5 x^2+2}}-\frac {3}{2} \left (2419 \int \frac {x^2}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {997 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {3 x^4+5 x^2+2}}\right )\right )-\frac {x \left (149 x^2+113\right )}{9 \left (3 x^4+5 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 1456

\(\displaystyle \frac {1}{9} \left (\frac {x \left (7257 x^2+6113\right )}{2 \sqrt {3 x^4+5 x^2+2}}-\frac {3}{2} \left (\frac {997 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {3 x^4+5 x^2+2}}+2419 \left (\frac {x \left (3 x^2+2\right )}{3 \sqrt {3 x^4+5 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{3 \sqrt {3 x^4+5 x^2+2}}\right )\right )\right )-\frac {x \left (149 x^2+113\right )}{9 \left (3 x^4+5 x^2+2\right )^{3/2}}\)

Input:

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

Output:

-1/9*(x*(113 + 149*x^2))/(2 + 5*x^2 + 3*x^4)^(3/2) + ((x*(6113 + 7257*x^2) 
)/(2*Sqrt[2 + 5*x^2 + 3*x^4]) - (3*(2419*((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[Arc 
Tan[x], -1/2])/(3*Sqrt[2 + 5*x^2 + 3*x^4])) + (997*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] 
))/2)/9
 

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 1492

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + 
 c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 
 - 4*a*c))   Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 
7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 LtQ[p, -1] && IntegerQ[2*p]
 

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 2206

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

Maple [A] (verified)

Time = 7.59 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.84

method	result	size
risch	\(\frac {x \left (7257 x^{6}+18208 x^{4}+14927 x^{2}+4000\right )}{6 \left (3 x^{4}+5 x^{2}+2\right )^{\frac {3}{2}}}+\frac {997 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{6 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {2419 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{18 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(135\)
elliptic	\(\frac {\left (-\frac {149}{81} x^{3}-\frac {113}{81} x \right ) \sqrt {3 x^{4}+5 x^{2}+2}}{\left (x^{4}+\frac {5}{3} x^{2}+\frac {2}{3}\right )^{2}}-\frac {6 \left (-\frac {2419}{36} x^{3}-\frac {6113}{108} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {997 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{6 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {2419 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{18 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(162\)
default	\(\frac {\left (-\frac {2}{9} x^{3}-\frac {5}{27} x \right ) \sqrt {3 x^{4}+5 x^{2}+2}}{\left (x^{4}+\frac {5}{3} x^{2}+\frac {2}{3}\right )^{2}}-\frac {6 \left (-\frac {97}{12} x^{3}-\frac {245}{36} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {997 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{6 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {2419 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{18 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {4 \left (\frac {5}{18} x^{3}+\frac {13}{54} x \right ) \sqrt {3 x^{4}+5 x^{2}+2}}{\left (x^{4}+\frac {5}{3} x^{2}+\frac {2}{3}\right )^{2}}-\frac {24 \left (\frac {115}{12} x^{3}+\frac {145}{18} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}-\frac {13 \left (\frac {5}{27} x^{3}+\frac {4}{27} x \right ) \sqrt {3 x^{4}+5 x^{2}+2}}{\left (x^{4}+\frac {5}{3} x^{2}+\frac {2}{3}\right )^{2}}+\frac {520 x^{3}+\frac {1313}{3} x}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {2 \left (-\frac {13}{81} x^{3}-\frac {10}{81} x \right ) \sqrt {3 x^{4}+5 x^{2}+2}}{\left (x^{4}+\frac {5}{3} x^{2}+\frac {2}{3}\right )^{2}}-\frac {12 \left (-\frac {97}{18} x^{3}-\frac {245}{54} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}\)	\(348\)

Input:

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

Output:

1/6*x*(7257*x^6+18208*x^4+14927*x^2+4000)/(3*x^4+5*x^2+2)^(3/2)+997/6*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 
))-2419/18*I*(x^2+1)^(1/2)*(6*x^2+4)^(1/2)/(3*x^4+5*x^2+2)^(1/2)*(Elliptic 
F(I*x,1/2*6^(1/2))-EllipticE(I*x,1/2*6^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.80 \[ \int \frac {\left (1+2 x^2\right ) \left (4-7 x^2+x^4\right )}{\left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=-\frac {2419 \, \sqrt {2} {\left (9 i \, x^{8} + 30 i \, x^{6} + 37 i \, x^{4} + 20 i \, x^{2} + 4 i\right )} E(\arcsin \left (i \, x\right )\,|\,\frac {3}{2}) + 5410 \, \sqrt {2} {\left (-9 i \, x^{8} - 30 i \, x^{6} - 37 i \, x^{4} - 20 i \, x^{2} - 4 i\right )} F(\arcsin \left (i \, x\right )\,|\,\frac {3}{2}) - 3 \, {\left (7257 \, x^{7} + 18208 \, x^{5} + 14927 \, x^{3} + 4000 \, x\right )} \sqrt {3 \, x^{4} + 5 \, x^{2} + 2}}{18 \, {\left (9 \, x^{8} + 30 \, x^{6} + 37 \, x^{4} + 20 \, x^{2} + 4\right )}} \] Input:

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

Output:

-1/18*(2419*sqrt(2)*(9*I*x^8 + 30*I*x^6 + 37*I*x^4 + 20*I*x^2 + 4*I)*ellip 
tic_e(arcsin(I*x), 3/2) + 5410*sqrt(2)*(-9*I*x^8 - 30*I*x^6 - 37*I*x^4 - 2 
0*I*x^2 - 4*I)*elliptic_f(arcsin(I*x), 3/2) - 3*(7257*x^7 + 18208*x^5 + 14 
927*x^3 + 4000*x)*sqrt(3*x^4 + 5*x^2 + 2))/(9*x^8 + 30*x^6 + 37*x^4 + 20*x 
^2 + 4)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (1+2 x^2\right ) \left (4-7 x^2+x^4\right )}{\left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\frac {-20 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{3}-21 \sqrt {3 x^{4}+5 x^{2}+2}\, x +3618 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{27 x^{12}+135 x^{10}+279 x^{8}+305 x^{6}+186 x^{4}+60 x^{2}+8}d x \right ) x^{8}+12060 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{27 x^{12}+135 x^{10}+279 x^{8}+305 x^{6}+186 x^{4}+60 x^{2}+8}d x \right ) x^{6}+14874 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{27 x^{12}+135 x^{10}+279 x^{8}+305 x^{6}+186 x^{4}+60 x^{2}+8}d x \right ) x^{4}+8040 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{27 x^{12}+135 x^{10}+279 x^{8}+305 x^{6}+186 x^{4}+60 x^{2}+8}d x \right ) x^{2}+1608 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{27 x^{12}+135 x^{10}+279 x^{8}+305 x^{6}+186 x^{4}+60 x^{2}+8}d x \right )-13365 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{4}}{27 x^{12}+135 x^{10}+279 x^{8}+305 x^{6}+186 x^{4}+60 x^{2}+8}d x \right ) x^{8}-44550 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{4}}{27 x^{12}+135 x^{10}+279 x^{8}+305 x^{6}+186 x^{4}+60 x^{2}+8}d x \right ) x^{6}-54945 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{4}}{27 x^{12}+135 x^{10}+279 x^{8}+305 x^{6}+186 x^{4}+60 x^{2}+8}d x \right ) x^{4}-29700 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{4}}{27 x^{12}+135 x^{10}+279 x^{8}+305 x^{6}+186 x^{4}+60 x^{2}+8}d x \right ) x^{2}-5940 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{4}}{27 x^{12}+135 x^{10}+279 x^{8}+305 x^{6}+186 x^{4}+60 x^{2}+8}d x \right )}{810 x^{8}+2700 x^{6}+3330 x^{4}+1800 x^{2}+360} \] Input:

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

Output:

( - 20*sqrt(3*x**4 + 5*x**2 + 2)*x**3 - 21*sqrt(3*x**4 + 5*x**2 + 2)*x + 3 
618*int(sqrt(3*x**4 + 5*x**2 + 2)/(27*x**12 + 135*x**10 + 279*x**8 + 305*x 
**6 + 186*x**4 + 60*x**2 + 8),x)*x**8 + 12060*int(sqrt(3*x**4 + 5*x**2 + 2 
)/(27*x**12 + 135*x**10 + 279*x**8 + 305*x**6 + 186*x**4 + 60*x**2 + 8),x) 
*x**6 + 14874*int(sqrt(3*x**4 + 5*x**2 + 2)/(27*x**12 + 135*x**10 + 279*x* 
*8 + 305*x**6 + 186*x**4 + 60*x**2 + 8),x)*x**4 + 8040*int(sqrt(3*x**4 + 5 
*x**2 + 2)/(27*x**12 + 135*x**10 + 279*x**8 + 305*x**6 + 186*x**4 + 60*x** 
2 + 8),x)*x**2 + 1608*int(sqrt(3*x**4 + 5*x**2 + 2)/(27*x**12 + 135*x**10 
+ 279*x**8 + 305*x**6 + 186*x**4 + 60*x**2 + 8),x) - 13365*int((sqrt(3*x** 
4 + 5*x**2 + 2)*x**4)/(27*x**12 + 135*x**10 + 279*x**8 + 305*x**6 + 186*x* 
*4 + 60*x**2 + 8),x)*x**8 - 44550*int((sqrt(3*x**4 + 5*x**2 + 2)*x**4)/(27 
*x**12 + 135*x**10 + 279*x**8 + 305*x**6 + 186*x**4 + 60*x**2 + 8),x)*x**6 
 - 54945*int((sqrt(3*x**4 + 5*x**2 + 2)*x**4)/(27*x**12 + 135*x**10 + 279* 
x**8 + 305*x**6 + 186*x**4 + 60*x**2 + 8),x)*x**4 - 29700*int((sqrt(3*x**4 
 + 5*x**2 + 2)*x**4)/(27*x**12 + 135*x**10 + 279*x**8 + 305*x**6 + 186*x** 
4 + 60*x**2 + 8),x)*x**2 - 5940*int((sqrt(3*x**4 + 5*x**2 + 2)*x**4)/(27*x 
**12 + 135*x**10 + 279*x**8 + 305*x**6 + 186*x**4 + 60*x**2 + 8),x))/(90*( 
9*x**8 + 30*x**6 + 37*x**4 + 20*x**2 + 4))