3.2.93 \(\int \frac {2+3 x^2}{x^4 \sqrt {3+5 x^2+x^4}} \, dx\) [193]

3.2.93.1 Optimal result

Integrand size = 25, antiderivative size = 302 \[ \int \frac {2+3 x^2}{x^4 \sqrt {3+5 x^2+x^4}} \, dx=\frac {7 x \left (5+\sqrt {13}+2 x^2\right )}{54 \sqrt {3+5 x^2+x^4}}-\frac {2 \sqrt {3+5 x^2+x^4}}{9 x^3}-\frac {7 \sqrt {3+5 x^2+x^4}}{27 x}-\frac {7 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{54 \sqrt {3+5 x^2+x^4}}-\frac {\sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{9 \sqrt {3+5 x^2+x^4}} \]

output

7/54*x*(5+2*x^2+13^(1/2))/(x^4+5*x^2+3)^(1/2)-2/9*(x^4+5*x^2+3)^(1/2)/x^3- 
7/27*(x^4+5*x^2+3)^(1/2)/x-1/27*(1/(36+x^2*(30+6*13^(1/2))))^(1/2)*(36+x^2 
*(30+6*13^(1/2)))^(1/2)*EllipticF(x*(30+6*13^(1/2))^(1/2)/(36+x^2*(30+6*13 
^(1/2)))^(1/2),1/6*(-78+30*13^(1/2))^(1/2))*(6+x^2*(5+13^(1/2)))*6^(1/2)/( 
5+13^(1/2))^(1/2)*((6+x^2*(5-13^(1/2)))/(6+x^2*(5+13^(1/2))))^(1/2)/(x^4+5 
*x^2+3)^(1/2)-7/324*(1/(36+x^2*(30+6*13^(1/2))))^(1/2)*(36+x^2*(30+6*13^(1 
/2)))^(1/2)*EllipticE(x*(30+6*13^(1/2))^(1/2)/(36+x^2*(30+6*13^(1/2)))^(1/ 
2),1/6*(-78+30*13^(1/2))^(1/2))*(6+x^2*(5+13^(1/2)))*(30+6*13^(1/2))^(1/2) 
*((6+x^2*(5-13^(1/2)))/(6+x^2*(5+13^(1/2))))^(1/2)/(x^4+5*x^2+3)^(1/2)
 

3.2.93.2 Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.24 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.78 \[ \int \frac {2+3 x^2}{x^4 \sqrt {3+5 x^2+x^4}} \, dx=\frac {-4 \left (18+51 x^2+41 x^4+7 x^6\right )+7 i \sqrt {2} \left (-5+\sqrt {13}\right ) x^3 \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )-i \sqrt {2} \left (-47+7 \sqrt {13}\right ) x^3 \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{108 x^3 \sqrt {3+5 x^2+x^4}} \]

input

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

output

(-4*(18 + 51*x^2 + 41*x^4 + 7*x^6) + (7*I)*Sqrt[2]*(-5 + Sqrt[13])*x^3*Sqr 
t[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*Elli 
pticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] - I*Sqrt 
[2]*(-47 + 7*Sqrt[13])*x^3*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*S 
qrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 1 
9/6 + (5*Sqrt[13])/6])/(108*x^3*Sqrt[3 + 5*x^2 + x^4])
 

3.2.93.3 Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1604, 25, 1604, 1503, 1412, 1455}

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.

\(\displaystyle \int \frac {3 x^2+2}{x^4 \sqrt {x^4+5 x^2+3}} \, dx\)

\(\Big \downarrow \) 1604

\(\displaystyle -\frac {1}{9} \int -\frac {7-2 x^2}{x^2 \sqrt {x^4+5 x^2+3}}dx-\frac {2 \sqrt {x^4+5 x^2+3}}{9 x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{9} \int \frac {7-2 x^2}{x^2 \sqrt {x^4+5 x^2+3}}dx-\frac {2 \sqrt {x^4+5 x^2+3}}{9 x^3}\)

\(\Big \downarrow \) 1604

\(\displaystyle \frac {1}{9} \left (-\frac {1}{3} \int \frac {6-7 x^2}{\sqrt {x^4+5 x^2+3}}dx-\frac {7 \sqrt {x^4+5 x^2+3}}{3 x}\right )-\frac {2 \sqrt {x^4+5 x^2+3}}{9 x^3}\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {1}{9} \left (\frac {1}{3} \left (7 \int \frac {x^2}{\sqrt {x^4+5 x^2+3}}dx-6 \int \frac {1}{\sqrt {x^4+5 x^2+3}}dx\right )-\frac {7 \sqrt {x^4+5 x^2+3}}{3 x}\right )-\frac {2 \sqrt {x^4+5 x^2+3}}{9 x^3}\)

\(\Big \downarrow \) 1412

\(\displaystyle \frac {1}{9} \left (\frac {1}{3} \left (7 \int \frac {x^2}{\sqrt {x^4+5 x^2+3}}dx-\frac {\sqrt {\frac {6}{5+\sqrt {13}}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}\right )-\frac {7 \sqrt {x^4+5 x^2+3}}{3 x}\right )-\frac {2 \sqrt {x^4+5 x^2+3}}{9 x^3}\)

\(\Big \downarrow \) 1455

\(\displaystyle \frac {1}{9} \left (\frac {1}{3} \left (7 \left (\frac {x \left (2 x^2+\sqrt {13}+5\right )}{2 \sqrt {x^4+5 x^2+3}}-\frac {\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{2 \sqrt {x^4+5 x^2+3}}\right )-\frac {\sqrt {\frac {6}{5+\sqrt {13}}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}\right )-\frac {7 \sqrt {x^4+5 x^2+3}}{3 x}\right )-\frac {2 \sqrt {x^4+5 x^2+3}}{9 x^3}\)

input

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

output

(-2*Sqrt[3 + 5*x^2 + x^4])/(9*x^3) + ((-7*Sqrt[3 + 5*x^2 + x^4])/(3*x) + ( 
7*((x*(5 + Sqrt[13] + 2*x^2))/(2*Sqrt[3 + 5*x^2 + x^4]) - (Sqrt[(5 + Sqrt[ 
13])/6]*Sqrt[(6 + (5 - Sqrt[13])*x^2)/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + 
Sqrt[13])*x^2)*EllipticE[ArcTan[Sqrt[(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[1 
3])/6])/(2*Sqrt[3 + 5*x^2 + x^4])) - (Sqrt[6/(5 + Sqrt[13])]*Sqrt[(6 + (5 
- Sqrt[13])*x^2)/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt[13])*x^2)*Ellipt 
icF[ArcTan[Sqrt[(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/Sqrt[3 + 5*x^ 
2 + x^4])/3)/9
 

3.2.93.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 1412

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] && 
!(PosQ[(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[ 
{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
 

rule 1455

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] &&  !(PosQ[ 
(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*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 1604

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

3.2.93.4 Maple [A] (verified)

Time = 2.11 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.75

method	result	size
default	\(-\frac {7 \sqrt {x^{4}+5 x^{2}+3}}{27 x}-\frac {28 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{3 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{9 x^{3}}-\frac {4 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{3 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}\)	\(228\)
risch	\(-\frac {7 x^{6}+41 x^{4}+51 x^{2}+18}{27 x^{3} \sqrt {x^{4}+5 x^{2}+3}}-\frac {4 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{3 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {28 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{3 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\)	\(228\)
elliptic	\(-\frac {7 \sqrt {x^{4}+5 x^{2}+3}}{27 x}-\frac {28 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{3 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{9 x^{3}}-\frac {4 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{3 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}\)	\(228\)

input

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

output

-7/27*(x^4+5*x^2+3)^(1/2)/x-28/3/(-30+6*13^(1/2))^(1/2)*(1-(-5/6+1/6*13^(1 
/2))*x^2)^(1/2)*(1-(-5/6-1/6*13^(1/2))*x^2)^(1/2)/(x^4+5*x^2+3)^(1/2)/(5+1 
3^(1/2))*(EllipticF(1/6*x*(-30+6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*39^(1/2)) 
-EllipticE(1/6*x*(-30+6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*39^(1/2)))-2/9*(x^ 
4+5*x^2+3)^(1/2)/x^3-4/3/(-30+6*13^(1/2))^(1/2)*(1-(-5/6+1/6*13^(1/2))*x^2 
)^(1/2)*(1-(-5/6-1/6*13^(1/2))*x^2)^(1/2)/(x^4+5*x^2+3)^(1/2)*EllipticF(1/ 
6*x*(-30+6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*39^(1/2))
 

3.2.93.5 Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.46 \[ \int \frac {2+3 x^2}{x^4 \sqrt {3+5 x^2+x^4}} \, dx=-\frac {7 \, {\left (\sqrt {13} \sqrt {6} \sqrt {3} x^{3} - 5 \, \sqrt {6} \sqrt {3} x^{3}\right )} \sqrt {\sqrt {13} - 5} E(\arcsin \left (\frac {1}{6} \, \sqrt {6} x \sqrt {\sqrt {13} - 5}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) - {\left (13 \, \sqrt {13} \sqrt {6} \sqrt {3} x^{3} - 5 \, \sqrt {6} \sqrt {3} x^{3}\right )} \sqrt {\sqrt {13} - 5} F(\arcsin \left (\frac {1}{6} \, \sqrt {6} x \sqrt {\sqrt {13} - 5}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) + 36 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (7 \, x^{2} + 6\right )}}{972 \, x^{3}} \]

input

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

output

-1/972*(7*(sqrt(13)*sqrt(6)*sqrt(3)*x^3 - 5*sqrt(6)*sqrt(3)*x^3)*sqrt(sqrt 
(13) - 5)*elliptic_e(arcsin(1/6*sqrt(6)*x*sqrt(sqrt(13) - 5)), 5/6*sqrt(13 
) + 19/6) - (13*sqrt(13)*sqrt(6)*sqrt(3)*x^3 - 5*sqrt(6)*sqrt(3)*x^3)*sqrt 
(sqrt(13) - 5)*elliptic_f(arcsin(1/6*sqrt(6)*x*sqrt(sqrt(13) - 5)), 5/6*sq 
rt(13) + 19/6) + 36*sqrt(x^4 + 5*x^2 + 3)*(7*x^2 + 6))/x^3
 

3.2.93.6 Sympy [F]

\[ \int \frac {2+3 x^2}{x^4 \sqrt {3+5 x^2+x^4}} \, dx=\int \frac {3 x^{2} + 2}{x^{4} \sqrt {x^{4} + 5 x^{2} + 3}}\, dx \]

input

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

output

Integral((3*x**2 + 2)/(x**4*sqrt(x**4 + 5*x**2 + 3)), x)
 

3.2.93.7 Maxima [F]

\[ \int \frac {2+3 x^2}{x^4 \sqrt {3+5 x^2+x^4}} \, dx=\int { \frac {3 \, x^{2} + 2}{\sqrt {x^{4} + 5 \, x^{2} + 3} x^{4}} \,d x } \]

input

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

output

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

3.2.93.8 Giac [F]

\[ \int \frac {2+3 x^2}{x^4 \sqrt {3+5 x^2+x^4}} \, dx=\int { \frac {3 \, x^{2} + 2}{\sqrt {x^{4} + 5 \, x^{2} + 3} x^{4}} \,d x } \]

input

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

output

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

3.2.93.9 Mupad [F(-1)]

Timed out. \[ \int \frac {2+3 x^2}{x^4 \sqrt {3+5 x^2+x^4}} \, dx=\int \frac {3\,x^2+2}{x^4\,\sqrt {x^4+5\,x^2+3}} \,d x \]

input

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

output

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