3.2.55 \(\int \frac {(2+3 x^2) \sqrt {3+5 x^2+x^4}}{x^4} \, dx\) [155]

3.2.55.1 Optimal result

Integrand size = 25, antiderivative size = 305 \[ \int \frac {\left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^4} \, dx=\frac {32 x \left (5+\sqrt {13}+2 x^2\right )}{9 \sqrt {3+5 x^2+x^4}}-\frac {64 \sqrt {3+5 x^2+x^4}}{9 x}-\frac {\left (2-9 x^2\right ) \sqrt {3+5 x^2+x^4}}{3 x^3}-\frac {16 \sqrt {\frac {2}{3} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{9 \sqrt {3+5 x^2+x^4}}+\frac {49 \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{3 \sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {3+5 x^2+x^4}} \]

output

32/9*x*(5+2*x^2+13^(1/2))/(x^4+5*x^2+3)^(1/2)-64/9*(x^4+5*x^2+3)^(1/2)/x-1 
/3*(-9*x^2+2)*(x^4+5*x^2+3)^(1/2)/x^3-16/27*(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)+49/3*(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+x^2*(5-13^( 
1/2)))/(6+x^2*(5+13^(1/2))))^(1/2)/(x^4+5*x^2+3)^(1/2)/(30+6*13^(1/2))^(1/ 
2)
 

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

Result contains complex when optimal does not.

Time = 9.68 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.78 \[ \int \frac {\left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^4} \, dx=\frac {-2 \left (18+141 x^2+191 x^4+37 x^6\right )+32 i \sqrt {2} \left (-5+\sqrt {13}\right ) x^3 \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )-i \sqrt {2} \left (-13+32 \sqrt {13}\right ) x^3 \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{18 x^3 \sqrt {3+5 x^2+x^4}} \]

input

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

output

(-2*(18 + 141*x^2 + 191*x^4 + 37*x^6) + (32*I)*Sqrt[2]*(-5 + Sqrt[13])*x^3 
*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]* 
EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] - I* 
Sqrt[2]*(-13 + 32*Sqrt[13])*x^3*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13 
])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]* 
x], 19/6 + (5*Sqrt[13])/6])/(18*x^3*Sqrt[3 + 5*x^2 + x^4])
 

3.2.55.3 Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1594, 25, 1604, 25, 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.

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

\(\Big \downarrow \) 1594

\(\displaystyle -\frac {1}{3} \int -\frac {49 x^2+64}{x^2 \sqrt {x^4+5 x^2+3}}dx-\frac {\sqrt {x^4+5 x^2+3} \left (2-9 x^2\right )}{3 x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{3} \int \frac {49 x^2+64}{x^2 \sqrt {x^4+5 x^2+3}}dx-\frac {\left (2-9 x^2\right ) \sqrt {x^4+5 x^2+3}}{3 x^3}\)

\(\Big \downarrow \) 1604

\(\displaystyle \frac {1}{3} \left (-\frac {1}{3} \int -\frac {64 x^2+147}{\sqrt {x^4+5 x^2+3}}dx-\frac {64 \sqrt {x^4+5 x^2+3}}{3 x}\right )-\frac {\left (2-9 x^2\right ) \sqrt {x^4+5 x^2+3}}{3 x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \int \frac {64 x^2+147}{\sqrt {x^4+5 x^2+3}}dx-\frac {64 \sqrt {x^4+5 x^2+3}}{3 x}\right )-\frac {\left (2-9 x^2\right ) \sqrt {x^4+5 x^2+3}}{3 x^3}\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (147 \int \frac {1}{\sqrt {x^4+5 x^2+3}}dx+64 \int \frac {x^2}{\sqrt {x^4+5 x^2+3}}dx\right )-\frac {64 \sqrt {x^4+5 x^2+3}}{3 x}\right )-\frac {\left (2-9 x^2\right ) \sqrt {x^4+5 x^2+3}}{3 x^3}\)

\(\Big \downarrow \) 1412

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (64 \int \frac {x^2}{\sqrt {x^4+5 x^2+3}}dx+\frac {49 \sqrt {\frac {3}{2 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}\right )-\frac {64 \sqrt {x^4+5 x^2+3}}{3 x}\right )-\frac {\left (2-9 x^2\right ) \sqrt {x^4+5 x^2+3}}{3 x^3}\)

\(\Big \downarrow \) 1455

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {49 \sqrt {\frac {3}{2 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}+64 \left (\frac {x \left (2 x^2+\sqrt {13}+5\right )}{2 \sqrt {x^4+5 x^2+3}}-\frac {\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{2 \sqrt {x^4+5 x^2+3}}\right )\right )-\frac {64 \sqrt {x^4+5 x^2+3}}{3 x}\right )-\frac {\left (2-9 x^2\right ) \sqrt {x^4+5 x^2+3}}{3 x^3}\)

input

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

output

-1/3*((2 - 9*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^3 + ((-64*Sqrt[3 + 5*x^2 + x^4] 
)/(3*x) + (64*((x*(5 + Sqrt[13] + 2*x^2))/(2*Sqrt[3 + 5*x^2 + x^4]) - (Sqr 
t[(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], (-1 
3 + 5*Sqrt[13])/6])/(2*Sqrt[3 + 5*x^2 + x^4])) + (49*Sqrt[3/(2*(5 + Sqrt[1 
3]))]*Sqrt[(6 + (5 - Sqrt[13])*x^2)/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sq 
rt[13])*x^2)*EllipticF[ArcTan[Sqrt[(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13] 
)/6])/Sqrt[3 + 5*x^2 + x^4])/3)/3
 

3.2.55.3.1 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 1594

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

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])
 

3.2.55.4 Maple [A] (verified)

Time = 2.15 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.75

method	result	size
default	\(-\frac {37 \sqrt {x^{4}+5 x^{2}+3}}{9 x}+\frac {98 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {256 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{3 x^{3}}\)	\(228\)
risch	\(-\frac {37 x^{6}+191 x^{4}+141 x^{2}+18}{9 x^{3} \sqrt {x^{4}+5 x^{2}+3}}+\frac {98 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {256 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\)	\(228\)
elliptic	\(-\frac {37 \sqrt {x^{4}+5 x^{2}+3}}{9 x}+\frac {98 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {256 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{3 x^{3}}\)	\(228\)

input

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

output

-37/9*(x^4+5*x^2+3)^(1/2)/x+98/(-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)*Ellipt 
icF(1/6*x*(-30+6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*39^(1/2))-256/(-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+13^(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)))-2/3*(x^4+5*x^2+3)^(1/2)/x^3
 

3.2.55.5 Fricas [F]

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

input

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

output

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

3.2.55.6 Sympy [F]

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

input

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

output

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

3.2.55.7 Maxima [F]

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

input

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

output

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

3.2.55.8 Giac [F]

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

input

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

output

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

3.2.55.9 Mupad [F(-1)]

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

input

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

output

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