3.4.52 \(\int \frac {1}{(3+2 x^2) (1+2 x^2+2 x^4)^{3/2}} \, dx\) [352]

3.4.52.1 Optimal result

Integrand size = 26, antiderivative size = 422 \[ \int \frac {1}{\left (3+2 x^2\right ) \left (1+2 x^2+2 x^4\right )^{3/2}} \, dx=-\frac {x \left (1+3 x^2\right )}{5 \sqrt {1+2 x^2+2 x^4}}+\frac {3 x \sqrt {1+2 x^2+2 x^4}}{5 \sqrt {2} \left (1+\sqrt {2} x^2\right )}+\frac {\arctan \left (\frac {\sqrt {\frac {5}{3}} x}{\sqrt {1+2 x^2+2 x^4}}\right )}{5 \sqrt {15}}-\frac {3 \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} E\left (2 \arctan \left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{5\ 2^{3/4} \sqrt {1+2 x^2+2 x^4}}+\frac {\left (2+\sqrt {2}\right ) \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{2\ 2^{3/4} \left (-2+3 \sqrt {2}\right ) \sqrt {1+2 x^2+2 x^4}}+\frac {\left (3+\sqrt {2}\right ) \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{24} \left (12-11 \sqrt {2}\right ),2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{15\ 2^{3/4} \left (2-3 \sqrt {2}\right ) \sqrt {1+2 x^2+2 x^4}} \]

output

1/75*arctan(1/3*x*15^(1/2)/(2*x^4+2*x^2+1)^(1/2))*15^(1/2)-1/5*x*(3*x^2+1) 
/(2*x^4+2*x^2+1)^(1/2)+3/10*x*(2*x^4+2*x^2+1)^(1/2)*2^(1/2)/(1+x^2*2^(1/2) 
)-3/10*(cos(2*arctan(2^(1/4)*x))^2)^(1/2)/cos(2*arctan(2^(1/4)*x))*Ellipti 
cE(sin(2*arctan(2^(1/4)*x)),1/2*(2-2^(1/2))^(1/2))*(1+x^2*2^(1/2))*((2*x^4 
+2*x^2+1)/(1+x^2*2^(1/2))^2)^(1/2)*2^(1/4)/(2*x^4+2*x^2+1)^(1/2)+1/30*(cos 
(2*arctan(2^(1/4)*x))^2)^(1/2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*a 
rctan(2^(1/4)*x)),1/2-11/24*2^(1/2),1/2*(2-2^(1/2))^(1/2))*(3+2^(1/2))*(1+ 
x^2*2^(1/2))*((2*x^4+2*x^2+1)/(1+x^2*2^(1/2))^2)^(1/2)*2^(1/4)/(2-3*2^(1/2 
))/(2*x^4+2*x^2+1)^(1/2)+1/4*(cos(2*arctan(2^(1/4)*x))^2)^(1/2)/cos(2*arct 
an(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),1/2*(2-2^(1/2))^(1/2))*( 
2+2^(1/2))*(1+x^2*2^(1/2))*((2*x^4+2*x^2+1)/(1+x^2*2^(1/2))^2)^(1/2)*2^(1/ 
4)/(-2+3*2^(1/2))/(2*x^4+2*x^2+1)^(1/2)
 

3.4.52.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.15 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.47 \[ \int \frac {1}{\left (3+2 x^2\right ) \left (1+2 x^2+2 x^4\right )^{3/2}} \, dx=\frac {-6 x-18 x^3-9 i \sqrt {1-i} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} E\left (\left .i \text {arcsinh}\left (\sqrt {1-i} x\right )\right |i\right )+(6+3 i) \sqrt {1-i} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {1-i} x\right ),i\right )+2 (1-i)^{3/2} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} \operatorname {EllipticPi}\left (\frac {1}{3}+\frac {i}{3},i \text {arcsinh}\left (\sqrt {1-i} x\right ),i\right )}{30 \sqrt {1+2 x^2+2 x^4}} \]

input

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

output

(-6*x - 18*x^3 - (9*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)* 
x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] + (6 + 3*I)*Sqrt[1 - I]*Sqrt[1 
 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], 
I] + 2*(1 - I)^(3/2)*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticP 
i[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/(30*Sqrt[1 + 2*x^2 + 2*x^4])
 

3.4.52.3 Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1547, 27, 2206, 27, 1511, 1416, 1509, 2220}

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.

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

\(\Big \downarrow \) 1547

\(\displaystyle \frac {4 \int \frac {\sqrt {2} x^2+1}{\left (2 x^2+3\right ) \sqrt {2 x^4+2 x^2+1}}dx}{5 \left (2-3 \sqrt {2}\right )}-\frac {\int -\frac {2 \left (-4 \sqrt {2} x^4-2 \left (2-\sqrt {2}\right ) x^2-5 \sqrt {2}+2\right )}{\left (2 x^4+2 x^2+1\right )^{3/2}}dx}{10 \left (2-3 \sqrt {2}\right )}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2206

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1511

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 1509

\(\displaystyle \frac {4 \int \frac {\sqrt {2} x^2+1}{\left (2 x^2+3\right ) \sqrt {2 x^4+2 x^2+1}}dx}{5 \left (2-3 \sqrt {2}\right )}+\frac {-\frac {5 \left (2+2 \sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{4 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}+3 \left (3-\sqrt {2}\right ) \left (\frac {\left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} E\left (2 \arctan \left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{\sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}-\frac {x \sqrt {2 x^4+2 x^2+1}}{\sqrt {2} x^2+1}\right )-\frac {x \left (3 \left (2-3 \sqrt {2}\right ) x^2-3 \sqrt {2}+2\right )}{\sqrt {2 x^4+2 x^2+1}}}{5 \left (2-3 \sqrt {2}\right )}\)

\(\Big \downarrow \) 2220

\(\displaystyle \frac {4 \left (\frac {\left (3+\sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{24} \left (12-11 \sqrt {2}\right ),2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{12\ 2^{3/4} \sqrt {2 x^4+2 x^2+1}}-\frac {\left (3-\sqrt {2}\right ) \arctan \left (\frac {\sqrt {\frac {5}{3}} x}{\sqrt {2 x^4+2 x^2+1}}\right )}{2 \sqrt {30}}\right )}{5 \left (2-3 \sqrt {2}\right )}+\frac {-\frac {5 \left (2+2 \sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{4 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}+3 \left (3-\sqrt {2}\right ) \left (\frac {\left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} E\left (2 \arctan \left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{\sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}-\frac {x \sqrt {2 x^4+2 x^2+1}}{\sqrt {2} x^2+1}\right )-\frac {x \left (3 \left (2-3 \sqrt {2}\right ) x^2-3 \sqrt {2}+2\right )}{\sqrt {2 x^4+2 x^2+1}}}{5 \left (2-3 \sqrt {2}\right )}\)

input

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

output

(-((x*(2 - 3*Sqrt[2] + 3*(2 - 3*Sqrt[2])*x^2))/Sqrt[1 + 2*x^2 + 2*x^4]) + 
3*(3 - Sqrt[2])*(-((x*Sqrt[1 + 2*x^2 + 2*x^4])/(1 + Sqrt[2]*x^2)) + ((1 + 
Sqrt[2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x^4)/(1 + Sqrt[2]*x^2)^2]*EllipticE[2*Arc 
Tan[2^(1/4)*x], (2 - Sqrt[2])/4])/(2^(1/4)*Sqrt[1 + 2*x^2 + 2*x^4])) - (5* 
(2 + 2*Sqrt[2])*(1 + Sqrt[2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x^4)/(1 + Sqrt[2]*x^ 
2)^2]*EllipticF[2*ArcTan[2^(1/4)*x], (2 - Sqrt[2])/4])/(4*2^(1/4)*Sqrt[1 + 
 2*x^2 + 2*x^4]))/(5*(2 - 3*Sqrt[2])) + (4*(-1/2*((3 - Sqrt[2])*ArcTan[(Sq 
rt[5/3]*x)/Sqrt[1 + 2*x^2 + 2*x^4]])/Sqrt[30] + ((3 + Sqrt[2])*(1 + Sqrt[2 
]*x^2)*Sqrt[(1 + 2*x^2 + 2*x^4)/(1 + Sqrt[2]*x^2)^2]*EllipticPi[(12 - 11*S 
qrt[2])/24, 2*ArcTan[2^(1/4)*x], (2 - Sqrt[2])/4])/(12*2^(3/4)*Sqrt[1 + 2* 
x^2 + 2*x^4])))/(5*(2 - 3*Sqrt[2]))
 

3.4.52.3.1 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 1416

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

rule 1509

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

rule 1511

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

rule 1547

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_)/((d_.) + (e_.)*(x_)^2), x_Sym 
bol] :> Simp[-(c*d^2 - b*d*e + a*e^2)^(p + 1/2)/(e^(2*p)*(Rt[c/a, 2]*d - e) 
)   Int[(1 + Rt[c/a, 2]*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] 
+ Simp[(c*d^2 - b*d*e + a*e^2)^(p + 1/2)/(Rt[c/a, 2]*d - e)   Int[(a + b*x^ 
2 + c*x^4)^p*ExpandToSum[((Rt[c/a, 2]*d - e)*(c*d^2 - b*d*e + a*e^2)^(-p - 
1/2) + ((1 + Rt[c/a, 2]*x^2)*(a + b*x^2 + c*x^4)^(-p - 1/2))/e^(2*p))/(d + 
e*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N 
eQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[p + 1/2, 0] && NeQ[c*d^2 - a*e^2, 0] & 
& PosQ[c/a]
 

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]
 

rule 2220

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A 
rcTan[Rt[-b + c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ 
-b + c*(d/e) + a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a 
+ b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*El 
lipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] 
 /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] & 
& EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[-b + c*(d/e) + a*(e/d)]
 

3.4.52.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.32 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.58

method	result	size
risch	\(-\frac {x \left (3 x^{2}+1\right )}{5 \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {2 \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{5 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {\left (-\frac {3}{10}+\frac {3 i}{10}\right ) \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, \left (F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )-E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )\right )}{\sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {2 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \Pi \left (x \sqrt {-1+i}, \frac {1}{3}+\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{15 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\)	\(246\)
default	\(-\frac {4 \left (\frac {3}{20} x^{3}+\frac {1}{20} x \right )}{\sqrt {2 x^{4}+2 x^{2}+1}}+\frac {\sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{10 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {3 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{10 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {3 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{10 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {3 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{10 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {2 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \Pi \left (x \sqrt {-1+i}, \frac {1}{3}+\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{15 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\)	\(366\)
elliptic	\(-\frac {4 \left (\frac {3}{20} x^{3}+\frac {1}{20} x \right )}{\sqrt {2 x^{4}+2 x^{2}+1}}+\frac {\sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{10 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {3 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{10 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {3 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{10 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {3 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{10 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {2 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \Pi \left (x \sqrt {-1+i}, \frac {1}{3}+\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{15 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\)	\(366\)

input

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

output

-1/5*x*(3*x^2+1)/(2*x^4+2*x^2+1)^(1/2)+2/5/(-1+I)^(1/2)*(1+(1-I)*x^2)^(1/2 
)*(1+(1+I)*x^2)^(1/2)/(2*x^4+2*x^2+1)^(1/2)*EllipticF(x*(-1+I)^(1/2),1/2*2 
^(1/2)+1/2*I*2^(1/2))+(-3/10+3/10*I)/(-1+I)^(1/2)*(1+(1-I)*x^2)^(1/2)*(1+( 
1+I)*x^2)^(1/2)/(2*x^4+2*x^2+1)^(1/2)*(EllipticF(x*(-1+I)^(1/2),1/2*2^(1/2 
)+1/2*I*2^(1/2))-EllipticE(x*(-1+I)^(1/2),1/2*2^(1/2)+1/2*I*2^(1/2)))+2/15 
/(-1+I)^(1/2)*(1-I*x^2+x^2)^(1/2)*(1+I*x^2+x^2)^(1/2)/(2*x^4+2*x^2+1)^(1/2 
)*EllipticPi(x*(-1+I)^(1/2),1/3+1/3*I,(-1-I)^(1/2)/(-1+I)^(1/2))
 

3.4.52.5 Fricas [F]

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

input

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

output

integral(sqrt(2*x^4 + 2*x^2 + 1)/(8*x^10 + 28*x^8 + 40*x^6 + 32*x^4 + 14*x 
^2 + 3), x)
 

3.4.52.6 Sympy [F]

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

input

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

output

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

3.4.52.7 Maxima [F]

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

input

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

output

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

3.4.52.8 Giac [F]

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

input

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

output

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

3.4.52.9 Mupad [F(-1)]

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

input

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

output

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