3.4.49 \(\int \frac {x^6}{(3+2 x^2) (1+2 x^2+2 x^4)^{3/2}} \, dx\) [349]

3.4.49.1 Optimal result

Integrand size = 29, antiderivative size = 423 \[ \int \frac {x^6}{\left (3+2 x^2\right ) \left (1+2 x^2+2 x^4\right )^{3/2}} \, dx=\frac {x \left (1-2 x^2\right )}{20 \sqrt {1+2 x^2+2 x^4}}+\frac {x \sqrt {1+2 x^2+2 x^4}}{10 \sqrt {2} \left (1+\sqrt {2} x^2\right )}-\frac {9}{40} \sqrt {\frac {3}{5}} \arctan \left (\frac {\sqrt {\frac {5}{3}} x}{\sqrt {1+2 x^2+2 x^4}}\right )-\frac {\left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} E\left (2 \arctan \left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{10\ 2^{3/4} \sqrt {1+2 x^2+2 x^4}}-\frac {\left (\sqrt [4]{2}+2^{3/4}\right ) \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{8 \left (-2+3 \sqrt {2}\right ) \sqrt {1+2 x^2+2 x^4}}-\frac {9 \left (3+\sqrt {2}\right ) \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{24} \left (12-11 \sqrt {2}\right ),2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{40\ 2^{3/4} \left (2-3 \sqrt {2}\right ) \sqrt {1+2 x^2+2 x^4}} \]

output

-9/200*arctan(1/3*x*15^(1/2)/(2*x^4+2*x^2+1)^(1/2))*15^(1/2)+1/20*x*(-2*x^ 
2+1)/(2*x^4+2*x^2+1)^(1/2)+1/20*x*(2*x^4+2*x^2+1)^(1/2)*2^(1/2)/(1+x^2*2^( 
1/2))-1/20*(cos(2*arctan(2^(1/4)*x))^2)^(1/2)/cos(2*arctan(2^(1/4)*x))*Ell 
ipticE(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)-9/80* 
(cos(2*arctan(2^(1/4)*x))^2)^(1/2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin 
(2*arctan(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)-1/8*(2^(1/4)+2^(3/4))*(cos(2*arctan(2^(1/4)*x 
))^2)^(1/2)/cos(2*arctan(2^(1/4)*x))*EllipticF(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+3*2^(1/2))/(2*x^4+2*x^2+1)^(1/2)
 

3.4.49.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 9.68 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.47 \[ \int \frac {x^6}{\left (3+2 x^2\right ) \left (1+2 x^2+2 x^4\right )^{3/2}} \, dx=\frac {2 x-4 x^3-2 i \sqrt {1-i} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} E\left (\left .i \text {arcsinh}\left (\sqrt {1-i} x\right )\right |i\right )+(8-6 i) \sqrt {1-i} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {1-i} x\right ),i\right )-9 (1-i)^{3/2} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} \operatorname {EllipticPi}\left (\frac {1}{3}+\frac {i}{3},i \text {arcsinh}\left (\sqrt {1-i} x\right ),i\right )}{40 \sqrt {1+2 x^2+2 x^4}} \]

input

Integrate[x^6/((3 + 2*x^2)*(1 + 2*x^2 + 2*x^4)^(3/2)),x]
 

output

(2*x - 4*x^3 - (2*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^ 
2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] + (8 - 6*I)*Sqrt[1 - I]*Sqrt[1 + 
 (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] 
 - 9*(1 - I)^(3/2)*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[ 
1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/(40*Sqrt[1 + 2*x^2 + 2*x^4])
 

3.4.49.3 Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {1638, 25, 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.

\(\displaystyle \int \frac {x^6}{\left (2 x^2+3\right ) \left (2 x^4+2 x^2+1\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1638

\(\displaystyle -\frac {27 \int \frac {\sqrt {2} x^2+1}{\left (2 x^2+3\right ) \sqrt {2 x^4+2 x^2+1}}dx}{10 \left (2-3 \sqrt {2}\right )}-\frac {\int -\frac {2 \left (5+6 \sqrt {2}\right ) x^4+3 \left (4+3 \sqrt {2}\right ) x^2+9}{\left (2 x^4+2 x^2+1\right )^{3/2}}dx}{10 \left (2-3 \sqrt {2}\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {2 \left (5+6 \sqrt {2}\right ) x^4+3 \left (4+3 \sqrt {2}\right ) x^2+9}{\left (2 x^4+2 x^2+1\right )^{3/2}}dx}{10 \left (2-3 \sqrt {2}\right )}-\frac {27 \int \frac {\sqrt {2} x^2+1}{\left (2 x^2+3\right ) \sqrt {2 x^4+2 x^2+1}}dx}{10 \left (2-3 \sqrt {2}\right )}\)

\(\Big \downarrow \) 2206

\(\displaystyle \frac {\frac {1}{4} \int \frac {2 \left (2 \left (2-3 \sqrt {2}\right ) x^2+3 \sqrt {2}+16\right )}{\sqrt {2 x^4+2 x^2+1}}dx+\frac {x \left (-2 \left (2-3 \sqrt {2}\right ) x^2-3 \sqrt {2}+2\right )}{2 \sqrt {2 x^4+2 x^2+1}}}{10 \left (2-3 \sqrt {2}\right )}-\frac {27 \int \frac {\sqrt {2} x^2+1}{\left (2 x^2+3\right ) \sqrt {2 x^4+2 x^2+1}}dx}{10 \left (2-3 \sqrt {2}\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{2} \int \frac {2 \left (2-3 \sqrt {2}\right ) x^2+3 \sqrt {2}+16}{\sqrt {2 x^4+2 x^2+1}}dx+\frac {x \left (-2 \left (2-3 \sqrt {2}\right ) x^2-3 \sqrt {2}+2\right )}{2 \sqrt {2 x^4+2 x^2+1}}}{10 \left (2-3 \sqrt {2}\right )}-\frac {27 \int \frac {\sqrt {2} x^2+1}{\left (2 x^2+3\right ) \sqrt {2 x^4+2 x^2+1}}dx}{10 \left (2-3 \sqrt {2}\right )}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {\frac {1}{2} \left (5 \left (2+\sqrt {2}\right ) \int \frac {1}{\sqrt {2 x^4+2 x^2+1}}dx+2 \left (3-\sqrt {2}\right ) \int \frac {1-\sqrt {2} x^2}{\sqrt {2 x^4+2 x^2+1}}dx\right )+\frac {x \left (-2 \left (2-3 \sqrt {2}\right ) x^2-3 \sqrt {2}+2\right )}{2 \sqrt {2 x^4+2 x^2+1}}}{10 \left (2-3 \sqrt {2}\right )}-\frac {27 \int \frac {\sqrt {2} x^2+1}{\left (2 x^2+3\right ) \sqrt {2 x^4+2 x^2+1}}dx}{10 \left (2-3 \sqrt {2}\right )}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {\frac {1}{2} \left (2 \left (3-\sqrt {2}\right ) \int \frac {1-\sqrt {2} x^2}{\sqrt {2 x^4+2 x^2+1}}dx+\frac {5 \left (2+\sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{2 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}\right )+\frac {x \left (-2 \left (2-3 \sqrt {2}\right ) x^2-3 \sqrt {2}+2\right )}{2 \sqrt {2 x^4+2 x^2+1}}}{10 \left (2-3 \sqrt {2}\right )}-\frac {27 \int \frac {\sqrt {2} x^2+1}{\left (2 x^2+3\right ) \sqrt {2 x^4+2 x^2+1}}dx}{10 \left (2-3 \sqrt {2}\right )}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {\frac {1}{2} \left (\frac {5 \left (2+\sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{2 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}+2 \left (3-\sqrt {2}\right ) \left (\frac {\left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} E\left (2 \arctan \left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{\sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}-\frac {x \sqrt {2 x^4+2 x^2+1}}{\sqrt {2} x^2+1}\right )\right )+\frac {x \left (-2 \left (2-3 \sqrt {2}\right ) x^2-3 \sqrt {2}+2\right )}{2 \sqrt {2 x^4+2 x^2+1}}}{10 \left (2-3 \sqrt {2}\right )}-\frac {27 \int \frac {\sqrt {2} x^2+1}{\left (2 x^2+3\right ) \sqrt {2 x^4+2 x^2+1}}dx}{10 \left (2-3 \sqrt {2}\right )}\)

\(\Big \downarrow \) 2220

\(\displaystyle \frac {\frac {1}{2} \left (\frac {5 \left (2+\sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{2 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}+2 \left (3-\sqrt {2}\right ) \left (\frac {\left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} E\left (2 \arctan \left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{\sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}-\frac {x \sqrt {2 x^4+2 x^2+1}}{\sqrt {2} x^2+1}\right )\right )+\frac {x \left (-2 \left (2-3 \sqrt {2}\right ) x^2-3 \sqrt {2}+2\right )}{2 \sqrt {2 x^4+2 x^2+1}}}{10 \left (2-3 \sqrt {2}\right )}-\frac {27 \left (\frac {\left (3+\sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{24} \left (12-11 \sqrt {2}\right ),2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{12\ 2^{3/4} \sqrt {2 x^4+2 x^2+1}}-\frac {\left (3-\sqrt {2}\right ) \arctan \left (\frac {\sqrt {\frac {5}{3}} x}{\sqrt {2 x^4+2 x^2+1}}\right )}{2 \sqrt {30}}\right )}{10 \left (2-3 \sqrt {2}\right )}\)

input

Int[x^6/((3 + 2*x^2)*(1 + 2*x^2 + 2*x^4)^(3/2)),x]
 

output

((x*(2 - 3*Sqrt[2] - 2*(2 - 3*Sqrt[2])*x^2))/(2*Sqrt[1 + 2*x^2 + 2*x^4]) + 
 (2*(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*A 
rcTan[2^(1/4)*x], (2 - Sqrt[2])/4])/(2^(1/4)*Sqrt[1 + 2*x^2 + 2*x^4])) + ( 
5*(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])/(2*2^(1/4)*Sqrt[1 + 
 2*x^2 + 2*x^4]))/2)/(10*(2 - 3*Sqrt[2])) - (27*(-1/2*((3 - Sqrt[2])*ArcTa 
n[(Sqrt[5/3]*x)/Sqrt[1 + 2*x^2 + 2*x^4]])/Sqrt[30] + ((3 + Sqrt[2])*(1 + S 
qrt[2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x^4)/(1 + Sqrt[2]*x^2)^2]*EllipticPi[(12 - 
 11*Sqrt[2])/24, 2*ArcTan[2^(1/4)*x], (2 - Sqrt[2])/4])/(12*2^(3/4)*Sqrt[1 
 + 2*x^2 + 2*x^4])))/(10*(2 - 3*Sqrt[2]))
 

3.4.49.3.1 Defintions of rubi rules used

rule 25

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

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 1638

Int[((x_)^(m_)*((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_))/((d_) + (e_.)*(x 
_)^2), x_Symbol] :> Simp[(-(-d/e)^(m/2))*((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)*x^m + ((-d/e)^(m/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] && ILtQ[p + 1/2, 0] && IGtQ[m/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)]
 

3.4.49.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.52 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.58

method	result	size
risch	\(-\frac {x \left (2 x^{2}-1\right )}{20 \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {2 \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{5 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {\left (-\frac {1}{20}+\frac {i}{20}\right ) \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, \left (F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )-E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )\right )}{\sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {9 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \Pi \left (x \sqrt {-1+i}, \frac {1}{3}+\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\)	\(246\)
elliptic	\(-\frac {4 \left (\frac {1}{40} x^{3}-\frac {1}{80} x \right )}{\sqrt {2 x^{4}+2 x^{2}+1}}+\frac {7 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {\sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {9 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \Pi \left (x \sqrt {-1+i}, \frac {1}{3}+\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\)	\(366\)
default	\(-\frac {9 x^{3}}{8 \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {7 \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{4 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {\left (-\frac {17}{16}+\frac {17 i}{16}\right ) \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, \left (F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )-E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )\right )}{\sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {2 \left (\frac {1}{8} x^{3}+\frac {1}{8} x \right )}{\sqrt {2 x^{4}+2 x^{2}+1}}+\frac {-\frac {3}{4} x^{3}-\frac {3}{8} x}{\sqrt {2 x^{4}+2 x^{2}+1}}+\frac {\frac {81}{40} x^{3}+\frac {27}{40} x}{\sqrt {2 x^{4}+2 x^{2}+1}}-\frac {27 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{80 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {81 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{80 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {81 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{80 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {81 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{80 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {9 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \Pi \left (x \sqrt {-1+i}, \frac {1}{3}+\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\)	\(586\)

input

int(x^6/(2*x^2+3)/(2*x^4+2*x^2+1)^(3/2),x,method=_RETURNVERBOSE)
 

output

-1/20*x*(2*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))+(-1/20+1/20*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)))-9/2 
0/(-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))
 

3.4.49.5 Fricas [F]

\[ \int \frac {x^6}{\left (3+2 x^2\right ) \left (1+2 x^2+2 x^4\right )^{3/2}} \, dx=\int { \frac {x^{6}}{{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac {3}{2}} {\left (2 \, x^{2} + 3\right )}} \,d x } \]

input

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

3.4.49.6 Sympy [F]

\[ \int \frac {x^6}{\left (3+2 x^2\right ) \left (1+2 x^2+2 x^4\right )^{3/2}} \, dx=\int \frac {x^{6}}{\left (2 x^{2} + 3\right ) \left (2 x^{4} + 2 x^{2} + 1\right )^{\frac {3}{2}}}\, dx \]

input

integrate(x**6/(2*x**2+3)/(2*x**4+2*x**2+1)**(3/2),x)
 

output

Integral(x**6/((2*x**2 + 3)*(2*x**4 + 2*x**2 + 1)**(3/2)), x)
 

3.4.49.7 Maxima [F]

\[ \int \frac {x^6}{\left (3+2 x^2\right ) \left (1+2 x^2+2 x^4\right )^{3/2}} \, dx=\int { \frac {x^{6}}{{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac {3}{2}} {\left (2 \, x^{2} + 3\right )}} \,d x } \]

input

integrate(x^6/(2*x^2+3)/(2*x^4+2*x^2+1)^(3/2),x, algorithm="maxima")
 

output

integrate(x^6/((2*x^4 + 2*x^2 + 1)^(3/2)*(2*x^2 + 3)), x)
 

3.4.49.8 Giac [F]

\[ \int \frac {x^6}{\left (3+2 x^2\right ) \left (1+2 x^2+2 x^4\right )^{3/2}} \, dx=\int { \frac {x^{6}}{{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac {3}{2}} {\left (2 \, x^{2} + 3\right )}} \,d x } \]

input

integrate(x^6/(2*x^2+3)/(2*x^4+2*x^2+1)^(3/2),x, algorithm="giac")
 

output

integrate(x^6/((2*x^4 + 2*x^2 + 1)^(3/2)*(2*x^2 + 3)), x)
 

3.4.49.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{\left (3+2 x^2\right ) \left (1+2 x^2+2 x^4\right )^{3/2}} \, dx=\int \frac {x^6}{\left (2\,x^2+3\right )\,{\left (2\,x^4+2\,x^2+1\right )}^{3/2}} \,d x \]

input

int(x^6/((2*x^2 + 3)*(2*x^2 + 2*x^4 + 1)^(3/2)),x)
 

output

int(x^6/((2*x^2 + 3)*(2*x^2 + 2*x^4 + 1)^(3/2)), x)