Integrand size = 26, antiderivative size = 428 \[ \int \frac {\left (1+2 x^2+2 x^4\right )^{3/2}}{3-2 x^2} \, dx=-\frac {1}{10} x \left (9+2 x^2\right ) \sqrt {1+2 x^2+2 x^4}-\frac {103 x \sqrt {1+2 x^2+2 x^4}}{10 \sqrt {2} \left (1+\sqrt {2} x^2\right )}+\frac {17}{8} \sqrt {\frac {17}{3}} \text {arctanh}\left (\frac {\sqrt {\frac {17}{3}} x}{\sqrt {1+2 x^2+2 x^4}}\right )+\frac {103 \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} E\left (2 \arctan \left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{10\ 2^{3/4} \sqrt {1+2 x^2+2 x^4}}-\frac {\left (66+383 \sqrt {2}\right ) \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{10\ 2^{3/4} \left (2+3 \sqrt {2}\right ) \sqrt {1+2 x^2+2 x^4}}-\frac {289 \left (3-\sqrt {2}\right ) \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{24} \left (12+11 \sqrt {2}\right ),2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{24\ 2^{3/4} \left (2+3 \sqrt {2}\right ) \sqrt {1+2 x^2+2 x^4}} \]
17/24*arctanh(1/3*x*51^(1/2)/(2*x^4+2*x^2+1)^(1/2))*51^(1/2)-1/10*x*(2*x^2 +9)*(2*x^4+2*x^2+1)^(1/2)-103/20*x*(2*x^4+2*x^2+1)^(1/2)*2^(1/2)/(1+x^2*2^ (1/2))+103/20*(cos(2*arctan(2^(1/4)*x))^2)^(1/2)/cos(2*arctan(2^(1/4)*x))* EllipticE(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)-28 9/48*(cos(2*arctan(2^(1/4)*x))^2)^(1/2)/cos(2*arctan(2^(1/4)*x))*EllipticP i(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)-1/20*(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))*(66+383*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)
Result contains complex when optimal does not.
Time = 10.20 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.49 \[ \int \frac {\left (1+2 x^2+2 x^4\right )^{3/2}}{3-2 x^2} \, dx=\frac {-108 x-240 x^3-264 x^5-48 x^7+618 i \sqrt {1-i} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} E\left (\left .i \text {arcsinh}\left (\sqrt {1-i} x\right )\right |i\right )-(1371-753 i) \sqrt {1-i} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {1-i} x\right ),i\right )+1445 (1-i)^{3/2} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} \operatorname {EllipticPi}\left (-\frac {1}{3}-\frac {i}{3},i \text {arcsinh}\left (\sqrt {1-i} x\right ),i\right )}{120 \sqrt {1+2 x^2+2 x^4}} \]
(-108*x - 240*x^3 - 264*x^5 - 48*x^7 + (618*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I )*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] - (137 1 - 753*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*Ellipti cF[I*ArcSinh[Sqrt[1 - I]*x], I] + 1445*(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] )/(120*Sqrt[1 + 2*x^2 + 2*x^4])
Time = 1.04 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {1530, 25, 27, 2207, 27, 2207, 27, 1511, 1416, 1509, 2222}
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.
| \(\displaystyle \int \frac {\left (2 x^4+2 x^2+1\right )^{3/2}}{3-2 x^2} \, dx\) |
| \(\Big \downarrow \) 1530 |
| \(\displaystyle \frac {289}{28} \int -\frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx-\frac {1}{112} \int -\frac {4 \left (-56 x^6-196 x^4-406 x^2-289 \sqrt {2}+202\right )}{\sqrt {2 x^4+2 x^2+1}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {289}{28} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx-\frac {1}{112} \int -\frac {4 \left (-56 x^6-196 x^4-406 x^2-289 \sqrt {2}+202\right )}{\sqrt {2 x^4+2 x^2+1}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \int \frac {-56 x^6-196 x^4-406 x^2-289 \sqrt {2}+202}{\sqrt {2 x^4+2 x^2+1}}dx-\frac {289}{28} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx\) |
| \(\Big \downarrow \) 2207 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{10} \int \frac {2 \left (-756 x^4-1946 x^2+5 \left (202-289 \sqrt {2}\right )\right )}{\sqrt {2 x^4+2 x^2+1}}dx-\frac {28}{5} x^3 \sqrt {2 x^4+2 x^2+1}\right )-\frac {289}{28} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{5} \int \frac {-756 x^4-1946 x^2+5 \left (202-289 \sqrt {2}\right )}{\sqrt {2 x^4+2 x^2+1}}dx-\frac {28}{5} x^3 \sqrt {2 x^4+2 x^2+1}\right )-\frac {289}{28} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx\) |
| \(\Big \downarrow \) 2207 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{5} \left (\frac {1}{6} \int \frac {6 \left (-1442 x^2-1445 \sqrt {2}+1136\right )}{\sqrt {2 x^4+2 x^2+1}}dx-126 x \sqrt {2 x^4+2 x^2+1}\right )-\frac {28}{5} x^3 \sqrt {2 x^4+2 x^2+1}\right )-\frac {289}{28} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{5} \left (\int \frac {-1442 x^2-1445 \sqrt {2}+1136}{\sqrt {2 x^4+2 x^2+1}}dx-126 x \sqrt {2 x^4+2 x^2+1}\right )-\frac {28}{5} x^3 \sqrt {2 x^4+2 x^2+1}\right )-\frac {289}{28} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{5} \left (2 \left (568-1083 \sqrt {2}\right ) \int \frac {1}{\sqrt {2 x^4+2 x^2+1}}dx+721 \sqrt {2} \int \frac {1-\sqrt {2} x^2}{\sqrt {2 x^4+2 x^2+1}}dx-126 \sqrt {2 x^4+2 x^2+1} x\right )-\frac {28}{5} x^3 \sqrt {2 x^4+2 x^2+1}\right )-\frac {289}{28} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{5} \left (721 \sqrt {2} \int \frac {1-\sqrt {2} x^2}{\sqrt {2 x^4+2 x^2+1}}dx+\frac {\left (568-1083 \sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{\sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}-126 \sqrt {2 x^4+2 x^2+1} x\right )-\frac {28}{5} x^3 \sqrt {2 x^4+2 x^2+1}\right )-\frac {289}{28} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{5} \left (\frac {\left (568-1083 \sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{\sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}+721 \sqrt {2} \left (\frac {\left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} E\left (2 \arctan \left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{\sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}-\frac {x \sqrt {2 x^4+2 x^2+1}}{\sqrt {2} x^2+1}\right )-126 \sqrt {2 x^4+2 x^2+1} x\right )-\frac {28}{5} x^3 \sqrt {2 x^4+2 x^2+1}\right )-\frac {289}{28} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx\) |
| \(\Big \downarrow \) 2222 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{5} \left (\frac {\left (568-1083 \sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{\sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}+721 \sqrt {2} \left (\frac {\left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} E\left (2 \arctan \left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{\sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}-\frac {x \sqrt {2 x^4+2 x^2+1}}{\sqrt {2} x^2+1}\right )-126 \sqrt {2 x^4+2 x^2+1} x\right )-\frac {28}{5} x^3 \sqrt {2 x^4+2 x^2+1}\right )-\frac {289}{28} \left (\frac {\left (3-\sqrt {2}\right )^2 \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{24} \left (12+11 \sqrt {2}\right ),2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{12 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}-\frac {7 \text {arctanh}\left (\frac {\sqrt {\frac {17}{3}} x}{\sqrt {2 x^4+2 x^2+1}}\right )}{2 \sqrt {51}}\right )\) |
((-28*x^3*Sqrt[1 + 2*x^2 + 2*x^4])/5 + (-126*x*Sqrt[1 + 2*x^2 + 2*x^4] + 7 21*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*ArcTan[2 ^(1/4)*x], (2 - Sqrt[2])/4])/(2^(1/4)*Sqrt[1 + 2*x^2 + 2*x^4])) + ((568 - 1083*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^(1/4)*Sqrt[1 + 2*x ^2 + 2*x^4]))/5)/28 - (289*((-7*ArcTanh[(Sqrt[17/3]*x)/Sqrt[1 + 2*x^2 + 2* x^4]])/(2*Sqrt[51]) + ((3 - Sqrt[2])^2*(1 + Sqrt[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^(1/4)*Sqrt[1 + 2*x^2 + 2*x^4])))/28
3.4.28.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[-(c*d^2 - b*d*e + a*e^2)^(p + 1/2)/(e^(2*p)*(c*d^2 - a*e^2)) Int[(a*d*Rt[c/a, 2] + a*e + (c*d + a*e*Rt[c/a, 2])*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] + Simp[1/(e^(2*p)*(c*d^2 - a*e^2)) Int[(1/Sqrt [a + b*x^2 + c*x^4])*ExpandToSum[(e^(2*p)*(c*d^2 - a*e^2)*(a + b*x^2 + c*x^ 4)^(p + 1/2) + (c*d^2 - b*d*e + a*e^2)^(p + 1/2)*(a*d*Rt[c/a, 2] + a*e + (c *d + a*e*Rt[c/a, 2])*x^2))/(d + e*x^2), x], 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] && IGtQ[p - 1/ 2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{n = Expon[Px, x^2], e = Coeff[Px, x^2, Expon[Px, x^2]]}, Simp[e*x^(2*n - 3)*(( a + b*x^2 + c*x^4)^(p + 1)/(c*(2*n + 4*p + 1))), x] + Simp[1/(c*(2*n + 4*p + 1)) Int[(a + b*x^2 + c*x^4)^p*ExpandToSum[c*(2*n + 4*p + 1)*Px - a*e*(2 *n - 3)*x^(2*n - 4) - b*e*(2*n + 2*p - 1)*x^(2*n - 2) - c*e*(2*n + 4*p + 1) *x^(2*n), x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Px, x^2] && Expon[ Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && !LtQ[p, -1]
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 rcTanh[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]))*Ell ipticPi[-(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] && NegQ[-b + c*(d/e) + a*(e/d)]
Result contains complex when optimal does not.
Time = 1.35 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.57
| method | result | size |
| risch | \(-\frac {x \left (2 x^{2}+9\right ) \sqrt {2 x^{4}+2 x^{2}+1}}{10}-\frac {457 \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {\left (\frac {103}{20}-\frac {103 i}{20}\right ) \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, \left (F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )-E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )\right )}{\sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {289 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \Pi \left (x \sqrt {-1+i}, -\frac {1}{3}-\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{12 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\) | \(246\) |
| default | \(-\frac {x^{3} \sqrt {2 x^{4}+2 x^{2}+1}}{5}-\frac {9 x \sqrt {2 x^{4}+2 x^{2}+1}}{10}-\frac {177 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{10 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {103 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {103 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {103 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {289 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \Pi \left (x \sqrt {-1+i}, -\frac {1}{3}-\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{12 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\) | \(377\) |
| elliptic | \(-\frac {x^{3} \sqrt {2 x^{4}+2 x^{2}+1}}{5}-\frac {9 x \sqrt {2 x^{4}+2 x^{2}+1}}{10}-\frac {177 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{10 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {103 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {103 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {103 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {289 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \Pi \left (x \sqrt {-1+i}, -\frac {1}{3}-\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{12 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\) | \(377\) |
-1/10*x*(2*x^2+9)*(2*x^4+2*x^2+1)^(1/2)-457/20/(-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))+(103/20-103/20*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) ))+289/12/(-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))
\[ \int \frac {\left (1+2 x^2+2 x^4\right )^{3/2}}{3-2 x^2} \, dx=\int { -\frac {{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac {3}{2}}}{2 \, x^{2} - 3} \,d x } \]
\[ \int \frac {\left (1+2 x^2+2 x^4\right )^{3/2}}{3-2 x^2} \, dx=- \int \frac {\sqrt {2 x^{4} + 2 x^{2} + 1}}{2 x^{2} - 3}\, dx - \int \frac {2 x^{2} \sqrt {2 x^{4} + 2 x^{2} + 1}}{2 x^{2} - 3}\, dx - \int \frac {2 x^{4} \sqrt {2 x^{4} + 2 x^{2} + 1}}{2 x^{2} - 3}\, dx \]
-Integral(sqrt(2*x**4 + 2*x**2 + 1)/(2*x**2 - 3), x) - Integral(2*x**2*sqr t(2*x**4 + 2*x**2 + 1)/(2*x**2 - 3), x) - Integral(2*x**4*sqrt(2*x**4 + 2* x**2 + 1)/(2*x**2 - 3), x)
\[ \int \frac {\left (1+2 x^2+2 x^4\right )^{3/2}}{3-2 x^2} \, dx=\int { -\frac {{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac {3}{2}}}{2 \, x^{2} - 3} \,d x } \]
\[ \int \frac {\left (1+2 x^2+2 x^4\right )^{3/2}}{3-2 x^2} \, dx=\int { -\frac {{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac {3}{2}}}{2 \, x^{2} - 3} \,d x } \]
Timed out. \[ \int \frac {\left (1+2 x^2+2 x^4\right )^{3/2}}{3-2 x^2} \, dx=-\int \frac {{\left (2\,x^4+2\,x^2+1\right )}^{3/2}}{2\,x^2-3} \,d x \]