3.4.27 \(\int \frac {x^2 (1+2 x^2+2 x^4)^{3/2}}{3-2 x^2} \, dx\) [327]

Optimal. Leaf size=463 \[ -\frac {213}{140} x \sqrt {1+2 x^2+2 x^4}-\frac {27}{70} x^3 \sqrt {1+2 x^2+2 x^4}-\frac {2211 x \sqrt {1+2 x^2+2 x^4}}{140 \sqrt {2} \left (1+\sqrt {2} x^2\right )}-\frac {1}{14} x \left (1+2 x^2+2 x^4\right )^{3/2}+\frac {17}{16} \sqrt {51} \tanh ^{-1}\left (\frac {\sqrt {\frac {17}{3}} x}{\sqrt {1+2 x^2+2 x^4}}\right )+\frac {2211 \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{140\ 2^{3/4} \sqrt {1+2 x^2+2 x^4}}-\frac {3 \left (514+2717 \sqrt {2}\right ) \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{140\ 2^{3/4} \left (2+3 \sqrt {2}\right ) \sqrt {1+2 x^2+2 x^4}}-\frac {289 \left (3-\sqrt {2}\right ) \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} \Pi \left (\frac {1}{24} \left (12+11 \sqrt {2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{16\ 2^{3/4} \left (2+3 \sqrt {2}\right ) \sqrt {1+2 x^2+2 x^4}} \]

[Out]

-1/14*x*(2*x^4+2*x^2+1)^(3/2)+17/16*arctanh(1/3*x*51^(1/2)/(2*x^4+2*x^2+1)^(1/2))*51^(1/2)-213/140*x*(2*x^4+2*
x^2+1)^(1/2)-27/70*x^3*(2*x^4+2*x^2+1)^(1/2)-2211/280*x*(2*x^4+2*x^2+1)^(1/2)*2^(1/2)/(1+x^2*2^(1/2))+2211/280
*(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)-289/32*(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)-3/280*(cos(2*arctan(2^(1/4)*x))^2)^(1/2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arc
tan(2^(1/4)*x)),1/2*(2-2^(1/2))^(1/2))*(514+2717*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)

________________________________________________________________________________________

Rubi [A]
time = 0.43, antiderivative size = 875, normalized size of antiderivative = 1.89, number of steps used = 19, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {1349, 1105, 1190, 1211, 1117, 1209, 1222, 1230, 1720} \begin {gather*} -\frac {1}{14} x \left (2 x^4+2 x^2+1\right )^{3/2}-\frac {3}{35} x \left (x^2+2\right ) \sqrt {2 x^4+2 x^2+1}-\frac {3}{20} x \left (2 x^2+9\right ) \sqrt {2 x^4+2 x^2+1}-\frac {6 \sqrt {2} x \sqrt {2 x^4+2 x^2+1}}{35 \left (\sqrt {2} x^2+1\right )}-\frac {309 x \sqrt {2 x^4+2 x^2+1}}{20 \sqrt {2} \left (\sqrt {2} x^2+1\right )}+\frac {17}{16} \sqrt {51} \tanh ^{-1}\left (\frac {\sqrt {\frac {17}{3}} x}{\sqrt {2 x^4+2 x^2+1}}\right )+\frac {6 \sqrt [4]{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}} E\left (2 \text {ArcTan}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{35 \sqrt {2 x^4+2 x^2+1}}+\frac {309 \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 \text {ArcTan}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{20\ 2^{3/4} \sqrt {2 x^4+2 x^2+1}}-\frac {3 \left (9+8 \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}} F\left (2 \text {ArcTan}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{20\ 2^{3/4} \sqrt {2 x^4+2 x^2+1}}-\frac {3 \left (3+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}} F\left (2 \text {ArcTan}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{70 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}-\frac {51 \left (5+\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}} F\left (2 \text {ArcTan}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{16 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}+\frac {867 \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}} F\left (2 \text {ArcTan}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{112 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}-\frac {289 \left (11-6 \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}} \Pi \left (\frac {1}{24} \left (12+11 \sqrt {2}\right );2 \text {ArcTan}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{224 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x*(2 + x^2)*Sqrt[1 + 2*x^2 + 2*x^4])/35 - (3*x*(9 + 2*x^2)*Sqrt[1 + 2*x^2 + 2*x^4])/20 - (309*x*Sqrt[1 + 2
*x^2 + 2*x^4])/(20*Sqrt[2]*(1 + Sqrt[2]*x^2)) - (6*Sqrt[2]*x*Sqrt[1 + 2*x^2 + 2*x^4])/(35*(1 + Sqrt[2]*x^2)) -
(x*(1 + 2*x^2 + 2*x^4)^(3/2))/14 + (17*Sqrt[51]*ArcTanh[(Sqrt[17/3]*x)/Sqrt[1 + 2*x^2 + 2*x^4]])/16 + (309*(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])
/(20*2^(3/4)*Sqrt[1 + 2*x^2 + 2*x^4]) + (6*2^(1/4)*(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])/(35*Sqrt[1 + 2*x^2 + 2*x^4]) + (867*(3 - Sqrt[2])*(1 + S
qrt[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])/(11
2*2^(1/4)*Sqrt[1 + 2*x^2 + 2*x^4]) - (51*(5 + 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])/(16*2^(1/4)*Sqrt[1 + 2*x^2 + 2*x^4]) - (3*(3 + 2*Sqr
t[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])/(70*2^(1/4)*Sqrt[1 + 2*x^2 + 2*x^4]) - (3*(9 + 8*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])/(20*2^(3/4)*Sqrt[1 + 2*x^2 + 2*x^4]) - (
289*(11 - 6*Sqrt[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])/(224*2^(1/4)*Sqrt[1 + 2*x^2 + 2*x^4])

Rule 1105

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*((a + b*x^2 + c*x^4)^p/(4*p + 1)), x] + Dis
t[2*(p/(4*p + 1)), Int[(2*a + b*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4
*a*c, 0] && GtQ[p, 0] && IntegerQ[2*p]

Rule 1117

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 1190

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*(2*b*e*p + c*d*(4*p +
3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c*x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Dist[2*(p/(c*(4*p + 1)*(4*p + 3
))), Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) - b^2*e*(2*p + 1))*x^2, x]*(a +
b*x^2 + c*x^4)^(p - 1), 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] && GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]

Rule 1209

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> 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 1211

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(
e + d*q)/q, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[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] && PosQ[c/a]

Rule 1222

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[-(e^2)^(-1), Int[(c*d -
b*e - c*e*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x], x] + Dist[(c*d^2 - b*d*e + a*e^2)/e^2, Int[(a + b*x^2 + c*x^4
)^(p - 1)/(d + e*x^2), 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]

Rule 1230

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]}, Di
st[(c*d + a*e*q)/(c*d^2 - a*e^2), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[(a*e*(e + d*q))/(c*d^2 - a*e^2)
, Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), 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] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]

Rule 1349

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x]
&& NeQ[b^2 - 4*a*c, 0] && (IGtQ[p, 0] || IGtQ[q, 0] || IntegersQ[m, q])

Rule 1720

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))*(ArcTan[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)*(A + B*x^2)*(Sqrt[A^2*((a + b*x^2 + c*x^4)/(a*(A + B*
x^2)^2))]/(4*d*e*A*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1
/2 - b*(A/(4*a*B))], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^
2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rubi steps

\begin {align*} \int \frac {x^2 \left (1+2 x^2+2 x^4\right )^{3/2}}{3-2 x^2} \, dx &=\int \left (-\frac {1}{2} \left (1+2 x^2+2 x^4\right )^{3/2}+\frac {3 \left (1+2 x^2+2 x^4\right )^{3/2}}{2 \left (3-2 x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \left (1+2 x^2+2 x^4\right )^{3/2} \, dx\right )+\frac {3}{2} \int \frac {\left (1+2 x^2+2 x^4\right )^{3/2}}{3-2 x^2} \, dx\\ &=-\frac {1}{14} x \left (1+2 x^2+2 x^4\right )^{3/2}-\frac {3}{14} \int \left (2+2 x^2\right ) \sqrt {1+2 x^2+2 x^4} \, dx-\frac {3}{8} \int \left (10+4 x^2\right ) \sqrt {1+2 x^2+2 x^4} \, dx+\frac {51}{4} \int \frac {\sqrt {1+2 x^2+2 x^4}}{3-2 x^2} \, dx\\ &=-\frac {3}{35} x \left (2+x^2\right ) \sqrt {1+2 x^2+2 x^4}-\frac {3}{20} x \left (9+2 x^2\right ) \sqrt {1+2 x^2+2 x^4}-\frac {1}{14} x \left (1+2 x^2+2 x^4\right )^{3/2}-\frac {1}{140} \int \frac {36+48 x^2}{\sqrt {1+2 x^2+2 x^4}} \, dx-\frac {1}{80} \int \frac {192+216 x^2}{\sqrt {1+2 x^2+2 x^4}} \, dx-\frac {51}{16} \int \frac {10+4 x^2}{\sqrt {1+2 x^2+2 x^4}} \, dx+\frac {867}{8} \int \frac {1}{\left (3-2 x^2\right ) \sqrt {1+2 x^2+2 x^4}} \, dx\\ &=-\frac {3}{35} x \left (2+x^2\right ) \sqrt {1+2 x^2+2 x^4}-\frac {3}{20} x \left (9+2 x^2\right ) \sqrt {1+2 x^2+2 x^4}-\frac {1}{14} x \left (1+2 x^2+2 x^4\right )^{3/2}+\frac {27 \int \frac {1-\sqrt {2} x^2}{\sqrt {1+2 x^2+2 x^4}} \, dx}{10 \sqrt {2}}+\frac {51 \int \frac {1-\sqrt {2} x^2}{\sqrt {1+2 x^2+2 x^4}} \, dx}{4 \sqrt {2}}+\frac {1}{35} \left (6 \sqrt {2}\right ) \int \frac {1-\sqrt {2} x^2}{\sqrt {1+2 x^2+2 x^4}} \, dx-\frac {1}{56} \left (867 \left (2-3 \sqrt {2}\right )\right ) \int \frac {1+\sqrt {2} x^2}{\left (3-2 x^2\right ) \sqrt {1+2 x^2+2 x^4}} \, dx+\frac {1}{56} \left (867 \left (3-\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+2 x^2+2 x^4}} \, dx-\frac {1}{8} \left (51 \left (5+\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+2 x^2+2 x^4}} \, dx-\frac {1}{35} \left (3 \left (3+2 \sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+2 x^2+2 x^4}} \, dx-\frac {1}{20} \left (3 \left (16+9 \sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+2 x^2+2 x^4}} \, dx\\ &=-\frac {3}{35} x \left (2+x^2\right ) \sqrt {1+2 x^2+2 x^4}-\frac {3}{20} x \left (9+2 x^2\right ) \sqrt {1+2 x^2+2 x^4}-\frac {309 x \sqrt {1+2 x^2+2 x^4}}{20 \sqrt {2} \left (1+\sqrt {2} x^2\right )}-\frac {6 \sqrt {2} x \sqrt {1+2 x^2+2 x^4}}{35 \left (1+\sqrt {2} x^2\right )}-\frac {1}{14} x \left (1+2 x^2+2 x^4\right )^{3/2}+\frac {17}{16} \sqrt {51} \tanh ^{-1}\left (\frac {\sqrt {\frac {17}{3}} x}{\sqrt {1+2 x^2+2 x^4}}\right )+\frac {309 \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 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{20\ 2^{3/4} \sqrt {1+2 x^2+2 x^4}}+\frac {6 \sqrt [4]{2} \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 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{35 \sqrt {1+2 x^2+2 x^4}}+\frac {867 \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}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{112 \sqrt [4]{2} \sqrt {1+2 x^2+2 x^4}}-\frac {51 \left (5+\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}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{16 \sqrt [4]{2} \sqrt {1+2 x^2+2 x^4}}-\frac {3 \left (3+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}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{70 \sqrt [4]{2} \sqrt {1+2 x^2+2 x^4}}-\frac {3 \left (9+8 \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}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{20\ 2^{3/4} \sqrt {1+2 x^2+2 x^4}}-\frac {289 \left (11-6 \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}} \Pi \left (\frac {1}{24} \left (12+11 \sqrt {2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{224 \sqrt [4]{2} \sqrt {1+2 x^2+2 x^4}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 7.27, size = 214, normalized size = 0.46 \begin {gather*} \frac {-892 x-2080 x^3-2456 x^5-752 x^7-160 x^9+4422 i \sqrt {1-i} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} E\left (\left .i \sinh ^{-1}\left (\sqrt {1-i} x\right )\right |i\right )-(9669-5247 i) \sqrt {1-i} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} F\left (\left .i \sinh ^{-1}\left (\sqrt {1-i} x\right )\right |i\right )+10115 (1-i)^{3/2} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} \Pi \left (-\frac {1}{3}-\frac {i}{3};\left .i \sinh ^{-1}\left (\sqrt {1-i} x\right )\right |i\right )}{560 \sqrt {1+2 x^2+2 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-892*x - 2080*x^3 - 2456*x^5 - 752*x^7 - 160*x^9 + (4422*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I
)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] - (9669 - 5247*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1
+ I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] + 10115*(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])/(560*Sqrt[1 + 2*x^2 + 2*x^4])

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Maple [C] Result contains complex when optimal does not.
time = 0.14, size = 547, normalized size = 1.18

method	result	size
risch	\(-\frac {x \left (20 x^{4}+74 x^{2}+223\right ) \sqrt {2 x^{4}+2 x^{2}+1}}{140}+\frac {\left (\frac {2211}{280}-\frac {2211 i}{280}\right ) \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, \left (\EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )-\EllipticE \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )\right )}{\sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {9669 \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, \EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{280 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {289 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticPi \left (x \sqrt {-1+i}, -\frac {1}{3}-\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{8 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\)	\(251\)
elliptic	\(-\frac {x^{5} \sqrt {2 x^{4}+2 x^{2}+1}}{7}-\frac {37 x^{3} \sqrt {2 x^{4}+2 x^{2}+1}}{70}-\frac {223 x \sqrt {2 x^{4}+2 x^{2}+1}}{140}-\frac {3729 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{140 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {2211 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{280 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {2211 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticE \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{280 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {2211 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticE \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{280 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {289 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticPi \left (x \sqrt {-1+i}, -\frac {1}{3}-\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{8 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\)	\(396\)
default	\(-\frac {x^{5} \sqrt {2 x^{4}+2 x^{2}+1}}{7}-\frac {37 x^{3} \sqrt {2 x^{4}+2 x^{2}+1}}{70}-\frac {223 x \sqrt {2 x^{4}+2 x^{2}+1}}{140}-\frac {9 \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, \EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{35 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {\left (\frac {6}{35}-\frac {6 i}{35}\right ) \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, \left (\EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )-\EllipticE \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )\right )}{\sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {531 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {309 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{40 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {309 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticE \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{40 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {309 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticE \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{40 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {289 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticPi \left (x \sqrt {-1+i}, -\frac {1}{3}-\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{8 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\)	\(547\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*x^5*(2*x^4+2*x^2+1)^(1/2)-37/70*x^3*(2*x^4+2*x^2+1)^(1/2)-223/140*x*(2*x^4+2*x^2+1)^(1/2)-9/35/(-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))+(6/35-6/35*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)))-531/20/(-1+I)^(1/2
)*(1+x^2-I*x^2)^(1/2)*(1+x^2+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))-309/40*I/(-1+I)^(1/2)*(1+x^2-I*x^2)^(1/2)*(1+x^2+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))-309/40/(-1+I)^(1/2)*(1+x^2-I*x^2)^(1/2)*(1+x^2+I*x^2)^(1/2)/(2*x^4+2*x^2+1)^(1
/2)*EllipticE(x*(-1+I)^(1/2),1/2*2^(1/2)+1/2*I*2^(1/2))+309/40*I/(-1+I)^(1/2)*(1+x^2-I*x^2)^(1/2)*(1+x^2+I*x^2
)^(1/2)/(2*x^4+2*x^2+1)^(1/2)*EllipticE(x*(-1+I)^(1/2),1/2*2^(1/2)+1/2*I*2^(1/2))+289/8/(-1+I)^(1/2)*(1+x^2-I*
x^2)^(1/2)*(1+x^2+I*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))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(2*x^6 + 2*x^4 + x^2)*sqrt(2*x^4 + 2*x^2 + 1)/(2*x^2 - 3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {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 - \int \frac {2 x^{6} \sqrt {2 x^{4} + 2 x^{2} + 1}}{2 x^{2} - 3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {x^2\,{\left (2\,x^4+2\,x^2+1\right )}^{3/2}}{2\,x^2-3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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