3.4.28 \(\int \frac {(1+2 x^2+2 x^4)^{3/2}}{3-2 x^2} \, dx\) [328]

Optimal. Leaf size=428 \[ -\frac {1}{10} x \left (9+2 x^2\right ) \sqrt {1+2 x^2+2 x^4}-\frac {103 x \sqrt {1+2 x^2+2 x^4}}{10 \sqrt {2} \left (1+\sqrt {2} x^2\right )}+\frac {17}{8} \sqrt {\frac {17}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {17}{3}} x}{\sqrt {1+2 x^2+2 x^4}}\right )+\frac {103 \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{10\ 2^{3/4} \sqrt {1+2 x^2+2 x^4}}-\frac {\left (66+383 \sqrt {2}\right ) \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{10\ 2^{3/4} \left (2+3 \sqrt {2}\right ) \sqrt {1+2 x^2+2 x^4}}-\frac {289 \left (3-\sqrt {2}\right ) \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} \Pi \left (\frac {1}{24} \left (12+11 \sqrt {2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{24\ 2^{3/4} \left (2+3 \sqrt {2}\right ) \sqrt {1+2 x^2+2 x^4}} \]

[Out]

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)-289/48*(cos(2*arctan(2^(1/4)*x))^2)^(1/2)/cos(2*arctan(2^(1/4)*x))*Ellip
ticPi(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)

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Rubi [A]
time = 0.22, antiderivative size = 602, normalized size of antiderivative = 1.41, number of steps used = 12, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1222, 1190, 1211, 1117, 1209, 1230, 1720} \begin {gather*} -\frac {\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 )}{10\ 2^{3/4} \sqrt {2 x^4+2 x^2+1}}-\frac {17 \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 )}{8 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}+\frac {289 \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 )}{56 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}+\frac {103 \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 )}{10\ 2^{3/4} \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 )}{336 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}-\frac {1}{10} \left (2 x^2+9\right ) \sqrt {2 x^4+2 x^2+1} x-\frac {103 \sqrt {2 x^4+2 x^2+1} x}{10 \sqrt {2} \left (\sqrt {2} x^2+1\right )}+\frac {17}{8} \sqrt {\frac {17}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {17}{3}} x}{\sqrt {2 x^4+2 x^2+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/10*(x*(9 + 2*x^2)*Sqrt[1 + 2*x^2 + 2*x^4]) - (103*x*Sqrt[1 + 2*x^2 + 2*x^4])/(10*Sqrt[2]*(1 + Sqrt[2]*x^2))
+ (17*Sqrt[17/3]*ArcTanh[(Sqrt[17/3]*x)/Sqrt[1 + 2*x^2 + 2*x^4]])/8 + (103*(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])/(10*2^(3/4)*Sqrt[1 + 2*x^2 + 2*
x^4]) + (289*(3 - 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])/(56*2^(1/4)*Sqrt[1 + 2*x^2 + 2*x^4]) - (17*(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])/(8*2^(1/4)*Sqrt[1 +
2*x^2 + 2*x^4]) - ((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])/(10*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])/(336*2^(1/4)*Sqrt[1 + 2*x^2 + 2*x^4])

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 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 {\left (1+2 x^2+2 x^4\right )^{3/2}}{3-2 x^2} \, dx &=-\left (\frac {1}{4} \int \left (10+4 x^2\right ) \sqrt {1+2 x^2+2 x^4} \, dx\right )+\frac {17}{2} \int \frac {\sqrt {1+2 x^2+2 x^4}}{3-2 x^2} \, dx\\ &=-\frac {1}{10} x \left (9+2 x^2\right ) \sqrt {1+2 x^2+2 x^4}-\frac {1}{120} \int \frac {192+216 x^2}{\sqrt {1+2 x^2+2 x^4}} \, dx-\frac {17}{8} \int \frac {10+4 x^2}{\sqrt {1+2 x^2+2 x^4}} \, dx+\frac {289}{4} \int \frac {1}{\left (3-2 x^2\right ) \sqrt {1+2 x^2+2 x^4}} \, dx\\ &=-\frac {1}{10} x \left (9+2 x^2\right ) \sqrt {1+2 x^2+2 x^4}+\frac {9 \int \frac {1-\sqrt {2} x^2}{\sqrt {1+2 x^2+2 x^4}} \, dx}{5 \sqrt {2}}+\frac {17 \int \frac {1-\sqrt {2} x^2}{\sqrt {1+2 x^2+2 x^4}} \, dx}{2 \sqrt {2}}-\frac {1}{28} \left (289 \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}{28} \left (289 \left (3-\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+2 x^2+2 x^4}} \, dx-\frac {1}{4} \left (17 \left (5+\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+2 x^2+2 x^4}} \, dx-\frac {1}{10} \left (16+9 \sqrt {2}\right ) \int \frac {1}{\sqrt {1+2 x^2+2 x^4}} \, 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}} \tanh ^{-1}\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 \tan ^{-1}\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 {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}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{56 \sqrt [4]{2} \sqrt {1+2 x^2+2 x^4}}-\frac {17 \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 )}{8 \sqrt [4]{2} \sqrt {1+2 x^2+2 x^4}}-\frac {\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 )}{10\ 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 )}{336 \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 = 10.12, size = 209, normalized size = 0.49 \begin {gather*} \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 \sinh ^{-1}\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} F\left (\left .i \sinh ^{-1}\left (\sqrt {1-i} x\right )\right |i\right )+1445 (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 )}{120 \sqrt {1+2 x^2+2 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-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]*Ellipti
cE[I*ArcSinh[Sqrt[1 - I]*x], I] - (1371 - 753*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*Ellip
ticF[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])

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Maple [C] Result contains complex when optimal does not.
time = 0.17, size = 377, normalized size = 0.88

method	result	size
risch	\(-\frac {x \left (2 x^{2}+9\right ) \sqrt {2 x^{4}+2 x^{2}+1}}{10}+\frac {\left (\frac {103}{20}-\frac {103 i}{20}\right ) \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, \left (\EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )-\EllipticE \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )\right )}{\sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {457 \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, \EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {289 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticPi \left (x \sqrt {-1+i}, -\frac {1}{3}-\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{12 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\)	\(246\)
default	\(-\frac {x^{3} \sqrt {2 x^{4}+2 x^{2}+1}}{5}-\frac {9 x \sqrt {2 x^{4}+2 x^{2}+1}}{10}-\frac {177 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{10 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {103 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {103 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticE \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {103 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticE \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {289 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticPi \left (x \sqrt {-1+i}, -\frac {1}{3}-\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{12 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\)	\(377\)
elliptic	\(-\frac {x^{3} \sqrt {2 x^{4}+2 x^{2}+1}}{5}-\frac {9 x \sqrt {2 x^{4}+2 x^{2}+1}}{10}-\frac {177 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{10 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {103 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {103 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticE \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {103 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticE \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{20 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {289 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticPi \left (x \sqrt {-1+i}, -\frac {1}{3}-\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{12 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\)	\(377\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*x^3*(2*x^4+2*x^2+1)^(1/2)-9/10*x*(2*x^4+2*x^2+1)^(1/2)-177/10/(-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))-103/20*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))-
103/20/(-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)*EllipticE(x*(-1+I)^(1/2),1/2
*2^(1/2)+1/2*I*2^(1/2))+103/20*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)*El
lipticE(x*(-1+I)^(1/2),1/2*2^(1/2)+1/2*I*2^(1/2))+289/12/(-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((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)/(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((2*x^4+2*x^2+1)^(3/2)/(-2*x^2+3),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((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)/(2*x^2 - 3), x)

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

[Out]

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

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