3.116 \(\int \frac{4+x^2+3 x^4+5 x^6}{x^6 (3+2 x^2+x^4)^2} \, dx\)

Optimal. Leaf size=245 \[ \frac{25 x \left (1-7 x^2\right )}{648 \left (x^4+2 x^2+3\right )}+\frac{13}{81 x^3}-\frac{4}{45 x^5}-\frac{\sqrt{\frac{1}{6} \left (1139381+688419 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{2592}+\frac{\sqrt{\frac{1}{6} \left (1139381+688419 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{2592}-\frac{13}{27 x}+\frac{\sqrt{\frac{1}{6} \left (688419 \sqrt{3}-1139381\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{1296}-\frac{\sqrt{\frac{1}{6} \left (688419 \sqrt{3}-1139381\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{1296} \]

[Out]

-4/(45*x^5) + 13/(81*x^3) - 13/(27*x) + (25*x*(1 - 7*x^2))/(648*(3 + 2*x^2 + x^4)) + (Sqrt[(-1139381 + 688419*
Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/1296 - (Sqrt[(-1139381 + 688419*Sqrt
[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/1296 - (Sqrt[(1139381 + 688419*Sqrt[3])/
6]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/2592 + (Sqrt[(1139381 + 688419*Sqrt[3])/6]*Log[Sqrt[3] + Sqr
t[2*(-1 + Sqrt[3])]*x + x^2])/2592

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Rubi [A]  time = 0.328981, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {1669, 1664, 1169, 634, 618, 204, 628} \[ \frac{25 x \left (1-7 x^2\right )}{648 \left (x^4+2 x^2+3\right )}+\frac{13}{81 x^3}-\frac{4}{45 x^5}-\frac{\sqrt{\frac{1}{6} \left (1139381+688419 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{2592}+\frac{\sqrt{\frac{1}{6} \left (1139381+688419 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{2592}-\frac{13}{27 x}+\frac{\sqrt{\frac{1}{6} \left (688419 \sqrt{3}-1139381\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{1296}-\frac{\sqrt{\frac{1}{6} \left (688419 \sqrt{3}-1139381\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{1296} \]

Antiderivative was successfully verified.

[In]

Int[(4 + x^2 + 3*x^4 + 5*x^6)/(x^6*(3 + 2*x^2 + x^4)^2),x]

[Out]

-4/(45*x^5) + 13/(81*x^3) - 13/(27*x) + (25*x*(1 - 7*x^2))/(648*(3 + 2*x^2 + x^4)) + (Sqrt[(-1139381 + 688419*
Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/1296 - (Sqrt[(-1139381 + 688419*Sqrt
[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/1296 - (Sqrt[(1139381 + 688419*Sqrt[3])/
6]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/2592 + (Sqrt[(1139381 + 688419*Sqrt[3])/6]*Log[Sqrt[3] + Sqr
t[2*(-1 + Sqrt[3])]*x + x^2])/2592

Rule 1669

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainde
r[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, S
imp[(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] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[x^m*(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[(2*a*(p + 1)*(b^2
- 4*a*c)*PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x])/x^m + (b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e)
/x^m + c*(4*p + 7)*(b*d - 2*a*e)*x^(2 - m), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[
Pq, x^2], 1] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && ILtQ[m/2, 0]

Rule 1664

Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x
)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]

Rule 1169

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

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{4+x^2+3 x^4+5 x^6}{x^6 \left (3+2 x^2+x^4\right )^2} \, dx &=\frac{25 x \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac{1}{48} \int \frac{64-\frac{80 x^2}{3}+\frac{400 x^4}{9}+\frac{1550 x^6}{27}-\frac{350 x^8}{27}}{x^6 \left (3+2 x^2+x^4\right )} \, dx\\ &=\frac{25 x \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac{1}{48} \int \left (\frac{64}{3 x^6}-\frac{208}{9 x^4}+\frac{208}{9 x^2}-\frac{2 \left (-463+487 x^2\right )}{27 \left (3+2 x^2+x^4\right )}\right ) \, dx\\ &=-\frac{4}{45 x^5}+\frac{13}{81 x^3}-\frac{13}{27 x}+\frac{25 x \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}-\frac{1}{648} \int \frac{-463+487 x^2}{3+2 x^2+x^4} \, dx\\ &=-\frac{4}{45 x^5}+\frac{13}{81 x^3}-\frac{13}{27 x}+\frac{25 x \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}-\frac{\int \frac{-463 \sqrt{2 \left (-1+\sqrt{3}\right )}-\left (-463-487 \sqrt{3}\right ) x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{1296 \sqrt{6 \left (-1+\sqrt{3}\right )}}-\frac{\int \frac{-463 \sqrt{2 \left (-1+\sqrt{3}\right )}+\left (-463-487 \sqrt{3}\right ) x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{1296 \sqrt{6 \left (-1+\sqrt{3}\right )}}\\ &=-\frac{4}{45 x^5}+\frac{13}{81 x^3}-\frac{13}{27 x}+\frac{25 x \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}-\frac{\left (1461-463 \sqrt{3}\right ) \int \frac{1}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{7776}+\frac{\left (-1461+463 \sqrt{3}\right ) \int \frac{1}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{7776}-\frac{\sqrt{\frac{1}{6} \left (1139381+688419 \sqrt{3}\right )} \int \frac{-\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{2592}+\frac{\sqrt{\frac{1}{6} \left (1139381+688419 \sqrt{3}\right )} \int \frac{\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{2592}\\ &=-\frac{4}{45 x^5}+\frac{13}{81 x^3}-\frac{13}{27 x}+\frac{25 x \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}-\frac{\sqrt{\frac{1}{6} \left (1139381+688419 \sqrt{3}\right )} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )}{2592}+\frac{\sqrt{\frac{1}{6} \left (1139381+688419 \sqrt{3}\right )} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )}{2592}+\frac{\left (1461-463 \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{1}{-2 \left (1+\sqrt{3}\right )-x^2} \, dx,x,\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x\right )}{3888}-\frac{\left (-1461+463 \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{1}{-2 \left (1+\sqrt{3}\right )-x^2} \, dx,x,-\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x\right )}{3888}\\ &=-\frac{4}{45 x^5}+\frac{13}{81 x^3}-\frac{13}{27 x}+\frac{25 x \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac{\sqrt{\frac{1}{6} \left (-1139381+688419 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{1296}-\frac{\sqrt{\frac{1}{6} \left (-1139381+688419 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{1296}-\frac{\sqrt{\frac{1}{6} \left (1139381+688419 \sqrt{3}\right )} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )}{2592}+\frac{\sqrt{\frac{1}{6} \left (1139381+688419 \sqrt{3}\right )} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )}{2592}\\ \end{align*}

Mathematica [C]  time = 0.303282, size = 140, normalized size = 0.57 \[ \frac{-\frac{4 \left (2435 x^8+2475 x^6+3928 x^4-984 x^2+864\right )}{x^5 \left (x^4+2 x^2+3\right )}-\frac{10 i \left (475 \sqrt{2}-487 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{\sqrt{1-i \sqrt{2}}}+\frac{10 i \left (475 \sqrt{2}+487 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{\sqrt{1+i \sqrt{2}}}}{12960} \]

Antiderivative was successfully verified.

[In]

Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^6*(3 + 2*x^2 + x^4)^2),x]

[Out]

((-4*(864 - 984*x^2 + 3928*x^4 + 2475*x^6 + 2435*x^8))/(x^5*(3 + 2*x^2 + x^4)) - ((10*I)*(-487*I + 475*Sqrt[2]
)*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + ((10*I)*(487*I + 475*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[
2]]])/Sqrt[1 + I*Sqrt[2]])/12960

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Maple [B]  time = 0.023, size = 424, normalized size = 1.7 \begin{align*} -{\frac{1}{27\,{x}^{4}+54\,{x}^{2}+81} \left ({\frac{175\,{x}^{3}}{24}}-{\frac{25\,x}{24}} \right ) }-{\frac{481\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{7776}}-{\frac{475\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{5184}}-{\frac{ \left ( -962+962\,\sqrt{3} \right ) \sqrt{3}}{3888\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-950+950\,\sqrt{3}}{2592\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{463\,\sqrt{3}}{1944\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{481\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{7776}}+{\frac{475\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{5184}}-{\frac{ \left ( -962+962\,\sqrt{3} \right ) \sqrt{3}}{3888\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-950+950\,\sqrt{3}}{2592\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{463\,\sqrt{3}}{1944\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{4}{45\,{x}^{5}}}+{\frac{13}{81\,{x}^{3}}}-{\frac{13}{27\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^6+3*x^4+x^2+4)/x^6/(x^4+2*x^2+3)^2,x)

[Out]

-1/27*(175/24*x^3-25/24*x)/(x^4+2*x^2+3)-481/7776*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*
3^(1/2)-475/5184*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)-481/3888/(2+2*3^(1/2))^(1/2)*arct
an((2*x-(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))*3^(1/2)-475/2592/(2+2*3^(1/2))^(1/2)*arctan(
(2*x-(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))+463/1944/(2+2*3^(1/2))^(1/2)*arctan((2*x-(-2+2*
3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*3^(1/2)+481/7776*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1
/2)*3^(1/2)+475/5184*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)-481/3888/(2+2*3^(1/2))^(1/2)*
arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))*3^(1/2)-475/2592/(2+2*3^(1/2))^(1/2)*arc
tan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))+463/1944/(2+2*3^(1/2))^(1/2)*arctan((2*x+(-
2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*3^(1/2)-4/45/x^5+13/81/x^3-13/27/x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2435 \, x^{8} + 2475 \, x^{6} + 3928 \, x^{4} - 984 \, x^{2} + 864}{3240 \,{\left (x^{9} + 2 \, x^{7} + 3 \, x^{5}\right )}} - \frac{1}{648} \, \int \frac{487 \, x^{2} - 463}{x^{4} + 2 \, x^{2} + 3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^6+3*x^4+x^2+4)/x^6/(x^4+2*x^2+3)^2,x, algorithm="maxima")

[Out]

-1/3240*(2435*x^8 + 2475*x^6 + 3928*x^4 - 984*x^2 + 864)/(x^9 + 2*x^7 + 3*x^5) - 1/648*integrate((487*x^2 - 46
3)/(x^4 + 2*x^2 + 3), x)

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Fricas [B]  time = 1.74124, size = 2392, normalized size = 9.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^6+3*x^4+x^2+4)/x^6/(x^4+2*x^2+3)^2,x, algorithm="fricas")

[Out]

-1/1478473537631040*(1111136748188760*x^8 + 1129389507912600*x^6 + 1792421004881088*x^4 - 4971380*216699003^(1
/4)*sqrt(2)*(x^9 + 2*x^7 + 3*x^5)*sqrt(-784371528639*sqrt(3) + 1421762158683)*arctan(1/61448662235687217564537
18*sqrt(704195977)*216699003^(3/4)*sqrt(57039874137*x^2 + 216699003^(1/4)*(463*sqrt(3)*x + 1461*x)*sqrt(-78437
1528639*sqrt(3) + 1421762158683) + 57039874137*sqrt(3))*(487*sqrt(3)*sqrt(2) + 463*sqrt(2))*sqrt(-784371528639
*sqrt(3) + 1421762158683) - 1/969563780580726*216699003^(3/4)*(487*sqrt(3)*sqrt(2)*x + 463*sqrt(2)*x)*sqrt(-78
4371528639*sqrt(3) + 1421762158683) - 1/2*sqrt(3)*sqrt(2) + 1/2*sqrt(2)) - 4971380*216699003^(1/4)*sqrt(2)*(x^
9 + 2*x^7 + 3*x^5)*sqrt(-784371528639*sqrt(3) + 1421762158683)*arctan(1/6144866223568721756453718*sqrt(7041959
77)*216699003^(3/4)*sqrt(57039874137*x^2 - 216699003^(1/4)*(463*sqrt(3)*x + 1461*x)*sqrt(-784371528639*sqrt(3)
+ 1421762158683) + 57039874137*sqrt(3))*(487*sqrt(3)*sqrt(2) + 463*sqrt(2))*sqrt(-784371528639*sqrt(3) + 1421
762158683) - 1/969563780580726*216699003^(3/4)*(487*sqrt(3)*sqrt(2)*x + 463*sqrt(2)*x)*sqrt(-784371528639*sqrt
(3) + 1421762158683) + 1/2*sqrt(3)*sqrt(2) - 1/2*sqrt(2)) - 5*216699003^(1/4)*(1139381*x^9 + 2278762*x^7 + 341
8143*x^5 + 688419*sqrt(3)*(x^9 + 2*x^7 + 3*x^5))*sqrt(-784371528639*sqrt(3) + 1421762158683)*log(57039874137*x
^2 + 216699003^(1/4)*(463*sqrt(3)*x + 1461*x)*sqrt(-784371528639*sqrt(3) + 1421762158683) + 57039874137*sqrt(3
)) + 5*216699003^(1/4)*(1139381*x^9 + 2278762*x^7 + 3418143*x^5 + 688419*sqrt(3)*(x^9 + 2*x^7 + 3*x^5))*sqrt(-
784371528639*sqrt(3) + 1421762158683)*log(57039874137*x^2 - 216699003^(1/4)*(463*sqrt(3)*x + 1461*x)*sqrt(-784
371528639*sqrt(3) + 1421762158683) + 57039874137*sqrt(3)) - 449017889206464*x^2 + 394259610034944)/(x^9 + 2*x^
7 + 3*x^5)

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Sympy [A]  time = 0.590268, size = 65, normalized size = 0.27 \begin{align*} \operatorname{RootSum}{\left (20639121408 t^{4} - 2333452288 t^{2} + 72233001, \left ( t \mapsto t \log{\left (- \frac{206821195776 t^{3}}{704195977} + \frac{38757503008 t}{2112587931} + x \right )} \right )\right )} - \frac{2435 x^{8} + 2475 x^{6} + 3928 x^{4} - 984 x^{2} + 864}{3240 x^{9} + 6480 x^{7} + 9720 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x**6+3*x**4+x**2+4)/x**6/(x**4+2*x**2+3)**2,x)

[Out]

RootSum(20639121408*_t**4 - 2333452288*_t**2 + 72233001, Lambda(_t, _t*log(-206821195776*_t**3/704195977 + 387
57503008*_t/2112587931 + x))) - (2435*x**8 + 2475*x**6 + 3928*x**4 - 984*x**2 + 864)/(3240*x**9 + 6480*x**7 +
9720*x**5)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{5 \, x^{6} + 3 \, x^{4} + x^{2} + 4}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{2} x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^6+3*x^4+x^2+4)/x^6/(x^4+2*x^2+3)^2,x, algorithm="giac")

[Out]

integrate((5*x^6 + 3*x^4 + x^2 + 4)/((x^4 + 2*x^2 + 3)^2*x^6), x)