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

Optimal. Leaf size=238 \[ \frac{25 x \left (5 x^2+7\right )}{216 \left (x^4+2 x^2+3\right )}-\frac{4}{27 x^3}+\frac{1}{864} \sqrt{\frac{1}{6} \left (56673 \sqrt{3}-6073\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{864} \sqrt{\frac{1}{6} \left (56673 \sqrt{3}-6073\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{13}{27 x}-\frac{1}{432} \sqrt{\frac{1}{6} \left (6073+56673 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{432} \sqrt{\frac{1}{6} \left (6073+56673 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

[Out]

-4/(27*x^3) + 13/(27*x) + (25*x*(7 + 5*x^2))/(216*(3 + 2*x^2 + x^4)) - (Sqrt[(6073 + 56673*Sqrt[3])/6]*ArcTan[
(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/432 + (Sqrt[(6073 + 56673*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-
1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/432 + (Sqrt[(-6073 + 56673*Sqrt[3])/6]*Log[Sqrt[3] - Sqrt[2*(-1 +
Sqrt[3])]*x + x^2])/864 - (Sqrt[(-6073 + 56673*Sqrt[3])/6]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/864

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Rubi [A]  time = 0.335361, antiderivative size = 238, 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} \[ \frac{25 x \left (5 x^2+7\right )}{216 \left (x^4+2 x^2+3\right )}-\frac{4}{27 x^3}+\frac{1}{864} \sqrt{\frac{1}{6} \left (56673 \sqrt{3}-6073\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{864} \sqrt{\frac{1}{6} \left (56673 \sqrt{3}-6073\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{13}{27 x}-\frac{1}{432} \sqrt{\frac{1}{6} \left (6073+56673 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{432} \sqrt{\frac{1}{6} \left (6073+56673 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-4/(27*x^3) + 13/(27*x) + (25*x*(7 + 5*x^2))/(216*(3 + 2*x^2 + x^4)) - (Sqrt[(6073 + 56673*Sqrt[3])/6]*ArcTan[
(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/432 + (Sqrt[(6073 + 56673*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-
1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/432 + (Sqrt[(-6073 + 56673*Sqrt[3])/6]*Log[Sqrt[3] - Sqrt[2*(-1 +
Sqrt[3])]*x + x^2])/864 - (Sqrt[(-6073 + 56673*Sqrt[3])/6]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/864

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^4 \left (3+2 x^2+x^4\right )^2} \, dx &=\frac{25 x \left (7+5 x^2\right )}{216 \left (3+2 x^2+x^4\right )}+\frac{1}{48} \int \frac{64-\frac{80 x^2}{3}+\frac{50 x^4}{9}+\frac{250 x^6}{9}}{x^4 \left (3+2 x^2+x^4\right )} \, dx\\ &=\frac{25 x \left (7+5 x^2\right )}{216 \left (3+2 x^2+x^4\right )}+\frac{1}{48} \int \left (\frac{64}{3 x^4}-\frac{208}{9 x^2}+\frac{2 \left (137+229 x^2\right )}{9 \left (3+2 x^2+x^4\right )}\right ) \, dx\\ &=-\frac{4}{27 x^3}+\frac{13}{27 x}+\frac{25 x \left (7+5 x^2\right )}{216 \left (3+2 x^2+x^4\right )}+\frac{1}{216} \int \frac{137+229 x^2}{3+2 x^2+x^4} \, dx\\ &=-\frac{4}{27 x^3}+\frac{13}{27 x}+\frac{25 x \left (7+5 x^2\right )}{216 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{137 \sqrt{2 \left (-1+\sqrt{3}\right )}-\left (137-229 \sqrt{3}\right ) x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{432 \sqrt{6 \left (-1+\sqrt{3}\right )}}+\frac{\int \frac{137 \sqrt{2 \left (-1+\sqrt{3}\right )}+\left (137-229 \sqrt{3}\right ) x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{432 \sqrt{6 \left (-1+\sqrt{3}\right )}}\\ &=-\frac{4}{27 x^3}+\frac{13}{27 x}+\frac{25 x \left (7+5 x^2\right )}{216 \left (3+2 x^2+x^4\right )}+\frac{1}{432} \sqrt{\frac{1}{6} \left (88046+31373 \sqrt{3}\right )} \int \frac{1}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx+\frac{1}{432} \sqrt{\frac{1}{6} \left (88046+31373 \sqrt{3}\right )} \int \frac{1}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx+\frac{1}{864} \sqrt{\frac{1}{6} \left (-6073+56673 \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-\frac{1}{864} \sqrt{\frac{1}{6} \left (-6073+56673 \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\\ &=-\frac{4}{27 x^3}+\frac{13}{27 x}+\frac{25 x \left (7+5 x^2\right )}{216 \left (3+2 x^2+x^4\right )}+\frac{1}{864} \sqrt{\frac{1}{6} \left (-6073+56673 \sqrt{3}\right )} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )-\frac{1}{864} \sqrt{\frac{1}{6} \left (-6073+56673 \sqrt{3}\right )} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )-\frac{1}{216} \sqrt{\frac{1}{6} \left (88046+31373 \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 )-\frac{1}{216} \sqrt{\frac{1}{6} \left (88046+31373 \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 )\\ &=-\frac{4}{27 x^3}+\frac{13}{27 x}+\frac{25 x \left (7+5 x^2\right )}{216 \left (3+2 x^2+x^4\right )}-\frac{1}{432} \sqrt{\frac{1}{6} \left (6073+56673 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{432} \sqrt{\frac{1}{6} \left (6073+56673 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{864} \sqrt{\frac{1}{6} \left (-6073+56673 \sqrt{3}\right )} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )-\frac{1}{864} \sqrt{\frac{1}{6} \left (-6073+56673 \sqrt{3}\right )} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )\\ \end{align*}

Mathematica [C]  time = 0.310279, size = 131, normalized size = 0.55 \[ \frac{1}{864} \left (\frac{4 \left (229 x^6+351 x^4+248 x^2-96\right )}{x^3 \left (x^4+2 x^2+3\right )}+\frac{2 \left (229+46 i \sqrt{2}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{\sqrt{1-i \sqrt{2}}}+\frac{2 \left (229-46 i \sqrt{2}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{\sqrt{1+i \sqrt{2}}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*(-96 + 248*x^2 + 351*x^4 + 229*x^6))/(x^3*(3 + 2*x^2 + x^4)) + (2*(229 + (46*I)*Sqrt[2])*ArcTan[x/Sqrt[1 -
I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + (2*(229 - (46*I)*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2
]])/864

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Maple [B]  time = 0.023, size = 419, normalized size = 1.8 \begin{align*}{\frac{1}{27\,{x}^{4}+54\,{x}^{2}+81} \left ({\frac{125\,{x}^{3}}{8}}+{\frac{175\,x}{8}} \right ) }+{\frac{275\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{5184}}+{\frac{23\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{864}}+{\frac{ \left ( -550+550\,\sqrt{3} \right ) \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{-46+46\,\sqrt{3}}{432\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{137\,\sqrt{3}}{648\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{275\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{5184}}-{\frac{23\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{864}}+{\frac{ \left ( -550+550\,\sqrt{3} \right ) \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{-46+46\,\sqrt{3}}{432\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{137\,\sqrt{3}}{648\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{4}{27\,{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^4/(x^4+2*x^2+3)^2,x)

[Out]

1/27*(125/8*x^3+175/8*x)/(x^4+2*x^2+3)+275/5184*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^
(1/2)+23/864*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)+275/2592/(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))*3^(1/2)+23/432/(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))+137/648/(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)-275/5184*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(
1/2)-23/864*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)+275/2592/(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))*3^(1/2)+23/432/(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))+137/648/(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)-4/27/x^3+13/27/x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{229 \, x^{6} + 351 \, x^{4} + 248 \, x^{2} - 96}{216 \,{\left (x^{7} + 2 \, x^{5} + 3 \, x^{3}\right )}} + \frac{1}{216} \, \int \frac{229 \, x^{2} + 137}{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^4/(x^4+2*x^2+3)^2,x, algorithm="maxima")

[Out]

1/216*(229*x^6 + 351*x^4 + 248*x^2 - 96)/(x^7 + 2*x^5 + 3*x^3) + 1/216*integrate((229*x^2 + 137)/(x^4 + 2*x^2
+ 3), x)

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Fricas [B]  time = 1.72255, size = 2205, normalized size = 9.26 \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^4/(x^4+2*x^2+3)^2,x, algorithm="fricas")

[Out]

1/2261454002496*(2397560030424*x^6 + 3674862754056*x^4 - 277108*118956627^(1/4)*sqrt(6297)*sqrt(2)*(x^7 + 2*x^
5 + 3*x^3)*sqrt(6073*sqrt(3) + 170019)*arctan(1/295480530439458889122*118956627^(3/4)*sqrt(81861)*sqrt(6297)*s
qrt(3*118956627^(1/4)*sqrt(6297)*(137*sqrt(3)*x - 687*x)*sqrt(6073*sqrt(3) + 170019) + 3926135421*x^2 + 392613
5421*sqrt(3))*(229*sqrt(3)*sqrt(2) - 137*sqrt(2))*sqrt(6073*sqrt(3) + 170019) - 1/16481916497358*118956627^(3/
4)*sqrt(6297)*(229*sqrt(3)*sqrt(2)*x - 137*sqrt(2)*x)*sqrt(6073*sqrt(3) + 170019) + 1/2*sqrt(3)*sqrt(2) - 1/2*
sqrt(2)) - 277108*118956627^(1/4)*sqrt(6297)*sqrt(2)*(x^7 + 2*x^5 + 3*x^3)*sqrt(6073*sqrt(3) + 170019)*arctan(
1/295480530439458889122*118956627^(3/4)*sqrt(81861)*sqrt(6297)*sqrt(-3*118956627^(1/4)*sqrt(6297)*(137*sqrt(3)
*x - 687*x)*sqrt(6073*sqrt(3) + 170019) + 3926135421*x^2 + 3926135421*sqrt(3))*(229*sqrt(3)*sqrt(2) - 137*sqrt
(2))*sqrt(6073*sqrt(3) + 170019) - 1/16481916497358*118956627^(3/4)*sqrt(6297)*(229*sqrt(3)*sqrt(2)*x - 137*sq
rt(2)*x)*sqrt(6073*sqrt(3) + 170019) - 1/2*sqrt(3)*sqrt(2) + 1/2*sqrt(2)) - 118956627^(1/4)*sqrt(6297)*(6073*x
^7 + 12146*x^5 + 18219*x^3 - 56673*sqrt(3)*(x^7 + 2*x^5 + 3*x^3))*sqrt(6073*sqrt(3) + 170019)*log(3*118956627^
(1/4)*sqrt(6297)*(137*sqrt(3)*x - 687*x)*sqrt(6073*sqrt(3) + 170019) + 3926135421*x^2 + 3926135421*sqrt(3)) +
118956627^(1/4)*sqrt(6297)*(6073*x^7 + 12146*x^5 + 18219*x^3 - 56673*sqrt(3)*(x^7 + 2*x^5 + 3*x^3))*sqrt(6073*
sqrt(3) + 170019)*log(-3*118956627^(1/4)*sqrt(6297)*(137*sqrt(3)*x - 687*x)*sqrt(6073*sqrt(3) + 170019) + 3926
135421*x^2 + 3926135421*sqrt(3)) + 2596484225088*x^2 - 1005090667776)/(x^7 + 2*x^5 + 3*x^3)

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Sympy [A]  time = 0.575375, size = 60, normalized size = 0.25 \begin{align*} \operatorname{RootSum}{\left (2293235712 t^{4} + 12437504 t^{2} + 4405801, \left ( t \mapsto t \log{\left (\frac{19707494400 t^{3}}{145412423} + \frac{357152768 t}{145412423} + x \right )} \right )\right )} + \frac{229 x^{6} + 351 x^{4} + 248 x^{2} - 96}{216 x^{7} + 432 x^{5} + 648 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(2293235712*_t**4 + 12437504*_t**2 + 4405801, Lambda(_t, _t*log(19707494400*_t**3/145412423 + 357152768
*_t/145412423 + x))) + (229*x**6 + 351*x**4 + 248*x**2 - 96)/(216*x**7 + 432*x**5 + 648*x**3)

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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^{4}}\,{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^4/(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^4), x)