### 3.123 $$\int \frac{4+x^2+3 x^4+5 x^6}{x^2 (3+2 x^2+x^4)^3} \, dx$$

Optimal. Leaf size=253 $-\frac{25 x \left (x^2+5\right )}{144 \left (x^4+2 x^2+3\right )^2}-\frac{x \left (242 x^2+325\right )}{1728 \left (x^4+2 x^2+3\right )}-\frac{\sqrt{\frac{1}{3} \left (55161 \sqrt{3}-59711\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{4608}+\frac{\sqrt{\frac{1}{3} \left (55161 \sqrt{3}-59711\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{4608}-\frac{4}{27 x}+\frac{\sqrt{\frac{1}{3} \left (59711+55161 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{2304}-\frac{\sqrt{\frac{1}{3} \left (59711+55161 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{2304}$

[Out]

-4/(27*x) - (25*x*(5 + x^2))/(144*(3 + 2*x^2 + x^4)^2) - (x*(325 + 242*x^2))/(1728*(3 + 2*x^2 + x^4)) + (Sqrt[
(59711 + 55161*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/2304 - (Sqrt[(59711 +
55161*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/2304 - (Sqrt[(-59711 + 55161*
Sqrt[3])/3]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/4608 + (Sqrt[(-59711 + 55161*Sqrt[3])/3]*Log[Sqrt[3
] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/4608

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Rubi [A]  time = 0.342795, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 13, 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} $-\frac{25 x \left (x^2+5\right )}{144 \left (x^4+2 x^2+3\right )^2}-\frac{x \left (242 x^2+325\right )}{1728 \left (x^4+2 x^2+3\right )}-\frac{\sqrt{\frac{1}{3} \left (55161 \sqrt{3}-59711\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{4608}+\frac{\sqrt{\frac{1}{3} \left (55161 \sqrt{3}-59711\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{4608}-\frac{4}{27 x}+\frac{\sqrt{\frac{1}{3} \left (59711+55161 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{2304}-\frac{\sqrt{\frac{1}{3} \left (59711+55161 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{2304}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-4/(27*x) - (25*x*(5 + x^2))/(144*(3 + 2*x^2 + x^4)^2) - (x*(325 + 242*x^2))/(1728*(3 + 2*x^2 + x^4)) + (Sqrt[
(59711 + 55161*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/2304 - (Sqrt[(59711 +
55161*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/2304 - (Sqrt[(-59711 + 55161*
Sqrt[3])/3]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/4608 + (Sqrt[(-59711 + 55161*Sqrt[3])/3]*Log[Sqrt[3
] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/4608

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^2 \left (3+2 x^2+x^4\right )^3} \, dx &=-\frac{25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}+\frac{1}{96} \int \frac{128+30 x^2-\frac{250 x^4}{3}}{x^2 \left (3+2 x^2+x^4\right )^2} \, dx\\ &=-\frac{25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{2048-\frac{56 x^2}{3}-\frac{1936 x^4}{3}}{x^2 \left (3+2 x^2+x^4\right )} \, dx}{4608}\\ &=-\frac{25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}+\frac{\int \left (\frac{2048}{3 x^2}-\frac{8 \left (173+166 x^2\right )}{3+2 x^2+x^4}\right ) \, dx}{4608}\\ &=-\frac{4}{27 x}-\frac{25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}-\frac{1}{576} \int \frac{173+166 x^2}{3+2 x^2+x^4} \, dx\\ &=-\frac{4}{27 x}-\frac{25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}-\frac{\int \frac{173 \sqrt{2 \left (-1+\sqrt{3}\right )}-\left (173-166 \sqrt{3}\right ) x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{1152 \sqrt{6 \left (-1+\sqrt{3}\right )}}-\frac{\int \frac{173 \sqrt{2 \left (-1+\sqrt{3}\right )}+\left (173-166 \sqrt{3}\right ) x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{1152 \sqrt{6 \left (-1+\sqrt{3}\right )}}\\ &=-\frac{4}{27 x}-\frac{25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}-\frac{\sqrt{\frac{1}{3} \left (-59711+55161 \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}{4608}+\frac{\sqrt{\frac{1}{3} \left (-59711+55161 \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}{4608}-\frac{\sqrt{\frac{1}{3} \left (112597+57436 \sqrt{3}\right )} \int \frac{1}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{2304}-\frac{\sqrt{\frac{1}{3} \left (112597+57436 \sqrt{3}\right )} \int \frac{1}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{2304}\\ &=-\frac{4}{27 x}-\frac{25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}-\frac{\sqrt{\frac{1}{3} \left (-59711+55161 \sqrt{3}\right )} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )}{4608}+\frac{\sqrt{\frac{1}{3} \left (-59711+55161 \sqrt{3}\right )} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )}{4608}+\frac{\sqrt{\frac{1}{3} \left (112597+57436 \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 )}{1152}+\frac{\sqrt{\frac{1}{3} \left (112597+57436 \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 )}{1152}\\ &=-\frac{4}{27 x}-\frac{25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}+\frac{\sqrt{\frac{1}{3} \left (59711+55161 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{2304}-\frac{\sqrt{\frac{1}{3} \left (59711+55161 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{2304}-\frac{\sqrt{\frac{1}{3} \left (-59711+55161 \sqrt{3}\right )} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )}{4608}+\frac{\sqrt{\frac{1}{3} \left (-59711+55161 \sqrt{3}\right )} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )}{4608}\\ \end{align*}

Mathematica [C]  time = 0.378739, size = 140, normalized size = 0.55 $\frac{-\frac{12 \left (166 x^8+611 x^6+1412 x^4+1849 x^2+768\right )}{x \left (x^4+2 x^2+3\right )^2}+\frac{3 i \left (7 \sqrt{2}+332 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{\sqrt{1-i \sqrt{2}}}-\frac{3 i \left (7 \sqrt{2}-332 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{\sqrt{1+i \sqrt{2}}}}{6912}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((-12*(768 + 1849*x^2 + 1412*x^4 + 611*x^6 + 166*x^8))/(x*(3 + 2*x^2 + x^4)^2) + ((3*I)*(332*I + 7*Sqrt[2])*Ar
cTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] - ((3*I)*(-332*I + 7*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/
Sqrt[1 + I*Sqrt[2]])/6912

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Maple [B]  time = 0.023, size = 424, normalized size = 1.7 \begin{align*} -{\frac{1}{27\, \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}} \left ({\frac{121\,{x}^{7}}{32}}+{\frac{809\,{x}^{5}}{64}}+{\frac{419\,{x}^{3}}{16}}+{\frac{2475\,x}{64}} \right ) }-{\frac{325\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{27648}}+{\frac{7\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{9216}}-{\frac{ \left ( -650+650\,\sqrt{3} \right ) \sqrt{3}}{13824\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-14+14\,\sqrt{3}}{4608\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{173\,\sqrt{3}}{1728\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{325\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{27648}}-{\frac{7\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{9216}}-{\frac{ \left ( -650+650\,\sqrt{3} \right ) \sqrt{3}}{13824\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-14+14\,\sqrt{3}}{4608\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{173\,\sqrt{3}}{1728\,\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}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(121/32*x^7+809/64*x^5+419/16*x^3+2475/64*x)/(x^4+2*x^2+3)^2-325/27648*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(
1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)+7/9216*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)-325/1382
4/(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)+7/4608/(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))-173/1728/(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)+325/27648*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))
^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)-7/9216*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)-325/13
824/(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)+7/4608/(
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))-173/1728/(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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{166 \, x^{8} + 611 \, x^{6} + 1412 \, x^{4} + 1849 \, x^{2} + 768}{576 \,{\left (x^{9} + 4 \, x^{7} + 10 \, x^{5} + 12 \, x^{3} + 9 \, x\right )}} - \frac{1}{576} \, \int \frac{166 \, x^{2} + 173}{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^2/(x^4+2*x^2+3)^3,x, algorithm="maxima")

[Out]

-1/576*(166*x^8 + 611*x^6 + 1412*x^4 + 1849*x^2 + 768)/(x^9 + 4*x^7 + 10*x^5 + 12*x^3 + 9*x) - 1/576*integrate
((166*x^2 + 173)/(x^4 + 2*x^2 + 3), x)

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Fricas [B]  time = 1.73342, size = 2529, normalized size = 10. \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^2/(x^4+2*x^2+3)^3,x, algorithm="fricas")

[Out]

-1/2978955242496*(858518351136*x^8 + 3159968147856*x^6 + 210956*1391283^(1/4)*sqrt(681)*sqrt(6)*sqrt(3)*sqrt(2
)*(x^9 + 4*x^7 + 10*x^5 + 12*x^3 + 9*x)*sqrt(59711*sqrt(3) + 165483)*arctan(1/15811665652336538898*sqrt(119717
53)*1391283^(3/4)*sqrt(681)*sqrt(6)*sqrt(1391283^(1/4)*sqrt(681)*sqrt(6)*(166*sqrt(3)*x - 173*x)*sqrt(59711*sq
rt(3) + 165483) + 107745777*x^2 + 107745777*sqrt(3))*(173*sqrt(3)*sqrt(2) - 498*sqrt(2))*sqrt(59711*sqrt(3) +
165483) - 1/440249244822*1391283^(3/4)*sqrt(681)*sqrt(6)*(173*sqrt(3)*sqrt(2)*x - 498*sqrt(2)*x)*sqrt(59711*sq
rt(3) + 165483) + 1/2*sqrt(3)*sqrt(2) - 1/2*sqrt(2)) + 210956*1391283^(1/4)*sqrt(681)*sqrt(6)*sqrt(3)*sqrt(2)*
(x^9 + 4*x^7 + 10*x^5 + 12*x^3 + 9*x)*sqrt(59711*sqrt(3) + 165483)*arctan(1/47434996957009616694*sqrt(11971753
)*1391283^(3/4)*sqrt(681)*sqrt(6)*sqrt(-9*1391283^(1/4)*sqrt(681)*sqrt(6)*(166*sqrt(3)*x - 173*x)*sqrt(59711*s
qrt(3) + 165483) + 969711993*x^2 + 969711993*sqrt(3))*(173*sqrt(3)*sqrt(2) - 498*sqrt(2))*sqrt(59711*sqrt(3) +
165483) - 1/440249244822*1391283^(3/4)*sqrt(681)*sqrt(6)*(173*sqrt(3)*sqrt(2)*x - 498*sqrt(2)*x)*sqrt(59711*s
qrt(3) + 165483) - 1/2*sqrt(3)*sqrt(2) + 1/2*sqrt(2)) + 7302577781952*x^4 - 1391283^(1/4)*sqrt(681)*sqrt(6)*(1
65483*x^9 + 661932*x^7 + 1654830*x^5 + 1985796*x^3 - 59711*sqrt(3)*(x^9 + 4*x^7 + 10*x^5 + 12*x^3 + 9*x) + 148
9347*x)*sqrt(59711*sqrt(3) + 165483)*log(9*1391283^(1/4)*sqrt(681)*sqrt(6)*(166*sqrt(3)*x - 173*x)*sqrt(59711*
sqrt(3) + 165483) + 969711993*x^2 + 969711993*sqrt(3)) + 1391283^(1/4)*sqrt(681)*sqrt(6)*(165483*x^9 + 661932*
x^7 + 1654830*x^5 + 1985796*x^3 - 59711*sqrt(3)*(x^9 + 4*x^7 + 10*x^5 + 12*x^3 + 9*x) + 1489347*x)*sqrt(59711*
sqrt(3) + 165483)*log(-9*1391283^(1/4)*sqrt(681)*sqrt(6)*(166*sqrt(3)*x - 173*x)*sqrt(59711*sqrt(3) + 165483)
+ 969711993*x^2 + 969711993*sqrt(3)) + 9562653200304*x^2 + 3971940323328)/(x^9 + 4*x^7 + 10*x^5 + 12*x^3 + 9*x
)

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Sympy [A]  time = 0.607374, size = 73, normalized size = 0.29 \begin{align*} - \frac{166 x^{8} + 611 x^{6} + 1412 x^{4} + 1849 x^{2} + 768}{576 x^{9} + 2304 x^{7} + 5760 x^{5} + 6912 x^{3} + 5184 x} + \operatorname{RootSum}{\left (4174708211712 t^{4} + 15652880384 t^{2} + 37564641, \left ( t \mapsto t \log{\left (- \frac{98146713600 t^{3}}{11971753} - \frac{9639364864 t}{323237331} + x \right )} \right )\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(166*x**8 + 611*x**6 + 1412*x**4 + 1849*x**2 + 768)/(576*x**9 + 2304*x**7 + 5760*x**5 + 6912*x**3 + 5184*x) +
RootSum(4174708211712*_t**4 + 15652880384*_t**2 + 37564641, Lambda(_t, _t*log(-98146713600*_t**3/11971753 - 9
639364864*_t/323237331 + x)))

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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 )}^{3} x^{2}}\,{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^2/(x^4+2*x^2+3)^3,x, algorithm="giac")

[Out]

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