3.109 \(\int \frac{x^8 (4+x^2+3 x^4+5 x^6)}{(3+2 x^2+x^4)^2} \, dx\)

Optimal. Leaf size=248 \[ \frac{5 x^7}{7}-\frac{17 x^5}{5}+\frac{19 x^3}{3}+\frac{25 \left (5 x^2+3\right ) x}{8 \left (x^4+2 x^2+3\right )}-\frac{1}{32} \sqrt{\frac{1}{2} \left (618291 \sqrt{3}-262771\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{1}{32} \sqrt{\frac{1}{2} \left (618291 \sqrt{3}-262771\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+38 x+\frac{1}{16} \sqrt{\frac{1}{2} \left (262771+618291 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{1}{16} \sqrt{\frac{1}{2} \left (262771+618291 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

[Out]

38*x + (19*x^3)/3 - (17*x^5)/5 + (5*x^7)/7 + (25*x*(3 + 5*x^2))/(8*(3 + 2*x^2 + x^4)) + (Sqrt[(262771 + 618291
*Sqrt[3])/2]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/16 - (Sqrt[(262771 + 618291*Sqrt[3]
)/2]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/16 - (Sqrt[(-262771 + 618291*Sqrt[3])/2]*Lo
g[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/32 + (Sqrt[(-262771 + 618291*Sqrt[3])/2]*Log[Sqrt[3] + Sqrt[2*(-1
+ Sqrt[3])]*x + x^2])/32

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Rubi [A]  time = 0.344904, antiderivative size = 248, 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 = {1668, 1676, 1169, 634, 618, 204, 628} \[ \frac{5 x^7}{7}-\frac{17 x^5}{5}+\frac{19 x^3}{3}+\frac{25 \left (5 x^2+3\right ) x}{8 \left (x^4+2 x^2+3\right )}-\frac{1}{32} \sqrt{\frac{1}{2} \left (618291 \sqrt{3}-262771\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{1}{32} \sqrt{\frac{1}{2} \left (618291 \sqrt{3}-262771\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+38 x+\frac{1}{16} \sqrt{\frac{1}{2} \left (262771+618291 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{1}{16} \sqrt{\frac{1}{2} \left (262771+618291 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(x^8*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^2,x]

[Out]

38*x + (19*x^3)/3 - (17*x^5)/5 + (5*x^7)/7 + (25*x*(3 + 5*x^2))/(8*(3 + 2*x^2 + x^4)) + (Sqrt[(262771 + 618291
*Sqrt[3])/2]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/16 - (Sqrt[(262771 + 618291*Sqrt[3]
)/2]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/16 - (Sqrt[(-262771 + 618291*Sqrt[3])/2]*Lo
g[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/32 + (Sqrt[(-262771 + 618291*Sqrt[3])/2]*Log[Sqrt[3] + Sqrt[2*(-1
+ Sqrt[3])]*x + x^2])/32

Rule 1668

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[(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] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p +
7)*(b*d - 2*a*e)*x^2, 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] && IGtQ[m/2, 0]

Rule 1676

Int[(Pq_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^2 + c*x^4), x], x
] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1

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{x^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx &=\frac{25 x \left (3+5 x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac{1}{48} \int \frac{-450-1650 x^2+1200 x^4-336 x^8+240 x^{10}}{3+2 x^2+x^4} \, dx\\ &=\frac{25 x \left (3+5 x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac{1}{48} \int \left (1824+912 x^2-816 x^4+240 x^6-\frac{6 \left (987+1339 x^2\right )}{3+2 x^2+x^4}\right ) \, dx\\ &=38 x+\frac{19 x^3}{3}-\frac{17 x^5}{5}+\frac{5 x^7}{7}+\frac{25 x \left (3+5 x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac{1}{8} \int \frac{987+1339 x^2}{3+2 x^2+x^4} \, dx\\ &=38 x+\frac{19 x^3}{3}-\frac{17 x^5}{5}+\frac{5 x^7}{7}+\frac{25 x \left (3+5 x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac{\int \frac{987 \sqrt{2 \left (-1+\sqrt{3}\right )}-\left (987-1339 \sqrt{3}\right ) x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{16 \sqrt{6 \left (-1+\sqrt{3}\right )}}-\frac{\int \frac{987 \sqrt{2 \left (-1+\sqrt{3}\right )}+\left (987-1339 \sqrt{3}\right ) x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{16 \sqrt{6 \left (-1+\sqrt{3}\right )}}\\ &=38 x+\frac{19 x^3}{3}-\frac{17 x^5}{5}+\frac{5 x^7}{7}+\frac{25 x \left (3+5 x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac{1}{32} \left (1339+329 \sqrt{3}\right ) \int \frac{1}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx-\frac{1}{32} \left (1339+329 \sqrt{3}\right ) \int \frac{1}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx-\frac{1}{32} \sqrt{\frac{1}{2} \left (-262771+618291 \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}{32} \sqrt{\frac{1}{2} \left (-262771+618291 \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\\ &=38 x+\frac{19 x^3}{3}-\frac{17 x^5}{5}+\frac{5 x^7}{7}+\frac{25 x \left (3+5 x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac{1}{32} \sqrt{\frac{1}{2} \left (-262771+618291 \sqrt{3}\right )} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{1}{32} \sqrt{\frac{1}{2} \left (-262771+618291 \sqrt{3}\right )} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )-\frac{1}{16} \left (-1339-329 \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}{16} \left (-1339-329 \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 )\\ &=38 x+\frac{19 x^3}{3}-\frac{17 x^5}{5}+\frac{5 x^7}{7}+\frac{25 x \left (3+5 x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac{1}{16} \sqrt{\frac{1}{2} \left (262771+618291 \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}{16} \sqrt{\frac{1}{2} \left (262771+618291 \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}{32} \sqrt{\frac{1}{2} \left (-262771+618291 \sqrt{3}\right )} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{1}{32} \sqrt{\frac{1}{2} \left (-262771+618291 \sqrt{3}\right )} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )\\ \end{align*}

Mathematica [C]  time = 0.177918, size = 145, normalized size = 0.58 \[ \frac{5 x^7}{7}-\frac{17 x^5}{5}+\frac{19 x^3}{3}+\frac{25 \left (5 x^2+3\right ) x}{8 \left (x^4+2 x^2+3\right )}+38 x-\frac{\left (1339 \sqrt{2}+352 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{16 \sqrt{2-2 i \sqrt{2}}}-\frac{\left (1339 \sqrt{2}-352 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{16 \sqrt{2+2 i \sqrt{2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^8*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^2,x]

[Out]

38*x + (19*x^3)/3 - (17*x^5)/5 + (5*x^7)/7 + (25*x*(3 + 5*x^2))/(8*(3 + 2*x^2 + x^4)) - ((352*I + 1339*Sqrt[2]
)*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/(16*Sqrt[2 - (2*I)*Sqrt[2]]) - ((-352*I + 1339*Sqrt[2])*ArcTan[x/Sqrt[1 + I*S
qrt[2]]])/(16*Sqrt[2 + (2*I)*Sqrt[2]])

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Maple [B]  time = 0.105, size = 427, normalized size = 1.7 \begin{align*}{\frac{5\,{x}^{7}}{7}}-{\frac{17\,{x}^{5}}{5}}+{\frac{19\,{x}^{3}}{3}}+38\,x-{\frac{1}{{x}^{4}+2\,{x}^{2}+3} \left ( -{\frac{125\,{x}^{3}}{8}}-{\frac{75\,x}{8}} \right ) }-{\frac{505\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{64}}-{\frac{11\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{4}}-{\frac{ \left ( -1010+1010\,\sqrt{3} \right ) \sqrt{3}}{32\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-22+22\,\sqrt{3}}{2\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{329\,\sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{505\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{64}}+{\frac{11\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{4}}-{\frac{ \left ( -1010+1010\,\sqrt{3} \right ) \sqrt{3}}{32\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-22+22\,\sqrt{3}}{2\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{329\,\sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^2,x)

[Out]

5/7*x^7-17/5*x^5+19/3*x^3+38*x-(-125/8*x^3-75/8*x)/(x^4+2*x^2+3)-505/64*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))
*(-2+2*3^(1/2))^(1/2)*3^(1/2)-11/4*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)-505/32/(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)-11/2/(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))-329/8/(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)+505/64*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1
/2))^(1/2)*3^(1/2)+11/4*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)-505/32/(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)-11/2/(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))-329/8/(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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{5}{7} \, x^{7} - \frac{17}{5} \, x^{5} + \frac{19}{3} \, x^{3} + 38 \, x + \frac{25 \,{\left (5 \, x^{3} + 3 \, x\right )}}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac{1}{8} \, \int \frac{1339 \, x^{2} + 987}{x^{4} + 2 \, x^{2} + 3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^2,x, algorithm="maxima")

[Out]

5/7*x^7 - 17/5*x^5 + 19/3*x^3 + 38*x + 25/8*(5*x^3 + 3*x)/(x^4 + 2*x^2 + 3) - 1/8*integrate((1339*x^2 + 987)/(
x^4 + 2*x^2 + 3), x)

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Fricas [B]  time = 1.7466, size = 2484, normalized size = 10.02 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^2,x, algorithm="fricas")

[Out]

1/338902147590720*(242072962564800*x^11 - 668121376678848*x^9 + 568064552152064*x^7 + 13714240239171136*x^5 -
102773860*14158657803^(1/4)*sqrt(68699)*sqrt(3)*sqrt(2)*(x^4 + 2*x^2 + 3)*sqrt(262771*sqrt(3) + 1854873)*arcta
n(1/3145089554732313026311937382*sqrt(50431867201)*14158657803^(3/4)*sqrt(68699)*sqrt(3*14158657803^(1/4)*sqrt
(68699)*(1339*sqrt(3)*x - 987*x)*sqrt(262771*sqrt(3) + 1854873) + 453886804809*x^2 + 453886804809*sqrt(3))*(32
9*sqrt(3)*sqrt(2) - 1339*sqrt(2))*sqrt(262771*sqrt(3) + 1854873) - 1/20787713069048994*14158657803^(3/4)*sqrt(
68699)*(329*sqrt(3)*sqrt(2)*x - 1339*sqrt(2)*x)*sqrt(262771*sqrt(3) + 1854873) + 1/2*sqrt(3)*sqrt(2) - 1/2*sqr
t(2)) - 102773860*14158657803^(1/4)*sqrt(68699)*sqrt(3)*sqrt(2)*(x^4 + 2*x^2 + 3)*sqrt(262771*sqrt(3) + 185487
3)*arctan(1/3145089554732313026311937382*sqrt(50431867201)*14158657803^(3/4)*sqrt(68699)*sqrt(-3*14158657803^(
1/4)*sqrt(68699)*(1339*sqrt(3)*x - 987*x)*sqrt(262771*sqrt(3) + 1854873) + 453886804809*x^2 + 453886804809*sqr
t(3))*(329*sqrt(3)*sqrt(2) - 1339*sqrt(2))*sqrt(262771*sqrt(3) + 1854873) - 1/20787713069048994*14158657803^(3
/4)*sqrt(68699)*(329*sqrt(3)*sqrt(2)*x - 1339*sqrt(2)*x)*sqrt(262771*sqrt(3) + 1854873) - 1/2*sqrt(3)*sqrt(2)
+ 1/2*sqrt(2)) + 35*14158657803^(1/4)*sqrt(68699)*(1854873*x^4 + 3709746*x^2 - 262771*sqrt(3)*(x^4 + 2*x^2 + 3
) + 5564619)*sqrt(262771*sqrt(3) + 1854873)*log(3*14158657803^(1/4)*sqrt(68699)*(1339*sqrt(3)*x - 987*x)*sqrt(
262771*sqrt(3) + 1854873) + 453886804809*x^2 + 453886804809*sqrt(3)) - 35*14158657803^(1/4)*sqrt(68699)*(18548
73*x^4 + 3709746*x^2 - 262771*sqrt(3)*(x^4 + 2*x^2 + 3) + 5564619)*sqrt(262771*sqrt(3) + 1854873)*log(-3*14158
657803^(1/4)*sqrt(68699)*(1339*sqrt(3)*x - 987*x)*sqrt(262771*sqrt(3) + 1854873) + 453886804809*x^2 + 45388680
4809*sqrt(3)) + 37491050077223400*x^3 + 41812052459005080*x)/(x^4 + 2*x^2 + 3)

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Sympy [A]  time = 0.537435, size = 71, normalized size = 0.29 \begin{align*} \frac{5 x^{7}}{7} - \frac{17 x^{5}}{5} + \frac{19 x^{3}}{3} + 38 x + \frac{125 x^{3} + 75 x}{8 x^{4} + 16 x^{2} + 24} + \operatorname{RootSum}{\left (1048576 t^{4} + 538155008 t^{2} + 1146851282043, \left ( t \mapsto t \log{\left (- \frac{16547840 t^{3}}{453886804809} - \frac{11974973632 t}{453886804809} + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(5*x**6+3*x**4+x**2+4)/(x**4+2*x**2+3)**2,x)

[Out]

5*x**7/7 - 17*x**5/5 + 19*x**3/3 + 38*x + (125*x**3 + 75*x)/(8*x**4 + 16*x**2 + 24) + RootSum(1048576*_t**4 +
538155008*_t**2 + 1146851282043, Lambda(_t, _t*log(-16547840*_t**3/453886804809 - 11974973632*_t/453886804809
+ x)))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x^{6} + 3 \, x^{4} + x^{2} + 4\right )} x^{8}}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^2,x, algorithm="giac")

[Out]

integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^8/(x^4 + 2*x^2 + 3)^2, x)