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

[Out]

38*x+19/3*x^3-17/5*x^5+5/7*x^7+25/8*x*(5*x^2+3)/(x^4+2*x^2+3)-1/64*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-52
5542+1236582*3^(1/2))^(1/2)+1/64*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-525542+1236582*3^(1/2))^(1/2)+1/32*a
rctan((-2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(525542+1236582*3^(1/2))^(1/2)-1/32*arctan((2*x+(-2+2*3
^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(525542+1236582*3^(1/2))^(1/2)

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Rubi [A]  time = 0.34, antiderivative size = 248, normalized size of antiderivative = 1.00, 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 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 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 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]

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 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 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

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*}

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Mathematica [C]  time = 0.17, 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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fricas [B]  time = 0.81, size = 519, normalized size = 2.09 \[ \frac {242072962564800 \, x^{11} - 668121376678848 \, x^{9} + 568064552152064 \, x^{7} + 13714240239171136 \, x^{5} - 102773860 \cdot 14158657803^{\frac {1}{4}} \sqrt {68699} \sqrt {3} \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {262771 \, \sqrt {3} + 1854873} \arctan \left (\frac {1}{3145089554732313026311937382} \, \sqrt {50431867201} 14158657803^{\frac {3}{4}} \sqrt {68699} \sqrt {3 \cdot 14158657803^{\frac {1}{4}} \sqrt {68699} {\left (1339 \, \sqrt {3} x - 987 \, x\right )} \sqrt {262771 \, \sqrt {3} + 1854873} + 453886804809 \, x^{2} + 453886804809 \, \sqrt {3}} {\left (329 \, \sqrt {3} \sqrt {2} - 1339 \, \sqrt {2}\right )} \sqrt {262771 \, \sqrt {3} + 1854873} - \frac {1}{20787713069048994} \cdot 14158657803^{\frac {3}{4}} \sqrt {68699} {\left (329 \, \sqrt {3} \sqrt {2} x - 1339 \, \sqrt {2} x\right )} \sqrt {262771 \, \sqrt {3} + 1854873} + \frac {1}{2} \, \sqrt {3} \sqrt {2} - \frac {1}{2} \, \sqrt {2}\right ) - 102773860 \cdot 14158657803^{\frac {1}{4}} \sqrt {68699} \sqrt {3} \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {262771 \, \sqrt {3} + 1854873} \arctan \left (\frac {1}{3145089554732313026311937382} \, \sqrt {50431867201} 14158657803^{\frac {3}{4}} \sqrt {68699} \sqrt {-3 \cdot 14158657803^{\frac {1}{4}} \sqrt {68699} {\left (1339 \, \sqrt {3} x - 987 \, x\right )} \sqrt {262771 \, \sqrt {3} + 1854873} + 453886804809 \, x^{2} + 453886804809 \, \sqrt {3}} {\left (329 \, \sqrt {3} \sqrt {2} - 1339 \, \sqrt {2}\right )} \sqrt {262771 \, \sqrt {3} + 1854873} - \frac {1}{20787713069048994} \cdot 14158657803^{\frac {3}{4}} \sqrt {68699} {\left (329 \, \sqrt {3} \sqrt {2} x - 1339 \, \sqrt {2} x\right )} \sqrt {262771 \, \sqrt {3} + 1854873} - \frac {1}{2} \, \sqrt {3} \sqrt {2} + \frac {1}{2} \, \sqrt {2}\right ) + 35 \cdot 14158657803^{\frac {1}{4}} \sqrt {68699} {\left (1854873 \, x^{4} + 3709746 \, x^{2} - 262771 \, \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 3\right )} + 5564619\right )} \sqrt {262771 \, \sqrt {3} + 1854873} \log \left (3 \cdot 14158657803^{\frac {1}{4}} \sqrt {68699} {\left (1339 \, \sqrt {3} x - 987 \, x\right )} \sqrt {262771 \, \sqrt {3} + 1854873} + 453886804809 \, x^{2} + 453886804809 \, \sqrt {3}\right ) - 35 \cdot 14158657803^{\frac {1}{4}} \sqrt {68699} {\left (1854873 \, x^{4} + 3709746 \, x^{2} - 262771 \, \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 3\right )} + 5564619\right )} \sqrt {262771 \, \sqrt {3} + 1854873} \log \left (-3 \cdot 14158657803^{\frac {1}{4}} \sqrt {68699} {\left (1339 \, \sqrt {3} x - 987 \, x\right )} \sqrt {262771 \, \sqrt {3} + 1854873} + 453886804809 \, x^{2} + 453886804809 \, \sqrt {3}\right ) + 37491050077223400 \, x^{3} + 41812052459005080 \, x}{338902147590720 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \]

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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giac [B]  time = 1.89, size = 585, normalized size = 2.36 \[ \frac {5}{7} \, x^{7} - \frac {17}{5} \, x^{5} + \frac {19}{3} \, x^{3} + \frac {1}{20736} \, \sqrt {2} {\left (1339 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 24102 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 24102 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 1339 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 35532 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 35532 \cdot 3^{\frac {1}{4}} \sqrt {-6 \, \sqrt {3} + 18}\right )} \arctan \left (\frac {3^{\frac {3}{4}} {\left (x + 3^{\frac {1}{4}} \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}\right )}}{3 \, \sqrt {\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}}\right ) + \frac {1}{20736} \, \sqrt {2} {\left (1339 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 24102 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 24102 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 1339 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 35532 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 35532 \cdot 3^{\frac {1}{4}} \sqrt {-6 \, \sqrt {3} + 18}\right )} \arctan \left (\frac {3^{\frac {3}{4}} {\left (x - 3^{\frac {1}{4}} \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}\right )}}{3 \, \sqrt {\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}}\right ) + \frac {1}{41472} \, \sqrt {2} {\left (24102 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 1339 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 1339 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 24102 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 35532 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 35532 \cdot 3^{\frac {1}{4}} \sqrt {6 \, \sqrt {3} + 18}\right )} \log \left (x^{2} + 2 \cdot 3^{\frac {1}{4}} x \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}} + \sqrt {3}\right ) - \frac {1}{41472} \, \sqrt {2} {\left (24102 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 1339 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 1339 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 24102 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 35532 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 35532 \cdot 3^{\frac {1}{4}} \sqrt {6 \, \sqrt {3} + 18}\right )} \log \left (x^{2} - 2 \cdot 3^{\frac {1}{4}} x \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}} + \sqrt {3}\right ) + 38 \, x + \frac {25 \, {\left (5 \, x^{3} + 3 \, x\right )}}{8 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \]

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]

5/7*x^7 - 17/5*x^5 + 19/3*x^3 + 1/20736*sqrt(2)*(1339*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 24102*3^(3/4)*s
qrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 24102*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 1339*3^(3/4)*(
-6*sqrt(3) + 18)^(3/2) - 35532*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) + 35532*3^(1/4)*sqrt(-6*sqrt(3) + 18))*arc
tan(1/3*3^(3/4)*(x + 3^(1/4)*sqrt(-1/6*sqrt(3) + 1/2))/sqrt(1/6*sqrt(3) + 1/2)) + 1/20736*sqrt(2)*(1339*3^(3/4
)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 24102*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 24102*3^(3/4)*(s
qrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 1339*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 35532*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3
) + 18) + 35532*3^(1/4)*sqrt(-6*sqrt(3) + 18))*arctan(1/3*3^(3/4)*(x - 3^(1/4)*sqrt(-1/6*sqrt(3) + 1/2))/sqrt(
1/6*sqrt(3) + 1/2)) + 1/41472*sqrt(2)*(24102*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 1339*3^(3/4
)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 1339*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 24102*3^(3/4)*sqrt(6*sqrt(3) + 18)*(
sqrt(3) - 3) - 35532*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) - 35532*3^(1/4)*sqrt(6*sqrt(3) + 18))*log(x^2 + 2*3
^(1/4)*x*sqrt(-1/6*sqrt(3) + 1/2) + sqrt(3)) - 1/41472*sqrt(2)*(24102*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sq
rt(3) + 18) - 1339*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 1339*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 24102*3^(3/
4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 35532*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) - 35532*3^(1/4)*sqrt(6*sqr
t(3) + 18))*log(x^2 - 2*3^(1/4)*x*sqrt(-1/6*sqrt(3) + 1/2) + sqrt(3)) + 38*x + 25/8*(5*x^3 + 3*x)/(x^4 + 2*x^2
+ 3)

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maple [B]  time = 0.12, size = 427, normalized size = 1.72 \[ \frac {5 x^{7}}{7}-\frac {17 x^{5}}{5}+\frac {19 x^{3}}{3}+38 x -\frac {505 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{32 \sqrt {2+2 \sqrt {3}}}-\frac {11 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{2 \sqrt {2+2 \sqrt {3}}}-\frac {329 \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{8 \sqrt {2+2 \sqrt {3}}}-\frac {505 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{32 \sqrt {2+2 \sqrt {3}}}-\frac {11 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{2 \sqrt {2+2 \sqrt {3}}}-\frac {329 \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{8 \sqrt {2+2 \sqrt {3}}}-\frac {505 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{64}-\frac {11 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{4}+\frac {505 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{64}+\frac {11 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{4}-\frac {-\frac {125}{8} x^{3}-\frac {75}{8} x}{x^{4}+2 x^{2}+3} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \frac {5}{7} \, x^{7} - \frac {17}{5} \, x^{5} + \frac {19}{3} \, x^{3} + 38 \, x + \frac {25 \, {\left (5 \, x^{3} + 3 \, x\right )}}{8 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac {1}{8} \, \int \frac {1339 \, x^{2} + 987}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]

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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mupad [B]  time = 0.11, size = 171, normalized size = 0.69 \[ 38\,x+\frac {\frac {125\,x^3}{8}+\frac {75\,x}{8}}{x^4+2\,x^2+3}+\frac {19\,x^3}{3}-\frac {17\,x^5}{5}+\frac {5\,x^7}{7}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-262771-\sqrt {2}\,734099{}\mathrm {i}}\,734099{}\mathrm {i}}{64\,\left (-\frac {1112159985}{64}+\frac {\sqrt {2}\,724555713{}\mathrm {i}}{128}\right )}+\frac {734099\,\sqrt {2}\,x\,\sqrt {-262771-\sqrt {2}\,734099{}\mathrm {i}}}{128\,\left (-\frac {1112159985}{64}+\frac {\sqrt {2}\,724555713{}\mathrm {i}}{128}\right )}\right )\,\sqrt {-262771-\sqrt {2}\,734099{}\mathrm {i}}\,1{}\mathrm {i}}{16}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-262771+\sqrt {2}\,734099{}\mathrm {i}}\,734099{}\mathrm {i}}{64\,\left (\frac {1112159985}{64}+\frac {\sqrt {2}\,724555713{}\mathrm {i}}{128}\right )}-\frac {734099\,\sqrt {2}\,x\,\sqrt {-262771+\sqrt {2}\,734099{}\mathrm {i}}}{128\,\left (\frac {1112159985}{64}+\frac {\sqrt {2}\,724555713{}\mathrm {i}}{128}\right )}\right )\,\sqrt {-262771+\sqrt {2}\,734099{}\mathrm {i}}\,1{}\mathrm {i}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

38*x + (atan((x*(- 2^(1/2)*734099i - 262771)^(1/2)*734099i)/(64*((2^(1/2)*724555713i)/128 - 1112159985/64)) +
(734099*2^(1/2)*x*(- 2^(1/2)*734099i - 262771)^(1/2))/(128*((2^(1/2)*724555713i)/128 - 1112159985/64)))*(- 2^(
1/2)*734099i - 262771)^(1/2)*1i)/16 - (atan((x*(2^(1/2)*734099i - 262771)^(1/2)*734099i)/(64*((2^(1/2)*7245557
13i)/128 + 1112159985/64)) - (734099*2^(1/2)*x*(2^(1/2)*734099i - 262771)^(1/2))/(128*((2^(1/2)*724555713i)/12
8 + 1112159985/64)))*(2^(1/2)*734099i - 262771)^(1/2)*1i)/16 + ((75*x)/8 + (125*x^3)/8)/(2*x^2 + x^4 + 3) + (1
9*x^3)/3 - (17*x^5)/5 + (5*x^7)/7

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sympy [A]  time = 0.61, size = 71, normalized size = 0.29 \[ \frac {5 x^{7}}{7} - \frac {17 x^{5}}{5} + \frac {19 x^{3}}{3} + 38 x + \frac {125 x^{3} + 75 x}{8 x^{4} + 16 x^{2} + 24} + \operatorname {RootSum} {\left (1048576 t^{4} + 538155008 t^{2} + 1146851282043, \left (t \mapsto t \log {\left (- \frac {16547840 t^{3}}{453886804809} - \frac {11974973632 t}{453886804809} + x \right )} \right )\right )} \]

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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