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

Optimal. Leaf size=225 \[ -\frac {1}{32} \sqrt {\frac {1}{6} \left (12899 \sqrt {3}-19291\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {1}{32} \sqrt {\frac {1}{6} \left (12899 \sqrt {3}-19291\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {25 \left (x^2+1\right ) x}{8 \left (x^4+2 x^2+3\right )}+5 x+\frac {1}{16} \sqrt {\frac {1}{6} \left (19291+12899 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{16} \sqrt {\frac {1}{6} \left (19291+12899 \sqrt {3}\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right ) \]

[Out]

5*x+25/8*x*(x^2+1)/(x^4+2*x^2+3)-1/192*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-115746+77394*3^(1/2))^(1/2)+1/
192*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-115746+77394*3^(1/2))^(1/2)+1/96*arctan((-2*x+(-2+2*3^(1/2))^(1/2
))/(2+2*3^(1/2))^(1/2))*(115746+77394*3^(1/2))^(1/2)-1/96*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2
))*(115746+77394*3^(1/2))^(1/2)

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Rubi [A]  time = 0.30, antiderivative size = 225, 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} \[ \frac {25 \left (x^2+1\right ) x}{8 \left (x^4+2 x^2+3\right )}-\frac {1}{32} \sqrt {\frac {1}{6} \left (12899 \sqrt {3}-19291\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {1}{32} \sqrt {\frac {1}{6} \left (12899 \sqrt {3}-19291\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+5 x+\frac {1}{16} \sqrt {\frac {1}{6} \left (19291+12899 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{16} \sqrt {\frac {1}{6} \left (19291+12899 \sqrt {3}\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

5*x + (25*x*(1 + x^2))/(8*(3 + 2*x^2 + x^4)) + (Sqrt[(19291 + 12899*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])]
- 2*x)/Sqrt[2*(1 + Sqrt[3])]])/16 - (Sqrt[(19291 + 12899*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sq
rt[2*(1 + Sqrt[3])]])/16 - (Sqrt[(-19291 + 12899*Sqrt[3])/6]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/32
+ (Sqrt[(-19291 + 12899*Sqrt[3])/6]*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^2 \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 (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{48} \int \frac {-150-186 x^2+240 x^4}{3+2 x^2+x^4} \, dx\\ &=\frac {25 x \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{48} \int \left (240-\frac {6 \left (145+111 x^2\right )}{3+2 x^2+x^4}\right ) \, dx\\ &=5 x+\frac {25 x \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {1}{8} \int \frac {145+111 x^2}{3+2 x^2+x^4} \, dx\\ &=5 x+\frac {25 x \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {\int \frac {145 \sqrt {2 \left (-1+\sqrt {3}\right )}-\left (145-111 \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 {145 \sqrt {2 \left (-1+\sqrt {3}\right )}+\left (145-111 \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 )}}\\ &=5 x+\frac {25 x \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {1}{96} \left (333+145 \sqrt {3}\right ) \int \frac {1}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx-\frac {1}{96} \left (333+145 \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}{6} \left (-19291+12899 \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}{6} \left (-19291+12899 \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\\ &=5 x+\frac {25 x \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {1}{32} \sqrt {\frac {1}{6} \left (-19291+12899 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{32} \sqrt {\frac {1}{6} \left (-19291+12899 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{48} \left (333+145 \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}{48} \left (333+145 \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 )\\ &=5 x+\frac {25 x \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{16} \sqrt {\frac {1}{6} \left (19291+12899 \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}{6} \left (19291+12899 \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}{6} \left (-19291+12899 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{32} \sqrt {\frac {1}{6} \left (-19291+12899 \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 = 0.16, size = 121, normalized size = 0.54 \[ \frac {25 \left (x^3+x\right )}{8 \left (x^4+2 x^2+3\right )}+5 x-\frac {\left (111 \sqrt {2}-34 i\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{16 \sqrt {2-2 i \sqrt {2}}}-\frac {\left (111 \sqrt {2}+34 i\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{16 \sqrt {2+2 i \sqrt {2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

5*x + (25*(x + x^3))/(8*(3 + 2*x^2 + x^4)) - ((-34*I + 111*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/(16*Sqrt[2
- (2*I)*Sqrt[2]]) - ((34*I + 111*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(16*Sqrt[2 + (2*I)*Sqrt[2]])

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fricas [B]  time = 0.84, size = 459, normalized size = 2.04 \[ \frac {98680445760 \, x^{5} + 31876 \cdot 499152603^{\frac {1}{4}} \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} \arctan \left (\frac {1}{2453286601800494203302} \cdot 499152603^{\frac {3}{4}} \sqrt {308376393} \sqrt {308376393 \, x^{2} + 499152603^{\frac {1}{4}} {\left (145 \, \sqrt {3} x - 333 \, x\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} + 308376393 \, \sqrt {3}} {\left (111 \, \sqrt {3} \sqrt {2} - 145 \, \sqrt {2}\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} - \frac {1}{7955494186614} \cdot 499152603^{\frac {3}{4}} {\left (111 \, \sqrt {3} \sqrt {2} x - 145 \, \sqrt {2} x\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} + \frac {1}{2} \, \sqrt {3} \sqrt {2} - \frac {1}{2} \, \sqrt {2}\right ) + 31876 \cdot 499152603^{\frac {1}{4}} \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} \arctan \left (\frac {1}{2453286601800494203302} \cdot 499152603^{\frac {3}{4}} \sqrt {308376393} \sqrt {308376393 \, x^{2} - 499152603^{\frac {1}{4}} {\left (145 \, \sqrt {3} x - 333 \, x\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} + 308376393 \, \sqrt {3}} {\left (111 \, \sqrt {3} \sqrt {2} - 145 \, \sqrt {2}\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} - \frac {1}{7955494186614} \cdot 499152603^{\frac {3}{4}} {\left (111 \, \sqrt {3} \sqrt {2} x - 145 \, \sqrt {2} x\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} - \frac {1}{2} \, \sqrt {3} \sqrt {2} + \frac {1}{2} \, \sqrt {2}\right ) + 259036170120 \, x^{3} + 499152603^{\frac {1}{4}} {\left (19291 \, x^{4} + 38582 \, x^{2} - 12899 \, \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 3\right )} + 57873\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} \log \left (308376393 \, x^{2} + 499152603^{\frac {1}{4}} {\left (145 \, \sqrt {3} x - 333 \, x\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} + 308376393 \, \sqrt {3}\right ) - 499152603^{\frac {1}{4}} {\left (19291 \, x^{4} + 38582 \, x^{2} - 12899 \, \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 3\right )} + 57873\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} \log \left (308376393 \, x^{2} - 499152603^{\frac {1}{4}} {\left (145 \, \sqrt {3} x - 333 \, x\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} + 308376393 \, \sqrt {3}\right ) + 357716615880 \, x}{19736089152 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19736089152*(98680445760*x^5 + 31876*499152603^(1/4)*sqrt(2)*(x^4 + 2*x^2 + 3)*sqrt(248834609*sqrt(3) + 4991
52603)*arctan(1/2453286601800494203302*499152603^(3/4)*sqrt(308376393)*sqrt(308376393*x^2 + 499152603^(1/4)*(1
45*sqrt(3)*x - 333*x)*sqrt(248834609*sqrt(3) + 499152603) + 308376393*sqrt(3))*(111*sqrt(3)*sqrt(2) - 145*sqrt
(2))*sqrt(248834609*sqrt(3) + 499152603) - 1/7955494186614*499152603^(3/4)*(111*sqrt(3)*sqrt(2)*x - 145*sqrt(2
)*x)*sqrt(248834609*sqrt(3) + 499152603) + 1/2*sqrt(3)*sqrt(2) - 1/2*sqrt(2)) + 31876*499152603^(1/4)*sqrt(2)*
(x^4 + 2*x^2 + 3)*sqrt(248834609*sqrt(3) + 499152603)*arctan(1/2453286601800494203302*499152603^(3/4)*sqrt(308
376393)*sqrt(308376393*x^2 - 499152603^(1/4)*(145*sqrt(3)*x - 333*x)*sqrt(248834609*sqrt(3) + 499152603) + 308
376393*sqrt(3))*(111*sqrt(3)*sqrt(2) - 145*sqrt(2))*sqrt(248834609*sqrt(3) + 499152603) - 1/7955494186614*4991
52603^(3/4)*(111*sqrt(3)*sqrt(2)*x - 145*sqrt(2)*x)*sqrt(248834609*sqrt(3) + 499152603) - 1/2*sqrt(3)*sqrt(2)
+ 1/2*sqrt(2)) + 259036170120*x^3 + 499152603^(1/4)*(19291*x^4 + 38582*x^2 - 12899*sqrt(3)*(x^4 + 2*x^2 + 3) +
57873)*sqrt(248834609*sqrt(3) + 499152603)*log(308376393*x^2 + 499152603^(1/4)*(145*sqrt(3)*x - 333*x)*sqrt(2
48834609*sqrt(3) + 499152603) + 308376393*sqrt(3)) - 499152603^(1/4)*(19291*x^4 + 38582*x^2 - 12899*sqrt(3)*(x
^4 + 2*x^2 + 3) + 57873)*sqrt(248834609*sqrt(3) + 499152603)*log(308376393*x^2 - 499152603^(1/4)*(145*sqrt(3)*
x - 333*x)*sqrt(248834609*sqrt(3) + 499152603) + 308376393*sqrt(3)) + 357716615880*x)/(x^4 + 2*x^2 + 3)

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giac [B]  time = 1.82, size = 566, normalized size = 2.52 \[ \frac {1}{6912} \, \sqrt {2} {\left (37 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 666 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 666 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 37 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 1740 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 1740 \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}{6912} \, \sqrt {2} {\left (37 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 666 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 666 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 37 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 1740 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 1740 \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}{13824} \, \sqrt {2} {\left (666 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 37 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 37 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 666 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 1740 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 1740 \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}{13824} \, \sqrt {2} {\left (666 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 37 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 37 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 666 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 1740 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 1740 \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 ) + 5 \, x + \frac {25 \, {\left (x^{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^2*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^2,x, algorithm="giac")

[Out]

1/6912*sqrt(2)*(37*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 666*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3)
- 3) - 666*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 37*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 1740*3^(1/4)*sqr
t(2)*sqrt(6*sqrt(3) + 18) + 1740*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)) + 1/6912*sqrt(2)*(37*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 666*3^(3/4)
*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 666*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 37*3^(3/4)*(-6
*sqrt(3) + 18)^(3/2) - 1740*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) + 1740*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)) + 1/13824*sqrt(2)*(666*3^(3/4)*sqr
t(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 37*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 37*3^(3/4)*(6*sqrt(3)
+ 18)^(3/2) + 666*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 1740*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) - 17
40*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)) - 1/13824*sqrt(2)*(
666*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 37*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 37*3^(3
/4)*(6*sqrt(3) + 18)^(3/2) + 666*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 1740*3^(1/4)*sqrt(2)*sqrt(-6*sqr
t(3) + 18) - 1740*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)) + 5*
x + 25/8*(x^3 + x)/(x^4 + 2*x^2 + 3)

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maple [B]  time = 0.03, size = 412, normalized size = 1.83 \[ 5 x -\frac {47 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{48 \sqrt {2+2 \sqrt {3}}}+\frac {17 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{32 \sqrt {2+2 \sqrt {3}}}-\frac {145 \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{24 \sqrt {2+2 \sqrt {3}}}-\frac {47 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{48 \sqrt {2+2 \sqrt {3}}}+\frac {17 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{32 \sqrt {2+2 \sqrt {3}}}-\frac {145 \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{24 \sqrt {2+2 \sqrt {3}}}-\frac {47 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{96}+\frac {17 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{64}+\frac {47 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{96}-\frac {17 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{64}-\frac {-\frac {25}{8} x^{3}-\frac {25}{8} x}{x^{4}+2 x^{2}+3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*x-(-25/8*x^3-25/8*x)/(x^4+2*x^2+3)-47/96*(-2+2*3^(1/2))^(1/2)*3^(1/2)*ln(x^2-(-2+2*3^(1/2))^(1/2)*x+3^(1/2))
+17/64*(-2+2*3^(1/2))^(1/2)*ln(x^2-(-2+2*3^(1/2))^(1/2)*x+3^(1/2))-47/48/(2+2*3^(1/2))^(1/2)*(-2+2*3^(1/2))*3^
(1/2)*arctan((2*x-(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))+17/32/(2+2*3^(1/2))^(1/2)*(-2+2*3^(1/2))*arctan((
2*x-(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))-145/24/(2+2*3^(1/2))^(1/2)*3^(1/2)*arctan((2*x-(-2+2*3^(1/2))^(
1/2))/(2+2*3^(1/2))^(1/2))+47/96*(-2+2*3^(1/2))^(1/2)*3^(1/2)*ln(x^2+(-2+2*3^(1/2))^(1/2)*x+3^(1/2))-17/64*(-2
+2*3^(1/2))^(1/2)*ln(x^2+(-2+2*3^(1/2))^(1/2)*x+3^(1/2))-47/48/(2+2*3^(1/2))^(1/2)*(-2+2*3^(1/2))*3^(1/2)*arct
an((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))+17/32/(2+2*3^(1/2))^(1/2)*(-2+2*3^(1/2))*arctan((2*x+(-2+2*
3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))-145/24/(2+2*3^(1/2))^(1/2)*3^(1/2)*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2
*3^(1/2))^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ 5 \, x + \frac {25 \, {\left (x^{3} + x\right )}}{8 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac {1}{8} \, \int \frac {111 \, x^{2} + 145}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*x + 25/8*(x^3 + x)/(x^4 + 2*x^2 + 3) - 1/8*integrate((111*x^2 + 145)/(x^4 + 2*x^2 + 3), x)

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mupad [B]  time = 0.96, size = 156, normalized size = 0.69 \[ 5\,x+\frac {\frac {25\,x^3}{8}+\frac {25\,x}{8}}{x^4+2\,x^2+3}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-57873-\sqrt {2}\,23907{}\mathrm {i}}\,7969{}\mathrm {i}}{576\,\left (-\frac {374543}{96}+\frac {\sqrt {2}\,1155505{}\mathrm {i}}{384}\right )}+\frac {7969\,\sqrt {2}\,x\,\sqrt {-57873-\sqrt {2}\,23907{}\mathrm {i}}}{1152\,\left (-\frac {374543}{96}+\frac {\sqrt {2}\,1155505{}\mathrm {i}}{384}\right )}\right )\,\sqrt {-57873-\sqrt {2}\,23907{}\mathrm {i}}\,1{}\mathrm {i}}{48}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-57873+\sqrt {2}\,23907{}\mathrm {i}}\,7969{}\mathrm {i}}{576\,\left (\frac {374543}{96}+\frac {\sqrt {2}\,1155505{}\mathrm {i}}{384}\right )}-\frac {7969\,\sqrt {2}\,x\,\sqrt {-57873+\sqrt {2}\,23907{}\mathrm {i}}}{1152\,\left (\frac {374543}{96}+\frac {\sqrt {2}\,1155505{}\mathrm {i}}{384}\right )}\right )\,\sqrt {-57873+\sqrt {2}\,23907{}\mathrm {i}}\,1{}\mathrm {i}}{48} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*x + ((25*x)/8 + (25*x^3)/8)/(2*x^2 + x^4 + 3) + (atan((x*(- 2^(1/2)*23907i - 57873)^(1/2)*7969i)/(576*((2^(1
/2)*1155505i)/384 - 374543/96)) + (7969*2^(1/2)*x*(- 2^(1/2)*23907i - 57873)^(1/2))/(1152*((2^(1/2)*1155505i)/
384 - 374543/96)))*(- 2^(1/2)*23907i - 57873)^(1/2)*1i)/48 - (atan((x*(2^(1/2)*23907i - 57873)^(1/2)*7969i)/(5
76*((2^(1/2)*1155505i)/384 + 374543/96)) - (7969*2^(1/2)*x*(2^(1/2)*23907i - 57873)^(1/2))/(1152*((2^(1/2)*115
5505i)/384 + 374543/96)))*(2^(1/2)*23907i - 57873)^(1/2)*1i)/48

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sympy [A]  time = 0.60, size = 51, normalized size = 0.23 \[ 5 x + \frac {25 x^{3} + 25 x}{8 x^{4} + 16 x^{2} + 24} + \operatorname {RootSum} {\left (3145728 t^{4} + 39507968 t^{2} + 166384201, \left (t \mapsto t \log {\left (- \frac {9240576 t^{3}}{102792131} - \frac {95003488 t}{102792131} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*x + (25*x**3 + 25*x)/(8*x**4 + 16*x**2 + 24) + RootSum(3145728*_t**4 + 39507968*_t**2 + 166384201, Lambda(_t
, _t*log(-9240576*_t**3/102792131 - 95003488*_t/102792131 + x)))

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