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

Optimal. Leaf size=232 \[ \frac {5 x^3}{3}-\frac {1}{32} \sqrt {\frac {1}{2} \left (26499 \sqrt {3}-14395\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {1}{32} \sqrt {\frac {1}{2} \left (26499 \sqrt {3}-14395\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {25 \left (x^2+3\right ) x}{8 \left (x^4+2 x^2+3\right )}-17 x-\frac {1}{16} \sqrt {\frac {1}{2} \left (14395+26499 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{16} \sqrt {\frac {1}{2} \left (14395+26499 \sqrt {3}\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right ) \]

[Out]

-17*x+5/3*x^3-25/8*x*(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))*(-28790+52998*3^(1/2))^
(1/2)+1/64*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-28790+52998*3^(1/2))^(1/2)-1/32*arctan((-2*x+(-2+2*3^(1/2)
)^(1/2))/(2+2*3^(1/2))^(1/2))*(28790+52998*3^(1/2))^(1/2)+1/32*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))
^(1/2))*(28790+52998*3^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.29, antiderivative size = 232, 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} \[ \frac {5 x^3}{3}-\frac {25 \left (x^2+3\right ) x}{8 \left (x^4+2 x^2+3\right )}-\frac {1}{32} \sqrt {\frac {1}{2} \left (26499 \sqrt {3}-14395\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {1}{32} \sqrt {\frac {1}{2} \left (26499 \sqrt {3}-14395\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-17 x-\frac {1}{16} \sqrt {\frac {1}{2} \left (14395+26499 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{16} \sqrt {\frac {1}{2} \left (14395+26499 \sqrt {3}\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-17*x + (5*x^3)/3 - (25*x*(3 + x^2))/(8*(3 + 2*x^2 + x^4)) - (Sqrt[(14395 + 26499*Sqrt[3])/2]*ArcTan[(Sqrt[2*(
-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/16 + (Sqrt[(14395 + 26499*Sqrt[3])/2]*ArcTan[(Sqrt[2*(-1 + Sqrt[
3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/16 - (Sqrt[(-14395 + 26499*Sqrt[3])/2]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])
]*x + x^2])/32 + (Sqrt[(-14395 + 26499*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^4 \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+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{48} \int \frac {450-150 x^2-336 x^4+240 x^6}{3+2 x^2+x^4} \, dx\\ &=-\frac {25 x \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{48} \int \left (-816+240 x^2+\frac {6 \left (483+127 x^2\right )}{3+2 x^2+x^4}\right ) \, dx\\ &=-17 x+\frac {5 x^3}{3}-\frac {25 x \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{8} \int \frac {483+127 x^2}{3+2 x^2+x^4} \, dx\\ &=-17 x+\frac {5 x^3}{3}-\frac {25 x \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {\int \frac {483 \sqrt {2 \left (-1+\sqrt {3}\right )}-\left (483-127 \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 {483 \sqrt {2 \left (-1+\sqrt {3}\right )}+\left (483-127 \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 )}}\\ &=-17 x+\frac {5 x^3}{3}-\frac {25 x \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{32} \left (127+161 \sqrt {3}\right ) \int \frac {1}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx+\frac {1}{32} \left (127+161 \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 (-14395+26499 \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 (-14395+26499 \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\\ &=-17 x+\frac {5 x^3}{3}-\frac {25 x \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {1}{32} \sqrt {\frac {1}{2} \left (-14395+26499 \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 (-14395+26499 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{16} \left (-127-161 \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 (-127-161 \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 )\\ &=-17 x+\frac {5 x^3}{3}-\frac {25 x \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {1}{16} \sqrt {\frac {1}{2} \left (14395+26499 \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 (14395+26499 \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 (-14395+26499 \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 (-14395+26499 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.16, size = 129, normalized size = 0.56 \[ \frac {5 x^3}{3}-\frac {25 \left (x^2+3\right ) x}{8 \left (x^4+2 x^2+3\right )}-17 x+\frac {\left (127 \sqrt {2}-356 i\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{16 \sqrt {2-2 i \sqrt {2}}}+\frac {\left (127 \sqrt {2}+356 i\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{16 \sqrt {2+2 i \sqrt {2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-17*x + (5*x^3)/3 - (25*x*(3 + x^2))/(8*(3 + 2*x^2 + x^4)) + ((-356*I + 127*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[
2]]])/(16*Sqrt[2 - (2*I)*Sqrt[2]]) + ((356*I + 127*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(16*Sqrt[2 + (2*I)*
Sqrt[2]])

________________________________________________________________________________________

fricas [B]  time = 1.09, size = 508, normalized size = 2.19 \[ \frac {2159655360 \, x^{7} - 17709173952 \, x^{5} - 123268 \cdot 143883^{\frac {1}{4}} \sqrt {219} \sqrt {3} \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {14395 \, \sqrt {3} + 79497} \arctan \left (\frac {1}{658350237832613766} \, \sqrt {24746051} 143883^{\frac {3}{4}} \sqrt {219} \sqrt {11 \cdot 143883^{\frac {1}{4}} \sqrt {219} {\left (127 \, \sqrt {3} x - 483 \, x\right )} \sqrt {14395 \, \sqrt {3} + 79497} + 222714459 \, x^{2} + 222714459 \, \sqrt {3}} {\left (161 \, \sqrt {3} \sqrt {2} - 127 \, \sqrt {2}\right )} \sqrt {14395 \, \sqrt {3} + 79497} - \frac {1}{8868084822} \cdot 143883^{\frac {3}{4}} \sqrt {219} {\left (161 \, \sqrt {3} \sqrt {2} x - 127 \, \sqrt {2} x\right )} \sqrt {14395 \, \sqrt {3} + 79497} + \frac {1}{2} \, \sqrt {3} \sqrt {2} - \frac {1}{2} \, \sqrt {2}\right ) - 123268 \cdot 143883^{\frac {1}{4}} \sqrt {219} \sqrt {3} \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {14395 \, \sqrt {3} + 79497} \arctan \left (\frac {1}{658350237832613766} \, \sqrt {24746051} 143883^{\frac {3}{4}} \sqrt {219} \sqrt {-11 \cdot 143883^{\frac {1}{4}} \sqrt {219} {\left (127 \, \sqrt {3} x - 483 \, x\right )} \sqrt {14395 \, \sqrt {3} + 79497} + 222714459 \, x^{2} + 222714459 \, \sqrt {3}} {\left (161 \, \sqrt {3} \sqrt {2} - 127 \, \sqrt {2}\right )} \sqrt {14395 \, \sqrt {3} + 79497} - \frac {1}{8868084822} \cdot 143883^{\frac {3}{4}} \sqrt {219} {\left (161 \, \sqrt {3} \sqrt {2} x - 127 \, \sqrt {2} x\right )} \sqrt {14395 \, \sqrt {3} + 79497} - \frac {1}{2} \, \sqrt {3} \sqrt {2} + \frac {1}{2} \, \sqrt {2}\right ) - 143883^{\frac {1}{4}} \sqrt {219} {\left (79497 \, x^{4} + 158994 \, x^{2} - 14395 \, \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 3\right )} + 238491\right )} \sqrt {14395 \, \sqrt {3} + 79497} \log \left (11 \cdot 143883^{\frac {1}{4}} \sqrt {219} {\left (127 \, \sqrt {3} x - 483 \, x\right )} \sqrt {14395 \, \sqrt {3} + 79497} + 222714459 \, x^{2} + 222714459 \, \sqrt {3}\right ) + 143883^{\frac {1}{4}} \sqrt {219} {\left (79497 \, x^{4} + 158994 \, x^{2} - 14395 \, \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 3\right )} + 238491\right )} \sqrt {14395 \, \sqrt {3} + 79497} \log \left (-11 \cdot 143883^{\frac {1}{4}} \sqrt {219} {\left (127 \, \sqrt {3} x - 483 \, x\right )} \sqrt {14395 \, \sqrt {3} + 79497} + 222714459 \, x^{2} + 222714459 \, \sqrt {3}\right ) - 41627357064 \, x^{3} - 78233515416 \, x}{1295793216 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1295793216*(2159655360*x^7 - 17709173952*x^5 - 123268*143883^(1/4)*sqrt(219)*sqrt(3)*sqrt(2)*(x^4 + 2*x^2 +
3)*sqrt(14395*sqrt(3) + 79497)*arctan(1/658350237832613766*sqrt(24746051)*143883^(3/4)*sqrt(219)*sqrt(11*14388
3^(1/4)*sqrt(219)*(127*sqrt(3)*x - 483*x)*sqrt(14395*sqrt(3) + 79497) + 222714459*x^2 + 222714459*sqrt(3))*(16
1*sqrt(3)*sqrt(2) - 127*sqrt(2))*sqrt(14395*sqrt(3) + 79497) - 1/8868084822*143883^(3/4)*sqrt(219)*(161*sqrt(3
)*sqrt(2)*x - 127*sqrt(2)*x)*sqrt(14395*sqrt(3) + 79497) + 1/2*sqrt(3)*sqrt(2) - 1/2*sqrt(2)) - 123268*143883^
(1/4)*sqrt(219)*sqrt(3)*sqrt(2)*(x^4 + 2*x^2 + 3)*sqrt(14395*sqrt(3) + 79497)*arctan(1/658350237832613766*sqrt
(24746051)*143883^(3/4)*sqrt(219)*sqrt(-11*143883^(1/4)*sqrt(219)*(127*sqrt(3)*x - 483*x)*sqrt(14395*sqrt(3) +
79497) + 222714459*x^2 + 222714459*sqrt(3))*(161*sqrt(3)*sqrt(2) - 127*sqrt(2))*sqrt(14395*sqrt(3) + 79497) -
1/8868084822*143883^(3/4)*sqrt(219)*(161*sqrt(3)*sqrt(2)*x - 127*sqrt(2)*x)*sqrt(14395*sqrt(3) + 79497) - 1/2
*sqrt(3)*sqrt(2) + 1/2*sqrt(2)) - 143883^(1/4)*sqrt(219)*(79497*x^4 + 158994*x^2 - 14395*sqrt(3)*(x^4 + 2*x^2
+ 3) + 238491)*sqrt(14395*sqrt(3) + 79497)*log(11*143883^(1/4)*sqrt(219)*(127*sqrt(3)*x - 483*x)*sqrt(14395*sq
rt(3) + 79497) + 222714459*x^2 + 222714459*sqrt(3)) + 143883^(1/4)*sqrt(219)*(79497*x^4 + 158994*x^2 - 14395*s
qrt(3)*(x^4 + 2*x^2 + 3) + 238491)*sqrt(14395*sqrt(3) + 79497)*log(-11*143883^(1/4)*sqrt(219)*(127*sqrt(3)*x -
483*x)*sqrt(14395*sqrt(3) + 79497) + 222714459*x^2 + 222714459*sqrt(3)) - 41627357064*x^3 - 78233515416*x)/(x
^4 + 2*x^2 + 3)

________________________________________________________________________________________

giac [B]  time = 1.85, size = 573, normalized size = 2.47 \[ \frac {5}{3} \, x^{3} - \frac {1}{20736} \, \sqrt {2} {\left (127 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2286 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 2286 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 127 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 17388 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 17388 \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 (127 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2286 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 2286 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 127 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 17388 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 17388 \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 (2286 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 127 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 127 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2286 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 17388 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 17388 \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 (2286 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 127 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 127 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2286 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 17388 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 17388 \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 ) - 17 \, x - \frac {25 \, {\left (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^4*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^2,x, algorithm="giac")

[Out]

5/3*x^3 - 1/20736*sqrt(2)*(127*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 2286*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) +
18)*(sqrt(3) - 3) - 2286*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 127*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 1
7388*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) + 17388*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)) - 1/20736*sqrt(2)*(127*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^
(3/2) + 2286*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 2286*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) +
18) + 127*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 17388*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) + 17388*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)*(2286*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 127*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2
) + 127*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 2286*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 17388*3^(1/4)*sqrt(
2)*sqrt(-6*sqrt(3) + 18) - 17388*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)*(2286*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 127*3^(3/4)*sqrt(2)*(
-6*sqrt(3) + 18)^(3/2) + 127*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 2286*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3)
- 17388*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) - 17388*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)) - 17*x - 25/8*(x^3 + 3*x)/(x^4 + 2*x^2 + 3)

________________________________________________________________________________________

maple [B]  time = 0.03, size = 416, normalized size = 1.79 \[ \frac {5 x^{3}}{3}-17 x -\frac {17 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{32 \sqrt {2+2 \sqrt {3}}}-\frac {89 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{16 \sqrt {2+2 \sqrt {3}}}+\frac {161 \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{8 \sqrt {2+2 \sqrt {3}}}-\frac {17 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{32 \sqrt {2+2 \sqrt {3}}}-\frac {89 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{16 \sqrt {2+2 \sqrt {3}}}+\frac {161 \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{8 \sqrt {2+2 \sqrt {3}}}-\frac {17 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{64}-\frac {89 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{32}+\frac {17 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{64}+\frac {89 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{32}+\frac {-\frac {25}{8} x^{3}-\frac {75}{8} x}{x^{4}+2 x^{2}+3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/3*x^3-17*x+(-25/8*x^3-75/8*x)/(x^4+2*x^2+3)-17/64*(-2+2*3^(1/2))^(1/2)*3^(1/2)*ln(x^2-(-2+2*3^(1/2))^(1/2)*x
+3^(1/2))-89/32*(-2+2*3^(1/2))^(1/2)*ln(x^2-(-2+2*3^(1/2))^(1/2)*x+3^(1/2))-17/32/(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))-89/16/(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))+161/8/(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))+17/64*(-2+2*3^(1/2))^(1/2)*3^(1/2)*ln(x^2+(-2+2*3^(1/2))^(1/2)*x+3^(1/2))+8
9/32*(-2+2*3^(1/2))^(1/2)*ln(x^2+(-2+2*3^(1/2))^(1/2)*x+3^(1/2))-17/32/(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))-89/16/(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))+161/8/(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))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {5}{3} \, x^{3} - 17 \, x - \frac {25 \, {\left (x^{3} + 3 \, x\right )}}{8 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac {1}{8} \, \int \frac {127 \, x^{2} + 483}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/3*x^3 - 17*x - 25/8*(x^3 + 3*x)/(x^4 + 2*x^2 + 3) + 1/8*integrate((127*x^2 + 483)/(x^4 + 2*x^2 + 3), x)

________________________________________________________________________________________

mupad [B]  time = 0.09, size = 162, normalized size = 0.70 \[ \frac {5\,x^3}{3}-\frac {\frac {25\,x^3}{8}+\frac {75\,x}{8}}{x^4+2\,x^2+3}-17\,x+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-14395-\sqrt {2}\,30817{}\mathrm {i}}\,30817{}\mathrm {i}}{64\,\left (-\frac {1571667}{64}+\frac {\sqrt {2}\,14884611{}\mathrm {i}}{128}\right )}-\frac {30817\,\sqrt {2}\,x\,\sqrt {-14395-\sqrt {2}\,30817{}\mathrm {i}}}{128\,\left (-\frac {1571667}{64}+\frac {\sqrt {2}\,14884611{}\mathrm {i}}{128}\right )}\right )\,\sqrt {-14395-\sqrt {2}\,30817{}\mathrm {i}}\,1{}\mathrm {i}}{16}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-14395+\sqrt {2}\,30817{}\mathrm {i}}\,30817{}\mathrm {i}}{64\,\left (\frac {1571667}{64}+\frac {\sqrt {2}\,14884611{}\mathrm {i}}{128}\right )}+\frac {30817\,\sqrt {2}\,x\,\sqrt {-14395+\sqrt {2}\,30817{}\mathrm {i}}}{128\,\left (\frac {1571667}{64}+\frac {\sqrt {2}\,14884611{}\mathrm {i}}{128}\right )}\right )\,\sqrt {-14395+\sqrt {2}\,30817{}\mathrm {i}}\,1{}\mathrm {i}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((x*(- 2^(1/2)*30817i - 14395)^(1/2)*30817i)/(64*((2^(1/2)*14884611i)/128 - 1571667/64)) - (30817*2^(1/2)
*x*(- 2^(1/2)*30817i - 14395)^(1/2))/(128*((2^(1/2)*14884611i)/128 - 1571667/64)))*(- 2^(1/2)*30817i - 14395)^
(1/2)*1i)/16 - ((75*x)/8 + (25*x^3)/8)/(2*x^2 + x^4 + 3) - 17*x - (atan((x*(2^(1/2)*30817i - 14395)^(1/2)*3081
7i)/(64*((2^(1/2)*14884611i)/128 + 1571667/64)) + (30817*2^(1/2)*x*(2^(1/2)*30817i - 14395)^(1/2))/(128*((2^(1
/2)*14884611i)/128 + 1571667/64)))*(2^(1/2)*30817i - 14395)^(1/2)*1i)/16 + (5*x^3)/3

________________________________________________________________________________________

sympy [A]  time = 0.61, size = 60, normalized size = 0.26 \[ \frac {5 x^{3}}{3} - 17 x + \frac {- 25 x^{3} - 75 x}{8 x^{4} + 16 x^{2} + 24} + \operatorname {RootSum} {\left (1048576 t^{4} + 29480960 t^{2} + 2106591003, \left (t \mapsto t \log {\left (\frac {557056 t^{3}}{816619683} + \frac {166600064 t}{816619683} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*x**3/3 - 17*x + (-25*x**3 - 75*x)/(8*x**4 + 16*x**2 + 24) + RootSum(1048576*_t**4 + 29480960*_t**2 + 2106591
003, Lambda(_t, _t*log(557056*_t**3/816619683 + 166600064*_t/816619683 + x)))

________________________________________________________________________________________