### 3.124 $$\int \frac{4+x^2+3 x^4+5 x^6}{x^4 (3+2 x^2+x^4)^3} \, dx$$

Optimal. Leaf size=262 $\frac{25 x \left (5 x^2+7\right )}{432 \left (x^4+2 x^2+3\right )^2}+\frac{x \left (1025 x^2+1474\right )}{5184 \left (x^4+2 x^2+3\right )}-\frac{4}{81 x^3}+\frac{\sqrt{\frac{1}{3} \left (11240451 \sqrt{3}-10004741\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{41472}-\frac{\sqrt{\frac{1}{3} \left (11240451 \sqrt{3}-10004741\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{41472}+\frac{7}{27 x}-\frac{\sqrt{\frac{1}{3} \left (10004741+11240451 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{20736}+\frac{\sqrt{\frac{1}{3} \left (10004741+11240451 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{20736}$

[Out]

-4/(81*x^3) + 7/(27*x) + (25*x*(7 + 5*x^2))/(432*(3 + 2*x^2 + x^4)^2) + (x*(1474 + 1025*x^2))/(5184*(3 + 2*x^2
+ x^4)) - (Sqrt[(10004741 + 11240451*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]]
)/20736 + (Sqrt[(10004741 + 11240451*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])
/20736 + (Sqrt[(-10004741 + 11240451*Sqrt[3])/3]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/41472 - (Sqrt[
(-10004741 + 11240451*Sqrt[3])/3]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/41472

________________________________________________________________________________________

Rubi [A]  time = 0.365888, antiderivative size = 262, 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} $\frac{25 x \left (5 x^2+7\right )}{432 \left (x^4+2 x^2+3\right )^2}+\frac{x \left (1025 x^2+1474\right )}{5184 \left (x^4+2 x^2+3\right )}-\frac{4}{81 x^3}+\frac{\sqrt{\frac{1}{3} \left (11240451 \sqrt{3}-10004741\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{41472}-\frac{\sqrt{\frac{1}{3} \left (11240451 \sqrt{3}-10004741\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{41472}+\frac{7}{27 x}-\frac{\sqrt{\frac{1}{3} \left (10004741+11240451 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{20736}+\frac{\sqrt{\frac{1}{3} \left (10004741+11240451 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{20736}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-4/(81*x^3) + 7/(27*x) + (25*x*(7 + 5*x^2))/(432*(3 + 2*x^2 + x^4)^2) + (x*(1474 + 1025*x^2))/(5184*(3 + 2*x^2
+ x^4)) - (Sqrt[(10004741 + 11240451*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]]
)/20736 + (Sqrt[(10004741 + 11240451*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])
/20736 + (Sqrt[(-10004741 + 11240451*Sqrt[3])/3]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/41472 - (Sqrt[
(-10004741 + 11240451*Sqrt[3])/3]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/41472

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^4 \left (3+2 x^2+x^4\right )^3} \, dx &=\frac{25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac{1}{96} \int \frac{128-\frac{160 x^2}{3}+50 x^4+\frac{1250 x^6}{9}}{x^4 \left (3+2 x^2+x^4\right )^2} \, dx\\ &=\frac{25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{2048-\frac{6656 x^2}{3}+\frac{2576 x^4}{9}+\frac{8200 x^6}{9}}{x^4 \left (3+2 x^2+x^4\right )} \, dx}{4608}\\ &=\frac{25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}+\frac{\int \left (\frac{2048}{3 x^4}-\frac{3584}{3 x^2}+\frac{8 \left (2242+2369 x^2\right )}{9 \left (3+2 x^2+x^4\right )}\right ) \, dx}{4608}\\ &=-\frac{4}{81 x^3}+\frac{7}{27 x}+\frac{25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{2242+2369 x^2}{3+2 x^2+x^4} \, dx}{5184}\\ &=-\frac{4}{81 x^3}+\frac{7}{27 x}+\frac{25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{2242 \sqrt{2 \left (-1+\sqrt{3}\right )}-\left (2242-2369 \sqrt{3}\right ) x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{10368 \sqrt{6 \left (-1+\sqrt{3}\right )}}+\frac{\int \frac{2242 \sqrt{2 \left (-1+\sqrt{3}\right )}+\left (2242-2369 \sqrt{3}\right ) x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{10368 \sqrt{6 \left (-1+\sqrt{3}\right )}}\\ &=-\frac{4}{81 x^3}+\frac{7}{27 x}+\frac{25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}+\frac{\left (2242-2369 \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}{20736 \sqrt{6 \left (-1+\sqrt{3}\right )}}+\frac{\left (7107+2242 \sqrt{3}\right ) \int \frac{1}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{62208}+\frac{\left (7107+2242 \sqrt{3}\right ) \int \frac{1}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{62208}+\frac{\left (-2242+2369 \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}{20736 \sqrt{6 \left (-1+\sqrt{3}\right )}}\\ &=-\frac{4}{81 x^3}+\frac{7}{27 x}+\frac{25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}+\frac{\sqrt{-\frac{10004741}{12}+\frac{3746817 \sqrt{3}}{4}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )}{20736}-\frac{\sqrt{-\frac{10004741}{12}+\frac{3746817 \sqrt{3}}{4}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )}{20736}-\frac{\left (7107+2242 \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 )}{31104}-\frac{\left (7107+2242 \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 )}{31104}\\ &=-\frac{4}{81 x^3}+\frac{7}{27 x}+\frac{25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}-\frac{\sqrt{\frac{1}{3} \left (10004741+11240451 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{20736}+\frac{\sqrt{\frac{1}{3} \left (10004741+11240451 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{20736}+\frac{\sqrt{-\frac{10004741}{12}+\frac{3746817 \sqrt{3}}{4}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )}{20736}-\frac{\sqrt{-\frac{10004741}{12}+\frac{3746817 \sqrt{3}}{4}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )}{20736}\\ \end{align*}

Mathematica [C]  time = 0.331905, size = 139, normalized size = 0.53 $\frac{\frac{4 \left (2369 x^{10}+8644 x^8+19939 x^6+20090 x^4+9024 x^2-2304\right )}{x^3 \left (x^4+2 x^2+3\right )^2}+\frac{\left (4738+127 i \sqrt{2}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{\sqrt{1-i \sqrt{2}}}+\frac{\left (4738-127 i \sqrt{2}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{\sqrt{1+i \sqrt{2}}}}{20736}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((4*(-2304 + 9024*x^2 + 20090*x^4 + 19939*x^6 + 8644*x^8 + 2369*x^10))/(x^3*(3 + 2*x^2 + x^4)^2) + ((4738 + (1
27*I)*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + ((4738 - (127*I)*Sqrt[2])*ArcTan[x/Sqrt[1
+ I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]])/20736

________________________________________________________________________________________

Maple [B]  time = 0.024, size = 429, normalized size = 1.6 \begin{align*}{\frac{1}{27\, \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}} \left ({\frac{1025\,{x}^{7}}{192}}+{\frac{881\,{x}^{5}}{48}}+{\frac{7523\,{x}^{3}}{192}}+{\frac{1087\,x}{32}} \right ) }+{\frac{4865\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{248832}}+{\frac{127\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{82944}}+{\frac{ \left ( -9730+9730\,\sqrt{3} \right ) \sqrt{3}}{124416\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-254+254\,\sqrt{3}}{41472\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{1121\,\sqrt{3}}{7776\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{4865\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{248832}}-{\frac{127\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{82944}}+{\frac{ \left ( -9730+9730\,\sqrt{3} \right ) \sqrt{3}}{124416\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-254+254\,\sqrt{3}}{41472\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{1121\,\sqrt{3}}{7776\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{4}{81\,{x}^{3}}}+{\frac{7}{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^4/(x^4+2*x^2+3)^3,x)

[Out]

1/27*(1025/192*x^7+881/48*x^5+7523/192*x^3+1087/32*x)/(x^4+2*x^2+3)^2+4865/248832*ln(x^2+3^(1/2)-x*(-2+2*3^(1/
2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)+127/82944*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)+
4865/124416/(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)+
127/41472/(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))+1121/7776/
(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)-4865/248832*ln(x^2+3^(1/2)+
x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)-127/82944*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^
(1/2))^(1/2)+4865/124416/(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)+127/41472/(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
))+1121/7776/(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)-4/81/x^3+7/27/
x

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2369 \, x^{10} + 8644 \, x^{8} + 19939 \, x^{6} + 20090 \, x^{4} + 9024 \, x^{2} - 2304}{5184 \,{\left (x^{11} + 4 \, x^{9} + 10 \, x^{7} + 12 \, x^{5} + 9 \, x^{3}\right )}} + \frac{1}{5184} \, \int \frac{2369 \, x^{2} + 2242}{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^4/(x^4+2*x^2+3)^3,x, algorithm="maxima")

[Out]

1/5184*(2369*x^10 + 8644*x^8 + 19939*x^6 + 20090*x^4 + 9024*x^2 - 2304)/(x^11 + 4*x^9 + 10*x^7 + 12*x^5 + 9*x^
3) + 1/5184*integrate((2369*x^2 + 2242)/(x^4 + 2*x^2 + 3), x)

________________________________________________________________________________________

Fricas [B]  time = 1.74976, size = 2952, normalized size = 11.27 \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^4/(x^4+2*x^2+3)^3,x, algorithm="fricas")

[Out]

1/135934787413472256*(62119890312985296*x^10 + 226662866975704896*x^8 + 522840224968600176*x^6 + 47239676*7132
36683^(1/4)*sqrt(15419)*sqrt(6)*sqrt(3)*sqrt(2)*(x^11 + 4*x^9 + 10*x^7 + 12*x^5 + 9*x^3)*sqrt(10004741*sqrt(3)
+ 33721353)*arctan(1/27609352591972558367520653346*sqrt(182097141061)*713236683^(3/4)*sqrt(15419)*sqrt(6)*sqr
t(3)*sqrt(713236683^(1/4)*sqrt(15419)*sqrt(6)*(2369*sqrt(3)*x - 2242*x)*sqrt(10004741*sqrt(3) + 33721353) + 54
6291423183*x^2 + 546291423183*sqrt(3))*(2242*sqrt(3)*sqrt(2) - 7107*sqrt(2))*sqrt(10004741*sqrt(3) + 33721353)
- 1/50539604724352062*713236683^(3/4)*sqrt(15419)*sqrt(6)*(2242*sqrt(3)*sqrt(2)*x - 7107*sqrt(2)*x)*sqrt(1000
4741*sqrt(3) + 33721353) + 1/2*sqrt(3)*sqrt(2) - 1/2*sqrt(2)) + 47239676*713236683^(1/4)*sqrt(15419)*sqrt(6)*s
qrt(3)*sqrt(2)*(x^11 + 4*x^9 + 10*x^7 + 12*x^5 + 9*x^3)*sqrt(10004741*sqrt(3) + 33721353)*arctan(1/82828057775
917675102561960038*sqrt(182097141061)*713236683^(3/4)*sqrt(15419)*sqrt(6)*sqrt(-27*713236683^(1/4)*sqrt(15419)
*sqrt(6)*(2369*sqrt(3)*x - 2242*x)*sqrt(10004741*sqrt(3) + 33721353) + 14749868425941*x^2 + 14749868425941*sqr
t(3))*(2242*sqrt(3)*sqrt(2) - 7107*sqrt(2))*sqrt(10004741*sqrt(3) + 33721353) - 1/50539604724352062*713236683^
(3/4)*sqrt(15419)*sqrt(6)*(2242*sqrt(3)*sqrt(2)*x - 7107*sqrt(2)*x)*sqrt(10004741*sqrt(3) + 33721353) - 1/2*sq
rt(3)*sqrt(2) + 1/2*sqrt(2)) + 526799745203830560*x^4 - 713236683^(1/4)*sqrt(15419)*sqrt(6)*(33721353*x^11 + 1
34885412*x^9 + 337213530*x^7 + 404656236*x^5 + 303492177*x^3 - 10004741*sqrt(3)*(x^11 + 4*x^9 + 10*x^7 + 12*x^
5 + 9*x^3))*sqrt(10004741*sqrt(3) + 33721353)*log(27*713236683^(1/4)*sqrt(15419)*sqrt(6)*(2369*sqrt(3)*x - 224
2*x)*sqrt(10004741*sqrt(3) + 33721353) + 14749868425941*x^2 + 14749868425941*sqrt(3)) + 713236683^(1/4)*sqrt(1
5419)*sqrt(6)*(33721353*x^11 + 134885412*x^9 + 337213530*x^7 + 404656236*x^5 + 303492177*x^3 - 10004741*sqrt(3
)*(x^11 + 4*x^9 + 10*x^7 + 12*x^5 + 9*x^3))*sqrt(10004741*sqrt(3) + 33721353)*log(-27*713236683^(1/4)*sqrt(154
19)*sqrt(6)*(2369*sqrt(3)*x - 2242*x)*sqrt(10004741*sqrt(3) + 33721353) + 14749868425941*x^2 + 14749868425941*
sqrt(3)) + 236627222534562816*x^2 - 60415461072654336)/(x^11 + 4*x^9 + 10*x^7 + 12*x^5 + 9*x^3)

________________________________________________________________________________________

Sympy [A]  time = 0.6269, size = 80, normalized size = 0.31 \begin{align*} \operatorname{RootSum}{\left (338151365148672 t^{4} + 2622682824704 t^{2} + 19257390441, \left ( t \mapsto t \log{\left (\frac{357010935644160 t^{3}}{182097141061} + \frac{26016957890816 t}{1638874269549} + x \right )} \right )\right )} + \frac{2369 x^{10} + 8644 x^{8} + 19939 x^{6} + 20090 x^{4} + 9024 x^{2} - 2304}{5184 x^{11} + 20736 x^{9} + 51840 x^{7} + 62208 x^{5} + 46656 x^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(338151365148672*_t**4 + 2622682824704*_t**2 + 19257390441, Lambda(_t, _t*log(357010935644160*_t**3/182
097141061 + 26016957890816*_t/1638874269549 + x))) + (2369*x**10 + 8644*x**8 + 19939*x**6 + 20090*x**4 + 9024*
x**2 - 2304)/(5184*x**11 + 20736*x**9 + 51840*x**7 + 62208*x**5 + 46656*x**3)

________________________________________________________________________________________

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^{4}}\,{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^4/(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^4), x)