### 3.114 $$\int \frac{4+x^2+3 x^4+5 x^6}{x^2 (3+2 x^2+x^4)^2} \, dx$$

Optimal. Leaf size=229 $-\frac{25 x \left (x^2+5\right )}{72 \left (x^4+2 x^2+3\right )}-\frac{1}{96} \sqrt{\frac{1}{6} \left (965+699 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{1}{96} \sqrt{\frac{1}{6} \left (965+699 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{4}{9 x}+\frac{1}{48} \sqrt{\frac{1}{6} \left (699 \sqrt{3}-965\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{1}{48} \sqrt{\frac{1}{6} \left (699 \sqrt{3}-965\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )$

[Out]

-4/(9*x) - (25*x*(5 + x^2))/(72*(3 + 2*x^2 + x^4)) + (Sqrt[(-965 + 699*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3
])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/48 - (Sqrt[(-965 + 699*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sq
rt[2*(1 + Sqrt[3])]])/48 - (Sqrt[(965 + 699*Sqrt[3])/6]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/96 + (S
qrt[(965 + 699*Sqrt[3])/6]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/96

________________________________________________________________________________________

Rubi [A]  time = 0.310033, antiderivative size = 229, 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 = {1669, 1664, 1169, 634, 618, 204, 628} $-\frac{25 x \left (x^2+5\right )}{72 \left (x^4+2 x^2+3\right )}-\frac{1}{96} \sqrt{\frac{1}{6} \left (965+699 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{1}{96} \sqrt{\frac{1}{6} \left (965+699 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{4}{9 x}+\frac{1}{48} \sqrt{\frac{1}{6} \left (699 \sqrt{3}-965\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{1}{48} \sqrt{\frac{1}{6} \left (699 \sqrt{3}-965\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-4/(9*x) - (25*x*(5 + x^2))/(72*(3 + 2*x^2 + x^4)) + (Sqrt[(-965 + 699*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3
])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/48 - (Sqrt[(-965 + 699*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sq
rt[2*(1 + Sqrt[3])]])/48 - (Sqrt[(965 + 699*Sqrt[3])/6]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/96 + (S
qrt[(965 + 699*Sqrt[3])/6]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/96

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^2 \left (3+2 x^2+x^4\right )^2} \, dx &=-\frac{25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}+\frac{1}{48} \int \frac{64+\frac{170 x^2}{3}-\frac{50 x^4}{3}}{x^2 \left (3+2 x^2+x^4\right )} \, dx\\ &=-\frac{25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}+\frac{1}{48} \int \left (\frac{64}{3 x^2}-\frac{2 \left (-7+19 x^2\right )}{3+2 x^2+x^4}\right ) \, dx\\ &=-\frac{4}{9 x}-\frac{25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}-\frac{1}{24} \int \frac{-7+19 x^2}{3+2 x^2+x^4} \, dx\\ &=-\frac{4}{9 x}-\frac{25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}-\frac{\int \frac{-7 \sqrt{2 \left (-1+\sqrt{3}\right )}-\left (-7-19 \sqrt{3}\right ) x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{48 \sqrt{6 \left (-1+\sqrt{3}\right )}}-\frac{\int \frac{-7 \sqrt{2 \left (-1+\sqrt{3}\right )}+\left (-7-19 \sqrt{3}\right ) x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{48 \sqrt{6 \left (-1+\sqrt{3}\right )}}\\ &=-\frac{4}{9 x}-\frac{25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}-\frac{1}{48} \sqrt{\frac{1}{6} \left (566-133 \sqrt{3}\right )} \int \frac{1}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx-\frac{1}{48} \sqrt{\frac{1}{6} \left (566-133 \sqrt{3}\right )} \int \frac{1}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx-\frac{1}{96} \sqrt{\frac{1}{6} \left (965+699 \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}{96} \sqrt{\frac{1}{6} \left (965+699 \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{4}{9 x}-\frac{25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}-\frac{1}{96} \sqrt{\frac{1}{6} \left (965+699 \sqrt{3}\right )} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{1}{96} \sqrt{\frac{1}{6} \left (965+699 \sqrt{3}\right )} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{1}{24} \sqrt{\frac{1}{6} \left (566-133 \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}{24} \sqrt{\frac{1}{6} \left (566-133 \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{4}{9 x}-\frac{25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}+\frac{1}{48} \sqrt{\frac{1}{6} \left (-965+699 \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}{48} \sqrt{\frac{1}{6} \left (-965+699 \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}{96} \sqrt{\frac{1}{6} \left (965+699 \sqrt{3}\right )} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{1}{96} \sqrt{\frac{1}{6} \left (965+699 \sqrt{3}\right )} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )\\ \end{align*}

Mathematica [C]  time = 0.184795, size = 126, normalized size = 0.55 $-\frac{25 x \left (x^2+5\right )}{72 \left (x^4+2 x^2+3\right )}-\frac{4}{9 x}-\frac{\left (19 \sqrt{2}+26 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{48 \sqrt{2-2 i \sqrt{2}}}-\frac{\left (19 \sqrt{2}-26 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{48 \sqrt{2+2 i \sqrt{2}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-4/(9*x) - (25*x*(5 + x^2))/(72*(3 + 2*x^2 + x^4)) - ((26*I + 19*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/(48*S
qrt[2 - (2*I)*Sqrt[2]]) - ((-26*I + 19*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(48*Sqrt[2 + (2*I)*Sqrt[2]])

________________________________________________________________________________________

Maple [B]  time = 0.023, size = 414, normalized size = 1.8 \begin{align*} -{\frac{1}{9\,{x}^{4}+18\,{x}^{2}+27} \left ({\frac{25\,{x}^{3}}{8}}+{\frac{125\,x}{8}} \right ) }-{\frac{\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{18}}-{\frac{13\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{192}}-{\frac{ \left ( -2+2\,\sqrt{3} \right ) \sqrt{3}}{9\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-26+26\,\sqrt{3}}{96\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{7\,\sqrt{3}}{72\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{18}}+{\frac{13\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{192}}-{\frac{ \left ( -2+2\,\sqrt{3} \right ) \sqrt{3}}{9\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-26+26\,\sqrt{3}}{96\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{7\,\sqrt{3}}{72\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{4}{9\,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^2/(x^4+2*x^2+3)^2,x)

[Out]

-1/9*(25/8*x^3+125/8*x)/(x^4+2*x^2+3)-1/18*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)
-13/192*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)-1/9/(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)-13/96/(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))+7/72/(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)+1/18*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)+13/192*ln(x^2
+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)-1/9/(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)-13/96/(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))+7/72/(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)-4/9/x

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{19 \, x^{4} + 63 \, x^{2} + 32}{24 \,{\left (x^{5} + 2 \, x^{3} + 3 \, x\right )}} - \frac{1}{24} \, \int \frac{19 \, x^{2} - 7}{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^2/(x^4+2*x^2+3)^2,x, algorithm="maxima")

[Out]

-1/24*(19*x^4 + 63*x^2 + 32)/(x^5 + 2*x^3 + 3*x) - 1/24*integrate((19*x^2 - 7)/(x^4 + 2*x^2 + 3), x)

________________________________________________________________________________________

Fricas [B]  time = 1.70574, size = 1871, normalized size = 8.17 \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^2/(x^4+2*x^2+3)^2,x, algorithm="fricas")

[Out]

-1/208156608*(164790648*x^4 - 2068*1465803^(1/4)*sqrt(2)*(x^5 + 2*x^3 + 3*x)*sqrt(-674535*sqrt(3) + 1465803)*a
rctan(1/547726639257666*1465803^(3/4)*sqrt(120461)*sqrt(1084149*x^2 + 1465803^(1/4)*(7*sqrt(3)*x + 57*x)*sqrt(
-674535*sqrt(3) + 1465803) + 1084149*sqrt(3))*(19*sqrt(3)*sqrt(2) + 7*sqrt(2))*sqrt(-674535*sqrt(3) + 1465803)
- 1/1515640302*1465803^(3/4)*(19*sqrt(3)*sqrt(2)*x + 7*sqrt(2)*x)*sqrt(-674535*sqrt(3) + 1465803) - 1/2*sqrt(
3)*sqrt(2) + 1/2*sqrt(2)) - 2068*1465803^(1/4)*sqrt(2)*(x^5 + 2*x^3 + 3*x)*sqrt(-674535*sqrt(3) + 1465803)*arc
tan(1/547726639257666*1465803^(3/4)*sqrt(120461)*sqrt(1084149*x^2 - 1465803^(1/4)*(7*sqrt(3)*x + 57*x)*sqrt(-6
74535*sqrt(3) + 1465803) + 1084149*sqrt(3))*(19*sqrt(3)*sqrt(2) + 7*sqrt(2))*sqrt(-674535*sqrt(3) + 1465803) -
1/1515640302*1465803^(3/4)*(19*sqrt(3)*sqrt(2)*x + 7*sqrt(2)*x)*sqrt(-674535*sqrt(3) + 1465803) + 1/2*sqrt(3)
*sqrt(2) - 1/2*sqrt(2)) - 1465803^(1/4)*(965*x^5 + 1930*x^3 + 699*sqrt(3)*(x^5 + 2*x^3 + 3*x) + 2895*x)*sqrt(-
674535*sqrt(3) + 1465803)*log(1084149*x^2 + 1465803^(1/4)*(7*sqrt(3)*x + 57*x)*sqrt(-674535*sqrt(3) + 1465803)
+ 1084149*sqrt(3)) + 1465803^(1/4)*(965*x^5 + 1930*x^3 + 699*sqrt(3)*(x^5 + 2*x^3 + 3*x) + 2895*x)*sqrt(-6745
35*sqrt(3) + 1465803)*log(1084149*x^2 - 1465803^(1/4)*(7*sqrt(3)*x + 57*x)*sqrt(-674535*sqrt(3) + 1465803) + 1
084149*sqrt(3)) + 546411096*x^2 + 277542144)/(x^5 + 2*x^3 + 3*x)

________________________________________________________________________________________

Sympy [A]  time = 0.550947, size = 53, normalized size = 0.23 \begin{align*} - \frac{19 x^{4} + 63 x^{2} + 32}{24 x^{5} + 48 x^{3} + 72 x} + \operatorname{RootSum}{\left (28311552 t^{4} - 1976320 t^{2} + 54289, \left ( t \mapsto t \log{\left (- \frac{28311552 t^{3}}{120461} + \frac{1103968 t}{120461} + x \right )} \right )\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(19*x**4 + 63*x**2 + 32)/(24*x**5 + 48*x**3 + 72*x) + RootSum(28311552*_t**4 - 1976320*_t**2 + 54289, Lambda(
_t, _t*log(-28311552*_t**3/120461 + 1103968*_t/120461 + x)))

________________________________________________________________________________________

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 )}^{2} x^{2}}\,{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^2/(x^4+2*x^2+3)^2,x, algorithm="giac")

[Out]

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