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

Optimal. Leaf size=248 \[ \frac{25 x \left (1-x^2\right )}{48 \left (x^4+2 x^2+3\right )^2}+\frac{x \left (51 x^2+64\right )}{192 \left (x^4+2 x^2+3\right )}+\frac{1}{512} \sqrt{\frac{1}{3} \left (1291+1019 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{512} \sqrt{\frac{1}{3} \left (1291+1019 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{256} \sqrt{\frac{1}{3} \left (1019 \sqrt{3}-1291\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{256} \sqrt{\frac{1}{3} \left (1019 \sqrt{3}-1291\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

[Out]

(25*x*(1 - x^2))/(48*(3 + 2*x^2 + x^4)^2) + (x*(64 + 51*x^2))/(192*(3 + 2*x^2 + x^4)) - (Sqrt[(-1291 + 1019*Sq
rt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 + (Sqrt[(-1291 + 1019*Sqrt[3])/3]*
ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 + (Sqrt[(1291 + 1019*Sqrt[3])/3]*Log[Sqrt[3]
- Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512 - (Sqrt[(1291 + 1019*Sqrt[3])/3]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*
x + x^2])/512

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Rubi [A]  time = 0.254342, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1678, 1178, 1169, 634, 618, 204, 628} \[ \frac{25 x \left (1-x^2\right )}{48 \left (x^4+2 x^2+3\right )^2}+\frac{x \left (51 x^2+64\right )}{192 \left (x^4+2 x^2+3\right )}+\frac{1}{512} \sqrt{\frac{1}{3} \left (1291+1019 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{512} \sqrt{\frac{1}{3} \left (1291+1019 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{256} \sqrt{\frac{1}{3} \left (1019 \sqrt{3}-1291\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{256} \sqrt{\frac{1}{3} \left (1019 \sqrt{3}-1291\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(25*x*(1 - x^2))/(48*(3 + 2*x^2 + x^4)^2) + (x*(64 + 51*x^2))/(192*(3 + 2*x^2 + x^4)) - (Sqrt[(-1291 + 1019*Sq
rt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 + (Sqrt[(-1291 + 1019*Sqrt[3])/3]*
ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 + (Sqrt[(1291 + 1019*Sqrt[3])/3]*Log[Sqrt[3]
- Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512 - (Sqrt[(1291 + 1019*Sqrt[3])/3]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*
x + x^2])/512

Rule 1678

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Pq, a +
b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[(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)*PolynomialQuot
ient[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] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a*b*e - d*(b^2 - 2*
a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), 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] && LtQ[p, -1] && IntegerQ[2*p]

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}{\left (3+2 x^2+x^4\right )^3} \, dx &=\frac{25 x \left (1-x^2\right )}{48 \left (3+2 x^2+x^4\right )^2}+\frac{1}{96} \int \frac{78+230 x^2}{\left (3+2 x^2+x^4\right )^2} \, dx\\ &=\frac{25 x \left (1-x^2\right )}{48 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (64+51 x^2\right )}{192 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{-288+1224 x^2}{3+2 x^2+x^4} \, dx}{4608}\\ &=\frac{25 x \left (1-x^2\right )}{48 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (64+51 x^2\right )}{192 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{-288 \sqrt{2 \left (-1+\sqrt{3}\right )}-\left (-288-1224 \sqrt{3}\right ) x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{9216 \sqrt{6 \left (-1+\sqrt{3}\right )}}+\frac{\int \frac{-288 \sqrt{2 \left (-1+\sqrt{3}\right )}+\left (-288-1224 \sqrt{3}\right ) x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{9216 \sqrt{6 \left (-1+\sqrt{3}\right )}}\\ &=\frac{25 x \left (1-x^2\right )}{48 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (64+51 x^2\right )}{192 \left (3+2 x^2+x^4\right )}+\frac{1}{768} \left (51-4 \sqrt{3}\right ) \int \frac{1}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx+\frac{1}{768} \left (51-4 \sqrt{3}\right ) \int \frac{1}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx+\frac{1}{512} \sqrt{\frac{1}{3} \left (1291+1019 \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}{512} \sqrt{\frac{1}{3} \left (1291+1019 \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{25 x \left (1-x^2\right )}{48 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (64+51 x^2\right )}{192 \left (3+2 x^2+x^4\right )}+\frac{1}{512} \sqrt{\frac{1}{3} \left (1291+1019 \sqrt{3}\right )} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )-\frac{1}{512} \sqrt{\frac{1}{3} \left (1291+1019 \sqrt{3}\right )} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{1}{384} \left (-51+4 \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}{384} \left (-51+4 \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{25 x \left (1-x^2\right )}{48 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (64+51 x^2\right )}{192 \left (3+2 x^2+x^4\right )}-\frac{1}{256} \sqrt{\frac{1}{3} \left (-1291+1019 \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}{256} \sqrt{\frac{1}{3} \left (-1291+1019 \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}{512} \sqrt{\frac{1}{3} \left (1291+1019 \sqrt{3}\right )} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )-\frac{1}{512} \sqrt{\frac{1}{3} \left (1291+1019 \sqrt{3}\right )} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )\\ \end{align*}

Mathematica [C]  time = 0.302399, size = 129, normalized size = 0.52 \[ \frac{1}{768} \left (\frac{4 x \left (51 x^6+166 x^4+181 x^2+292\right )}{\left (x^4+2 x^2+3\right )^2}+\frac{3 \left (34+21 i \sqrt{2}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{\sqrt{1-i \sqrt{2}}}+\frac{3 \left (34-21 i \sqrt{2}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{\sqrt{1+i \sqrt{2}}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*x*(292 + 181*x^2 + 166*x^4 + 51*x^6))/(3 + 2*x^2 + x^4)^2 + (3*(34 + (21*I)*Sqrt[2])*ArcTan[x/Sqrt[1 - I*S
qrt[2]]])/Sqrt[1 - I*Sqrt[2]] + (3*(34 - (21*I)*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]])/7
68

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Maple [B]  time = 0.022, size = 418, normalized size = 1.7 \begin{align*}{\frac{1}{ \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}} \left ({\frac{17\,{x}^{7}}{64}}+{\frac{83\,{x}^{5}}{96}}+{\frac{181\,{x}^{3}}{192}}+{\frac{73\,x}{48}} \right ) }+{\frac{55\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{3072}}+{\frac{21\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}+{\frac{ \left ( -110+110\,\sqrt{3} \right ) \sqrt{3}}{1536\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-42+42\,\sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{\sqrt{3}}{48\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{55\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{3072}}-{\frac{21\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}+{\frac{ \left ( -110+110\,\sqrt{3} \right ) \sqrt{3}}{1536\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-42+42\,\sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{\sqrt{3}}{48\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) } \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+2*x^2+3)^3,x)

[Out]

(17/64*x^7+83/96*x^5+181/192*x^3+73/48*x)/(x^4+2*x^2+3)^2+55/3072*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2
*3^(1/2))^(1/2)*3^(1/2)+21/1024*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)+55/1536/(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)+21/512/(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))-1/48/(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)-55/3072*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1
/2))^(1/2)*3^(1/2)-21/1024*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)+55/1536/(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)+21/512/(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))-1/48/(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., size = 0, normalized size = 0. \begin{align*} \frac{51 \, x^{7} + 166 \, x^{5} + 181 \, x^{3} + 292 \, x}{192 \,{\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} + \frac{1}{64} \, \int \frac{17 \, x^{2} - 4}{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+2*x^2+3)^3,x, algorithm="maxima")

[Out]

1/192*(51*x^7 + 166*x^5 + 181*x^3 + 292*x)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) + 1/64*integrate((17*x^2 - 4)/(
x^4 + 2*x^2 + 3), x)

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Fricas [B]  time = 1.68028, size = 2261, normalized size = 9.12 \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+2*x^2+3)^3,x, algorithm="fricas")

[Out]

1/7991829504*(2122829712*x^7 + 6909602592*x^5 - 3404*3115083^(1/4)*sqrt(6)*sqrt(3)*sqrt(2)*(x^8 + 4*x^6 + 10*x
^4 + 12*x^2 + 9)*sqrt(-1315529*sqrt(3) + 3115083)*arctan(1/41378565634793586*3115083^(3/4)*sqrt(2601507)*sqrt(
6)*sqrt(3115083^(1/4)*sqrt(6)*(17*sqrt(3)*x + 4*x)*sqrt(-1315529*sqrt(3) + 3115083) + 2601507*x^2 + 2601507*sq
rt(3))*(4*sqrt(3)*sqrt(2) + 51*sqrt(2))*sqrt(-1315529*sqrt(3) + 3115083) - 1/15905613798*3115083^(3/4)*sqrt(6)
*(4*sqrt(3)*sqrt(2)*x + 51*sqrt(2)*x)*sqrt(-1315529*sqrt(3) + 3115083) - 1/2*sqrt(3)*sqrt(2) + 1/2*sqrt(2)) -
3404*3115083^(1/4)*sqrt(6)*sqrt(3)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)*sqrt(-1315529*sqrt(3) + 3115083
)*arctan(1/41378565634793586*3115083^(3/4)*sqrt(2601507)*sqrt(6)*sqrt(-3115083^(1/4)*sqrt(6)*(17*sqrt(3)*x + 4
*x)*sqrt(-1315529*sqrt(3) + 3115083) + 2601507*x^2 + 2601507*sqrt(3))*(4*sqrt(3)*sqrt(2) + 51*sqrt(2))*sqrt(-1
315529*sqrt(3) + 3115083) - 1/15905613798*3115083^(3/4)*sqrt(6)*(4*sqrt(3)*sqrt(2)*x + 51*sqrt(2)*x)*sqrt(-131
5529*sqrt(3) + 3115083) + 1/2*sqrt(3)*sqrt(2) - 1/2*sqrt(2)) - 3115083^(1/4)*sqrt(6)*(3057*x^8 + 12228*x^6 + 3
0570*x^4 + 36684*x^2 + 1291*sqrt(3)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) + 27513)*sqrt(-1315529*sqrt(3) + 31150
83)*log(3115083^(1/4)*sqrt(6)*(17*sqrt(3)*x + 4*x)*sqrt(-1315529*sqrt(3) + 3115083) + 2601507*x^2 + 2601507*sq
rt(3)) + 3115083^(1/4)*sqrt(6)*(3057*x^8 + 12228*x^6 + 30570*x^4 + 36684*x^2 + 1291*sqrt(3)*(x^8 + 4*x^6 + 10*
x^4 + 12*x^2 + 9) + 27513)*sqrt(-1315529*sqrt(3) + 3115083)*log(-3115083^(1/4)*sqrt(6)*(17*sqrt(3)*x + 4*x)*sq
rt(-1315529*sqrt(3) + 3115083) + 2601507*x^2 + 2601507*sqrt(3)) + 7533964272*x^3 + 12154240704*x)/(x^8 + 4*x^6
+ 10*x^4 + 12*x^2 + 9)

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Sympy [A]  time = 0.571005, size = 68, normalized size = 0.27 \begin{align*} \frac{51 x^{7} + 166 x^{5} + 181 x^{3} + 292 x}{192 x^{8} + 768 x^{6} + 1920 x^{4} + 2304 x^{2} + 1728} + \operatorname{RootSum}{\left (51539607552 t^{4} - 338427904 t^{2} + 1038361, \left ( t \mapsto t \log{\left (\frac{5536481280 t^{3}}{867169} - \frac{19920128 t}{867169} + 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**4+2*x**2+3)**3,x)

[Out]

(51*x**7 + 166*x**5 + 181*x**3 + 292*x)/(192*x**8 + 768*x**6 + 1920*x**4 + 2304*x**2 + 1728) + RootSum(5153960
7552*_t**4 - 338427904*_t**2 + 1038361, Lambda(_t, _t*log(5536481280*_t**3/867169 - 19920128*_t/867169 + x)))

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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}}\,{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+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)