### 3.119 $$\int \frac{x^6 (4+x^2+3 x^4+5 x^6)}{(3+2 x^2+x^4)^3} \, dx$$

Optimal. Leaf size=235 $\frac{7 \left (58 x^2+11\right ) x}{64 \left (x^4+2 x^2+3\right )}+\frac{25 \left (3-x^2\right ) x}{16 \left (x^4+2 x^2+3\right )^2}-\frac{1}{512} \sqrt{1176531 \sqrt{3}-827621} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{1}{512} \sqrt{1176531 \sqrt{3}-827621} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+5 x+\frac{1}{256} \sqrt{827621+1176531 \sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{1}{256} \sqrt{827621+1176531 \sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )$

[Out]

5*x + (25*x*(3 - x^2))/(16*(3 + 2*x^2 + x^4)^2) + (7*x*(11 + 58*x^2))/(64*(3 + 2*x^2 + x^4)) + (Sqrt[827621 +
1176531*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 - (Sqrt[827621 + 1176531*Sq
rt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 - (Sqrt[-827621 + 1176531*Sqrt[3]]*Lo
g[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512 + (Sqrt[-827621 + 1176531*Sqrt[3]]*Log[Sqrt[3] + Sqrt[2*(-1 +
Sqrt[3])]*x + x^2])/512

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Rubi [A]  time = 0.300497, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 31, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.258, Rules used = {1668, 1678, 1676, 1169, 634, 618, 204, 628} $\frac{7 \left (58 x^2+11\right ) x}{64 \left (x^4+2 x^2+3\right )}+\frac{25 \left (3-x^2\right ) x}{16 \left (x^4+2 x^2+3\right )^2}-\frac{1}{512} \sqrt{1176531 \sqrt{3}-827621} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{1}{512} \sqrt{1176531 \sqrt{3}-827621} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+5 x+\frac{1}{256} \sqrt{827621+1176531 \sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{1}{256} \sqrt{827621+1176531 \sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

5*x + (25*x*(3 - x^2))/(16*(3 + 2*x^2 + x^4)^2) + (7*x*(11 + 58*x^2))/(64*(3 + 2*x^2 + x^4)) + (Sqrt[827621 +
1176531*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 - (Sqrt[827621 + 1176531*Sq
rt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 - (Sqrt[-827621 + 1176531*Sqrt[3]]*Lo
g[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512 + (Sqrt[-827621 + 1176531*Sqrt[3]]*Log[Sqrt[3] + Sqrt[2*(-1 +
Sqrt[3])]*x + x^2])/512

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 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 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

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{x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx &=\frac{25 x \left (3-x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{1}{96} \int \frac{-450+1650 x^2-672 x^6+480 x^8}{\left (3+2 x^2+x^4\right )^2} \, dx\\ &=\frac{25 x \left (3-x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{7 x \left (11+58 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{-12744-49104 x^2+23040 x^4}{3+2 x^2+x^4} \, dx}{4608}\\ &=\frac{25 x \left (3-x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{7 x \left (11+58 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{\int \left (23040-\frac{72 \left (1137+1322 x^2\right )}{3+2 x^2+x^4}\right ) \, dx}{4608}\\ &=5 x+\frac{25 x \left (3-x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{7 x \left (11+58 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac{1}{64} \int \frac{1137+1322 x^2}{3+2 x^2+x^4} \, dx\\ &=5 x+\frac{25 x \left (3-x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{7 x \left (11+58 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac{\int \frac{1137 \sqrt{2 \left (-1+\sqrt{3}\right )}-\left (1137-1322 \sqrt{3}\right ) x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{128 \sqrt{6 \left (-1+\sqrt{3}\right )}}-\frac{\int \frac{1137 \sqrt{2 \left (-1+\sqrt{3}\right )}+\left (1137-1322 \sqrt{3}\right ) x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{128 \sqrt{6 \left (-1+\sqrt{3}\right )}}\\ &=5 x+\frac{25 x \left (3-x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{7 x \left (11+58 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac{1}{256} \left (1322+379 \sqrt{3}\right ) \int \frac{1}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx-\frac{1}{256} \left (1322+379 \sqrt{3}\right ) \int \frac{1}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx-\frac{1}{512} \sqrt{-827621+1176531 \sqrt{3}} \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{-827621+1176531 \sqrt{3}} \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 (3-x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{7 x \left (11+58 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac{1}{512} \sqrt{-827621+1176531 \sqrt{3}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{1}{512} \sqrt{-827621+1176531 \sqrt{3}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )-\frac{1}{128} \left (-1322-379 \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}{128} \left (-1322-379 \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 (3-x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{7 x \left (11+58 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{1}{256} \sqrt{827621+1176531 \sqrt{3}} \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{827621+1176531 \sqrt{3}} \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{-827621+1176531 \sqrt{3}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{1}{512} \sqrt{-827621+1176531 \sqrt{3}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )\\ \end{align*}

Mathematica [C]  time = 0.330087, size = 138, normalized size = 0.59 $\frac{1}{256} \left (\frac{4 x \left (320 x^8+1686 x^6+4089 x^4+5112 x^2+3411\right )}{\left (x^4+2 x^2+3\right )^2}-\frac{i \left (185 \sqrt{2}-2644 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{\sqrt{1-i \sqrt{2}}}+\frac{i \left (185 \sqrt{2}+2644 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{\sqrt{1+i \sqrt{2}}}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((4*x*(3411 + 5112*x^2 + 4089*x^4 + 1686*x^6 + 320*x^8))/(3 + 2*x^2 + x^4)^2 - (I*(-2644*I + 185*Sqrt[2])*ArcT
an[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + (I*(2644*I + 185*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt
[1 + I*Sqrt[2]])/256

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Maple [B]  time = 0.022, size = 422, normalized size = 1.8 \begin{align*} 5\,x-{\frac{1}{ \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}} \left ( -{\frac{203\,{x}^{7}}{32}}-{\frac{889\,{x}^{5}}{64}}-{\frac{159\,{x}^{3}}{8}}-{\frac{531\,x}{64}} \right ) }-{\frac{943\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}-{\frac{185\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}-{\frac{ \left ( -1886+1886\,\sqrt{3} \right ) \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{-370+370\,\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{379\,\sqrt{3}}{64\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{943\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}+{\frac{185\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}-{\frac{ \left ( -1886+1886\,\sqrt{3} \right ) \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{-370+370\,\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{379\,\sqrt{3}}{64\,\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(x^6*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^3,x)

[Out]

5*x-(-203/32*x^7-889/64*x^5-159/8*x^3-531/64*x)/(x^4+2*x^2+3)^2-943/1024*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2)
)*(-2+2*3^(1/2))^(1/2)*3^(1/2)-185/1024*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)-943/512/(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))*3^(1/2)-185/512/(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))-379/64/(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)+943/1024*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2)
)*(-2+2*3^(1/2))^(1/2)*3^(1/2)+185/1024*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)-943/512/(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))*3^(1/2)-185/512/(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))-379/64/(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*} 5 \, x + \frac{406 \, x^{7} + 889 \, x^{5} + 1272 \, x^{3} + 531 \, x}{64 \,{\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} - \frac{1}{64} \, \int \frac{1322 \, x^{2} + 1137}{x^{4} + 2 \, x^{2} + 3}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*x + 1/64*(406*x^7 + 889*x^5 + 1272*x^3 + 531*x)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) - 1/64*integrate((1322*x
^2 + 1137)/(x^4 + 2*x^2 + 3), x)

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Fricas [B]  time = 1.77273, size = 2670, normalized size = 11.36 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4759173538071552*(23795867690357760*x^9 + 125374477893572448*x^7 + 304066571830852752*x^5 - 10534088*4152675
581883^(1/4)*sqrt(3)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)*sqrt(973721762751*sqrt(3) + 4152675581883)*arctan(1/8
471206900375217227324302495633*4152675581883^(3/4)*sqrt(516403378697)*sqrt(4647630408273*x^2 + 4152675581883^(
1/4)*(1322*sqrt(3)*sqrt(2)*x - 1137*sqrt(2)*x)*sqrt(973721762751*sqrt(3) + 4152675581883) + 4647630408273*sqrt
(3))*sqrt(973721762751*sqrt(3) + 4152675581883)*(379*sqrt(3) - 1322) - 1/5468081251875840963*4152675581883^(3/
4)*(379*sqrt(3)*x - 1322*x)*sqrt(973721762751*sqrt(3) + 4152675581883) + 1/2*sqrt(3)*sqrt(2) - 1/2*sqrt(2)) -
10534088*4152675581883^(1/4)*sqrt(3)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)*sqrt(973721762751*sqrt(3) + 415267558
1883)*arctan(1/8471206900375217227324302495633*4152675581883^(3/4)*sqrt(516403378697)*sqrt(4647630408273*x^2 -
4152675581883^(1/4)*(1322*sqrt(3)*sqrt(2)*x - 1137*sqrt(2)*x)*sqrt(973721762751*sqrt(3) + 4152675581883) + 46
47630408273*sqrt(3))*sqrt(973721762751*sqrt(3) + 4152675581883)*(379*sqrt(3) - 1322) - 1/5468081251875840963*4
152675581883^(3/4)*(379*sqrt(3)*x - 1322*x)*sqrt(973721762751*sqrt(3) + 4152675581883) - 1/2*sqrt(3)*sqrt(2) +
1/2*sqrt(2)) + 380138986353465216*x^3 - 4152675581883^(1/4)*(827621*sqrt(3)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 1
2*x^2 + 9) - 3529593*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9))*sqrt(973721762751*sqrt(3) + 4152675581883)*l
og(4647630408273*x^2 + 4152675581883^(1/4)*(1322*sqrt(3)*sqrt(2)*x - 1137*sqrt(2)*x)*sqrt(973721762751*sqrt(3)
+ 4152675581883) + 4647630408273*sqrt(3)) + 4152675581883^(1/4)*(827621*sqrt(3)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4
+ 12*x^2 + 9) - 3529593*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9))*sqrt(973721762751*sqrt(3) + 415267558188
3)*log(4647630408273*x^2 - 4152675581883^(1/4)*(1322*sqrt(3)*sqrt(2)*x - 1137*sqrt(2)*x)*sqrt(973721762751*sqr
t(3) + 4152675581883) + 4647630408273*sqrt(3)) + 253649077161907248*x)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)

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Sympy [A]  time = 0.586753, size = 71, normalized size = 0.3 \begin{align*} 5 x + \frac{406 x^{7} + 889 x^{5} + 1272 x^{3} + 531 x}{64 x^{8} + 256 x^{6} + 640 x^{4} + 768 x^{2} + 576} + \operatorname{RootSum}{\left (17179869184 t^{4} + 216955879424 t^{2} + 4152675581883, \left ( t \mapsto t \log{\left (- \frac{31641829376 t^{3}}{1549210136091} - \frac{455309168896 t}{1549210136091} + x \right )} \right )\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*x + (406*x**7 + 889*x**5 + 1272*x**3 + 531*x)/(64*x**8 + 256*x**6 + 640*x**4 + 768*x**2 + 576) + RootSum(171
79869184*_t**4 + 216955879424*_t**2 + 4152675581883, Lambda(_t, _t*log(-31641829376*_t**3/1549210136091 - 4553
09168896*_t/1549210136091 + x)))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x^{6} + 3 \, x^{4} + x^{2} + 4\right )} x^{6}}{{\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(x^6*(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^6/(x^4 + 2*x^2 + 3)^3, x)