### 3.121 $$\int \frac{x^2 (4+x^2+3 x^4+5 x^6)}{(3+2 x^2+x^4)^3} \, dx$$

Optimal. Leaf size=246 $\frac{25 x \left (x^2+1\right )}{16 \left (x^4+2 x^2+3\right )^2}-\frac{x \left (88 x^2+353\right )}{192 \left (x^4+2 x^2+3\right )}-\frac{11 \sqrt{\frac{1}{3} \left (1825+1089 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{1536}+\frac{11 \sqrt{\frac{1}{3} \left (1825+1089 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{1536}-\frac{11}{768} \sqrt{\frac{1}{3} \left (1089 \sqrt{3}-1825\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{11}{768} \sqrt{\frac{1}{3} \left (1089 \sqrt{3}-1825\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )$

[Out]

(25*x*(1 + x^2))/(16*(3 + 2*x^2 + x^4)^2) - (x*(353 + 88*x^2))/(192*(3 + 2*x^2 + x^4)) - (11*Sqrt[(-1825 + 108
9*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/768 + (11*Sqrt[(-1825 + 1089*Sqrt[
3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/768 - (11*Sqrt[(1825 + 1089*Sqrt[3])/3]*L
og[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/1536 + (11*Sqrt[(1825 + 1089*Sqrt[3])/3]*Log[Sqrt[3] + Sqrt[2*(-
1 + Sqrt[3])]*x + x^2])/1536

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Rubi [A]  time = 0.284363, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 31, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.226, Rules used = {1668, 1678, 1169, 634, 618, 204, 628} $\frac{25 x \left (x^2+1\right )}{16 \left (x^4+2 x^2+3\right )^2}-\frac{x \left (88 x^2+353\right )}{192 \left (x^4+2 x^2+3\right )}-\frac{11 \sqrt{\frac{1}{3} \left (1825+1089 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{1536}+\frac{11 \sqrt{\frac{1}{3} \left (1825+1089 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{1536}-\frac{11}{768} \sqrt{\frac{1}{3} \left (1089 \sqrt{3}-1825\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{11}{768} \sqrt{\frac{1}{3} \left (1089 \sqrt{3}-1825\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(25*x*(1 + x^2))/(16*(3 + 2*x^2 + x^4)^2) - (x*(353 + 88*x^2))/(192*(3 + 2*x^2 + x^4)) - (11*Sqrt[(-1825 + 108
9*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/768 + (11*Sqrt[(-1825 + 1089*Sqrt[
3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/768 - (11*Sqrt[(1825 + 1089*Sqrt[3])/3]*L
og[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/1536 + (11*Sqrt[(1825 + 1089*Sqrt[3])/3]*Log[Sqrt[3] + Sqrt[2*(-
1 + Sqrt[3])]*x + x^2])/1536

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 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^2 \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 (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{1}{96} \int \frac{-150+78 x^2+480 x^4}{\left (3+2 x^2+x^4\right )^2} \, dx\\ &=\frac{25 x \left (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{6072-2112 x^2}{3+2 x^2+x^4} \, dx}{4608}\\ &=\frac{25 x \left (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{6072 \sqrt{2 \left (-1+\sqrt{3}\right )}-\left (6072+2112 \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{6072 \sqrt{2 \left (-1+\sqrt{3}\right )}+\left (6072+2112 \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 )}{16 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}-\frac{\left (11 \left (24-23 \sqrt{3}\right )\right ) \int \frac{1}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{2304}-\frac{\left (11 \left (24-23 \sqrt{3}\right )\right ) \int \frac{1}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{2304}-\frac{\left (11 \left (23+8 \sqrt{3}\right )\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}{768 \sqrt{6 \left (-1+\sqrt{3}\right )}}+\frac{\left (11 \left (23+8 \sqrt{3}\right )\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}{768 \sqrt{6 \left (-1+\sqrt{3}\right )}}\\ &=\frac{25 x \left (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}-\frac{11}{768} \sqrt{\frac{1825}{12}+\frac{363 \sqrt{3}}{4}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{11}{768} \sqrt{\frac{1825}{12}+\frac{363 \sqrt{3}}{4}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{\left (11 \left (24-23 \sqrt{3}\right )\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 )}{1152}+\frac{\left (11 \left (24-23 \sqrt{3}\right )\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 )}{1152}\\ &=\frac{25 x \left (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}-\frac{11}{768} \sqrt{\frac{1}{3} \left (-1825+1089 \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{11}{768} \sqrt{\frac{1}{3} \left (-1825+1089 \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{11}{768} \sqrt{\frac{1825}{12}+\frac{363 \sqrt{3}}{4}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{11}{768} \sqrt{\frac{1825}{12}+\frac{363 \sqrt{3}}{4}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )\\ \end{align*}

Mathematica [C]  time = 0.299272, size = 133, normalized size = 0.54 $\frac{1}{768} \left (-\frac{4 x \left (88 x^6+529 x^4+670 x^2+759\right )}{\left (x^4+2 x^2+3\right )^2}-\frac{11 i \left (31 \sqrt{2}-16 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{\sqrt{1-i \sqrt{2}}}+\frac{11 i \left (31 \sqrt{2}+16 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{\sqrt{1+i \sqrt{2}}}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((-4*x*(759 + 670*x^2 + 529*x^4 + 88*x^6))/(3 + 2*x^2 + x^4)^2 - ((11*I)*(-16*I + 31*Sqrt[2])*ArcTan[x/Sqrt[1
- I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + ((11*I)*(16*I + 31*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqr
t[2]])/768

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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{11\,{x}^{7}}{24}}-{\frac{529\,{x}^{5}}{192}}-{\frac{335\,{x}^{3}}{96}}-{\frac{253\,x}{64}} \right ) }-{\frac{517\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{9216}}-{\frac{341\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{3072}}-{\frac{ \left ( -1034+1034\,\sqrt{3} \right ) \sqrt{3}}{4608\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-682+682\,\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{253\,\sqrt{3}}{576\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{517\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{9216}}+{\frac{341\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{3072}}-{\frac{ \left ( -1034+1034\,\sqrt{3} \right ) \sqrt{3}}{4608\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-682+682\,\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{253\,\sqrt{3}}{576\,\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^2*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^3,x)

[Out]

(-11/24*x^7-529/192*x^5-335/96*x^3-253/64*x)/(x^4+2*x^2+3)^2-517/9216*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(
-2+2*3^(1/2))^(1/2)*3^(1/2)-341/3072*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)-517/4608/(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)-341/1536/(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))+253/576/(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)+517/9216*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2)
)*(-2+2*3^(1/2))^(1/2)*3^(1/2)+341/3072*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)-517/4608/(
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)-341/1536/(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))+253/576/(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{88 \, x^{7} + 529 \, x^{5} + 670 \, x^{3} + 759 \, x}{192 \,{\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} - \frac{11}{192} \, \int \frac{8 \, x^{2} - 23}{x^{4} + 2 \, x^{2} + 3}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*(88*x^7 + 529*x^5 + 670*x^3 + 759*x)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) - 11/192*integrate((8*x^2 - 23
)/(x^4 + 2*x^2 + 3), x)

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Fricas [B]  time = 1.73407, size = 2074, normalized size = 8.43 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27952128*(12811392*x^7 + 77013936*x^5 + 1348*sqrt(6)*3^(3/4)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)*sq
rt(-1987425*sqrt(3) + 3557763)*arctan(1/2226179538*sqrt(3707)*sqrt(6)*3^(3/4)*sqrt(sqrt(6)*3^(1/4)*(8*sqrt(3)*
x + 23*x)*sqrt(-1987425*sqrt(3) + 3557763) + 33363*x^2 + 33363*sqrt(3))*(23*sqrt(3)*sqrt(2) + 24*sqrt(2))*sqrt
(-1987425*sqrt(3) + 3557763) - 1/200178*sqrt(6)*3^(3/4)*(23*sqrt(3)*sqrt(2)*x + 24*sqrt(2)*x)*sqrt(-1987425*sq
rt(3) + 3557763) - 1/2*sqrt(3)*sqrt(2) + 1/2*sqrt(2)) + 1348*sqrt(6)*3^(3/4)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 1
2*x^2 + 9)*sqrt(-1987425*sqrt(3) + 3557763)*arctan(1/2226179538*sqrt(3707)*sqrt(6)*3^(3/4)*sqrt(-sqrt(6)*3^(1/
4)*(8*sqrt(3)*x + 23*x)*sqrt(-1987425*sqrt(3) + 3557763) + 33363*x^2 + 33363*sqrt(3))*(23*sqrt(3)*sqrt(2) + 24
*sqrt(2))*sqrt(-1987425*sqrt(3) + 3557763) - 1/200178*sqrt(6)*3^(3/4)*(23*sqrt(3)*sqrt(2)*x + 24*sqrt(2)*x)*sq
rt(-1987425*sqrt(3) + 3557763) + 1/2*sqrt(3)*sqrt(2) - 1/2*sqrt(2)) - sqrt(6)*3^(1/4)*(3267*x^8 + 13068*x^6 +
32670*x^4 + 39204*x^2 + 1825*sqrt(3)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) + 29403)*sqrt(-1987425*sqrt(3) + 3557
763)*log(sqrt(6)*3^(1/4)*(8*sqrt(3)*x + 23*x)*sqrt(-1987425*sqrt(3) + 3557763) + 33363*x^2 + 33363*sqrt(3)) +
sqrt(6)*3^(1/4)*(3267*x^8 + 13068*x^6 + 32670*x^4 + 39204*x^2 + 1825*sqrt(3)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 +
9) + 29403)*sqrt(-1987425*sqrt(3) + 3557763)*log(-sqrt(6)*3^(1/4)*(8*sqrt(3)*x + 23*x)*sqrt(-1987425*sqrt(3) +
3557763) + 33363*x^2 + 33363*sqrt(3)) + 97541280*x^3 + 110498256*x)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)

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Sympy [A]  time = 0.592378, size = 68, normalized size = 0.28 \begin{align*} - \frac{88 x^{7} + 529 x^{5} + 670 x^{3} + 759 x}{192 x^{8} + 768 x^{6} + 1920 x^{4} + 2304 x^{2} + 1728} + \operatorname{RootSum}{\left (463856467968 t^{4} - 57887948800 t^{2} + 1929229929, \left ( t \mapsto t \log{\left (\frac{14193524736 t^{3}}{54274187} - \frac{17989888 t}{1345641} + x \right )} \right )\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(88*x**7 + 529*x**5 + 670*x**3 + 759*x)/(192*x**8 + 768*x**6 + 1920*x**4 + 2304*x**2 + 1728) + RootSum(463856
467968*_t**4 - 57887948800*_t**2 + 1929229929, Lambda(_t, _t*log(14193524736*_t**3/54274187 - 17989888*_t/1345
641 + 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^{2}}{{\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^2*(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^2/(x^4 + 2*x^2 + 3)^3, x)