3.118 \(\int \frac{x^8 (4+x^2+3 x^4+5 x^6)}{(3+2 x^2+x^4)^3} \, dx\)

Optimal. Leaf size=242 \[ \frac{5 x^3}{3}-\frac{\left (835 x^2+1468\right ) x}{64 \left (x^4+2 x^2+3\right )}+\frac{25 \left (5 x^2+3\right ) x}{16 \left (x^4+2 x^2+3\right )^2}-\frac{21}{512} \sqrt{22721 \sqrt{3}-34271} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{21}{512} \sqrt{22721 \sqrt{3}-34271} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-27 x-\frac{21}{256} \sqrt{34271+22721 \sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{21}{256} \sqrt{34271+22721 \sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

[Out]

-27*x + (5*x^3)/3 + (25*x*(3 + 5*x^2))/(16*(3 + 2*x^2 + x^4)^2) - (x*(1468 + 835*x^2))/(64*(3 + 2*x^2 + x^4))
- (21*Sqrt[34271 + 22721*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 + (21*Sqrt
[34271 + 22721*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 - (21*Sqrt[-34271 +
22721*Sqrt[3]]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512 + (21*Sqrt[-34271 + 22721*Sqrt[3]]*Log[Sqrt[
3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512

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Rubi [A]  time = 0.309967, antiderivative size = 242, 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} \[ \frac{5 x^3}{3}-\frac{\left (835 x^2+1468\right ) x}{64 \left (x^4+2 x^2+3\right )}+\frac{25 \left (5 x^2+3\right ) x}{16 \left (x^4+2 x^2+3\right )^2}-\frac{21}{512} \sqrt{22721 \sqrt{3}-34271} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{21}{512} \sqrt{22721 \sqrt{3}-34271} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-27 x-\frac{21}{256} \sqrt{34271+22721 \sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{21}{256} \sqrt{34271+22721 \sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(x^8*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^3,x]

[Out]

-27*x + (5*x^3)/3 + (25*x*(3 + 5*x^2))/(16*(3 + 2*x^2 + x^4)^2) - (x*(1468 + 835*x^2))/(64*(3 + 2*x^2 + x^4))
- (21*Sqrt[34271 + 22721*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 + (21*Sqrt
[34271 + 22721*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 - (21*Sqrt[-34271 +
22721*Sqrt[3]]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512 + (21*Sqrt[-34271 + 22721*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^8 \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+5 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{1}{96} \int \frac{-450-1050 x^2+2400 x^4-672 x^8+480 x^{10}}{\left (3+2 x^2+x^4\right )^2} \, dx\\ &=\frac{25 x \left (3+5 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (1468+835 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{98496+27432 x^2-78336 x^4+23040 x^6}{3+2 x^2+x^4} \, dx}{4608}\\ &=\frac{25 x \left (3+5 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (1468+835 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{\int \left (-124416+23040 x^2+\frac{1512 \left (312+137 x^2\right )}{3+2 x^2+x^4}\right ) \, dx}{4608}\\ &=-27 x+\frac{5 x^3}{3}+\frac{25 x \left (3+5 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (1468+835 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{21}{64} \int \frac{312+137 x^2}{3+2 x^2+x^4} \, dx\\ &=-27 x+\frac{5 x^3}{3}+\frac{25 x \left (3+5 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (1468+835 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{1}{256} \left (7 \sqrt{3 \left (1+\sqrt{3}\right )}\right ) \int \frac{312 \sqrt{2 \left (-1+\sqrt{3}\right )}-\left (312-137 \sqrt{3}\right ) x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx+\frac{1}{256} \left (7 \sqrt{3 \left (1+\sqrt{3}\right )}\right ) \int \frac{312 \sqrt{2 \left (-1+\sqrt{3}\right )}+\left (312-137 \sqrt{3}\right ) x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx\\ &=-27 x+\frac{5 x^3}{3}+\frac{25 x \left (3+5 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (1468+835 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac{1}{512} \left (21 \sqrt{-34271+22721 \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} \left (21 \sqrt{-34271+22721 \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}{256} \left (21 \sqrt{51217+28496 \sqrt{3}}\right ) \int \frac{1}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx+\frac{1}{256} \left (21 \sqrt{51217+28496 \sqrt{3}}\right ) \int \frac{1}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx\\ &=-27 x+\frac{5 x^3}{3}+\frac{25 x \left (3+5 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (1468+835 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac{21}{512} \sqrt{-34271+22721 \sqrt{3}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{21}{512} \sqrt{-34271+22721 \sqrt{3}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )-\frac{1}{128} \left (21 \sqrt{51217+28496 \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 (21 \sqrt{51217+28496 \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 )\\ &=-27 x+\frac{5 x^3}{3}+\frac{25 x \left (3+5 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (1468+835 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac{21}{256} \sqrt{34271+22721 \sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{21}{256} \sqrt{34271+22721 \sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{21}{512} \sqrt{-34271+22721 \sqrt{3}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{21}{512} \sqrt{-34271+22721 \sqrt{3}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )\\ \end{align*}

Mathematica [C]  time = 0.210776, size = 155, normalized size = 0.64 \[ \frac{5 x^3}{3}-\frac{\left (835 x^2+1468\right ) x}{64 \left (x^4+2 x^2+3\right )}+\frac{25 \left (5 x^2+3\right ) x}{16 \left (x^4+2 x^2+3\right )^2}-27 x+\frac{21 \left (137 \sqrt{2}-175 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{128 \sqrt{2-2 i \sqrt{2}}}+\frac{21 \left (137 \sqrt{2}+175 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{128 \sqrt{2+2 i \sqrt{2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^8*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^3,x]

[Out]

-27*x + (5*x^3)/3 + (25*x*(3 + 5*x^2))/(16*(3 + 2*x^2 + x^4)^2) - (x*(1468 + 835*x^2))/(64*(3 + 2*x^2 + x^4))
+ (21*(-175*I + 137*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/(128*Sqrt[2 - (2*I)*Sqrt[2]]) + (21*(175*I + 137*S
qrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(128*Sqrt[2 + (2*I)*Sqrt[2]])

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Maple [B]  time = 0.022, size = 426, normalized size = 1.8 \begin{align*}{\frac{5\,{x}^{3}}{3}}-27\,x+{\frac{1}{ \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}} \left ( -{\frac{835\,{x}^{7}}{64}}-{\frac{1569\,{x}^{5}}{32}}-{\frac{4941\,{x}^{3}}{64}}-{\frac{513\,x}{8}} \right ) }+{\frac{693\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}-{\frac{3675\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}+{\frac{ \left ( -1386+1386\,\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{-7350+7350\,\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{273\,\sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{693\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}+{\frac{3675\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}+{\frac{ \left ( -1386+1386\,\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{-7350+7350\,\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{273\,\sqrt{3}}{8\,\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^8*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^3,x)

[Out]

5/3*x^3-27*x+(-835/64*x^7-1569/32*x^5-4941/64*x^3-513/8*x)/(x^4+2*x^2+3)^2+693/1024*ln(x^2+3^(1/2)-x*(-2+2*3^(
1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)-3675/1024*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2
)+693/512/(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)-36
75/512/(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))+273/8/(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)-693/1024*ln(x^2+3^(1/2)+x*(-2+2*3^
(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)+3675/1024*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/
2)+693/512/(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)-3
675/512/(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))+273/8/(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{5}{3} \, x^{3} - 27 \, x - \frac{835 \, x^{7} + 3138 \, x^{5} + 4941 \, x^{3} + 4104 \, x}{64 \,{\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} + \frac{21}{64} \, \int \frac{137 \, x^{2} + 312}{x^{4} + 2 \, x^{2} + 3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^3,x, algorithm="maxima")

[Out]

5/3*x^3 - 27*x - 1/64*(835*x^7 + 3138*x^5 + 4941*x^3 + 4104*x)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) + 21/64*int
egrate((137*x^2 + 312)/(x^4 + 2*x^2 + 3), x)

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Fricas [B]  time = 1.8097, size = 2421, normalized size = 10. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^3,x, algorithm="fricas")

[Out]

1/954779317248*(1591298862080*x^11 - 19413846117376*x^9 - 99660064046704*x^7 - 285508852710816*x^5 - 2298072*1
548731523^(1/4)*sqrt(3)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)*sqrt(778671391*sqrt(3) + 1548731523)*arctan(1/1975
3021371716480527209*1548731523^(3/4)*sqrt(932401677)*sqrt(932401677*x^2 + 1548731523^(1/4)*(137*sqrt(3)*sqrt(2
)*x - 312*sqrt(2)*x)*sqrt(778671391*sqrt(3) + 1548731523) + 932401677*sqrt(3))*sqrt(778671391*sqrt(3) + 154873
1523)*(104*sqrt(3) - 137) - 1/21185098503117*1548731523^(3/4)*(104*sqrt(3)*x - 137*x)*sqrt(778671391*sqrt(3) +
1548731523) + 1/2*sqrt(3)*sqrt(2) - 1/2*sqrt(2)) - 2298072*1548731523^(1/4)*sqrt(3)*(x^8 + 4*x^6 + 10*x^4 + 1
2*x^2 + 9)*sqrt(778671391*sqrt(3) + 1548731523)*arctan(1/19753021371716480527209*1548731523^(3/4)*sqrt(9324016
77)*sqrt(932401677*x^2 - 1548731523^(1/4)*(137*sqrt(3)*sqrt(2)*x - 312*sqrt(2)*x)*sqrt(778671391*sqrt(3) + 154
8731523) + 932401677*sqrt(3))*sqrt(778671391*sqrt(3) + 1548731523)*(104*sqrt(3) - 137) - 1/21185098503117*1548
731523^(3/4)*(104*sqrt(3)*x - 137*x)*sqrt(778671391*sqrt(3) + 1548731523) - 1/2*sqrt(3)*sqrt(2) + 1/2*sqrt(2))
- 368738756006544*x^3 + 21*1548731523^(1/4)*(34271*sqrt(3)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) - 6816
3*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9))*sqrt(778671391*sqrt(3) + 1548731523)*log(932401677*x^2 + 154873
1523^(1/4)*(137*sqrt(3)*sqrt(2)*x - 312*sqrt(2)*x)*sqrt(778671391*sqrt(3) + 1548731523) + 932401677*sqrt(3)) -
21*1548731523^(1/4)*(34271*sqrt(3)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) - 68163*sqrt(2)*(x^8 + 4*x^6 +
10*x^4 + 12*x^2 + 9))*sqrt(778671391*sqrt(3) + 1548731523)*log(932401677*x^2 - 1548731523^(1/4)*(137*sqrt(3)*
sqrt(2)*x - 312*sqrt(2)*x)*sqrt(778671391*sqrt(3) + 1548731523) + 932401677*sqrt(3)) - 293236597809792*x)/(x^8
+ 4*x^6 + 10*x^4 + 12*x^2 + 9)

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Sympy [A]  time = 0.606788, size = 80, normalized size = 0.33 \begin{align*} \frac{5 x^{3}}{3} - 27 x - \frac{835 x^{7} + 3138 x^{5} + 4941 x^{3} + 4104 x}{64 x^{8} + 256 x^{6} + 640 x^{4} + 768 x^{2} + 576} + 21 \operatorname{RootSum}{\left (17179869184 t^{4} + 8983937024 t^{2} + 1548731523, \left ( t \mapsto t \log{\left (- \frac{1107296256 t^{3}}{310800559} + \frac{438857984 t}{310800559} + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(5*x**6+3*x**4+x**2+4)/(x**4+2*x**2+3)**3,x)

[Out]

5*x**3/3 - 27*x - (835*x**7 + 3138*x**5 + 4941*x**3 + 4104*x)/(64*x**8 + 256*x**6 + 640*x**4 + 768*x**2 + 576)
+ 21*RootSum(17179869184*_t**4 + 8983937024*_t**2 + 1548731523, Lambda(_t, _t*log(-1107296256*_t**3/310800559
+ 438857984*_t/310800559 + 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^{8}}{{\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^8*(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^8/(x^4 + 2*x^2 + 3)^3, x)