### 3.117 $$\int \frac{x^{10} (4+x^2+3 x^4+5 x^6)}{(3+2 x^2+x^4)^3} \, dx$$

Optimal. Leaf size=243 $x^5-9 x^3+\frac{\left (252 x^2+3305\right ) x}{64 \left (x^4+2 x^2+3\right )}-\frac{25 \left (7 x^2+15\right ) x}{16 \left (x^4+2 x^2+3\right )^2}+\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+58 x+\frac{3}{256} \sqrt{7678611 \sqrt{3}-8595619} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{3}{256} \sqrt{7678611 \sqrt{3}-8595619} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )$

[Out]

58*x - 9*x^3 + x^5 - (25*x*(15 + 7*x^2))/(16*(3 + 2*x^2 + x^4)^2) + (x*(3305 + 252*x^2))/(64*(3 + 2*x^2 + x^4)
) + (3*Sqrt[-8595619 + 7678611*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 - (3
*Sqrt[-8595619 + 7678611*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 + (3*Sqrt[
8595619 + 7678611*Sqrt[3]]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512 - (3*Sqrt[8595619 + 7678611*Sqrt
[3]]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512

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Rubi [A]  time = 0.359897, antiderivative size = 243, 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} $x^5-9 x^3+\frac{\left (252 x^2+3305\right ) x}{64 \left (x^4+2 x^2+3\right )}-\frac{25 \left (7 x^2+15\right ) x}{16 \left (x^4+2 x^2+3\right )^2}+\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+58 x+\frac{3}{256} \sqrt{7678611 \sqrt{3}-8595619} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{3}{256} \sqrt{7678611 \sqrt{3}-8595619} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(x^10*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^3,x]

[Out]

58*x - 9*x^3 + x^5 - (25*x*(15 + 7*x^2))/(16*(3 + 2*x^2 + x^4)^2) + (x*(3305 + 252*x^2))/(64*(3 + 2*x^2 + x^4)
) + (3*Sqrt[-8595619 + 7678611*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 - (3
*Sqrt[-8595619 + 7678611*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 + (3*Sqrt[
8595619 + 7678611*Sqrt[3]]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512 - (3*Sqrt[8595619 + 7678611*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^{10} \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 (15+7 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{1}{96} \int \frac{2250-2850 x^2-4800 x^4+2400 x^6-672 x^{10}+480 x^{12}}{\left (3+2 x^2+x^4\right )^2} \, dx\\ &=-\frac{25 x \left (15+7 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (3305+252 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{-201960+193248 x^2+87552 x^4-78336 x^6+23040 x^8}{3+2 x^2+x^4} \, dx}{4608}\\ &=-\frac{25 x \left (15+7 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (3305+252 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{\int \left (267264-124416 x^2+23040 x^4-\frac{216 \left (4647-148 x^2\right )}{3+2 x^2+x^4}\right ) \, dx}{4608}\\ &=58 x-9 x^3+x^5-\frac{25 x \left (15+7 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (3305+252 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac{3}{64} \int \frac{4647-148 x^2}{3+2 x^2+x^4} \, dx\\ &=58 x-9 x^3+x^5-\frac{25 x \left (15+7 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (3305+252 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac{1}{256} \sqrt{3 \left (1+\sqrt{3}\right )} \int \frac{4647 \sqrt{2 \left (-1+\sqrt{3}\right )}-\left (4647+148 \sqrt{3}\right ) x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx-\frac{1}{256} \sqrt{3 \left (1+\sqrt{3}\right )} \int \frac{4647 \sqrt{2 \left (-1+\sqrt{3}\right )}+\left (4647+148 \sqrt{3}\right ) x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx\\ &=58 x-9 x^3+x^5-\frac{25 x \left (15+7 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (3305+252 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac{1}{256} \left (3 \sqrt{7220107-458504 \sqrt{3}}\right ) \int \frac{1}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx-\frac{1}{256} \left (3 \sqrt{7220107-458504 \sqrt{3}}\right ) \int \frac{1}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx+\frac{1}{512} \left (3 \sqrt{8595619+7678611 \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 (3 \sqrt{8595619+7678611 \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\\ &=58 x-9 x^3+x^5-\frac{25 x \left (15+7 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (3305+252 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )-\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{1}{128} \left (3 \sqrt{7220107-458504 \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 (3 \sqrt{7220107-458504 \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 )\\ &=58 x-9 x^3+x^5-\frac{25 x \left (15+7 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (3305+252 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{3}{256} \sqrt{-8595619+7678611 \sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{3}{256} \sqrt{-8595619+7678611 \sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )-\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )\\ \end{align*}

Mathematica [C]  time = 0.222089, size = 156, normalized size = 0.64 $x^5-9 x^3+\frac{\left (252 x^2+3305\right ) x}{64 \left (x^4+2 x^2+3\right )}-\frac{25 \left (7 x^2+15\right ) x}{16 \left (x^4+2 x^2+3\right )^2}+58 x+\frac{3 \left (148 \sqrt{2}+4795 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{128 \sqrt{2-2 i \sqrt{2}}}+\frac{3 \left (148 \sqrt{2}-4795 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{128 \sqrt{2+2 i \sqrt{2}}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(x^10*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^3,x]

[Out]

58*x - 9*x^3 + x^5 - (25*x*(15 + 7*x^2))/(16*(3 + 2*x^2 + x^4)^2) + (x*(3305 + 252*x^2))/(64*(3 + 2*x^2 + x^4)
) + (3*(4795*I + 148*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/(128*Sqrt[2 - (2*I)*Sqrt[2]]) + (3*(-4795*I + 148
*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(128*Sqrt[2 + (2*I)*Sqrt[2]])

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Maple [B]  time = 0.02, size = 429, normalized size = 1.8 \begin{align*}{x}^{5}-9\,{x}^{3}+58\,x+{\frac{1}{ \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}} \left ({\frac{63\,{x}^{7}}{16}}+{\frac{3809\,{x}^{5}}{64}}+{\frac{3333\,{x}^{3}}{32}}+{\frac{8415\,x}{64}} \right ) }+{\frac{5091\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}+{\frac{14385\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}+{\frac{ \left ( -10182+10182\,\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{-28770+28770\,\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{4647\,\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{5091\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}-{\frac{14385\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}+{\frac{ \left ( -10182+10182\,\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{-28770+28770\,\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{4647\,\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^10*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^3,x)

[Out]

x^5-9*x^3+58*x+(63/16*x^7+3809/64*x^5+3333/32*x^3+8415/64*x)/(x^4+2*x^2+3)^2+5091/1024*ln(x^2+3^(1/2)-x*(-2+2*
3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)+14385/1024*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^
(1/2)+5091/512/(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)+14385/512/(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))-4647/64
/(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)-5091/1024*ln(x^2+3^(1/2)+x
*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)-14385/1024*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^
(1/2))^(1/2)+5091/512/(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)+14385/512/(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))-
4647/64/(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*} x^{5} - 9 \, x^{3} + 58 \, x + \frac{252 \, x^{7} + 3809 \, x^{5} + 6666 \, x^{3} + 8415 \, x}{64 \,{\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} + \frac{3}{64} \, \int \frac{148 \, x^{2} - 4647}{x^{4} + 2 \, x^{2} + 3}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^3,x, algorithm="maxima")

[Out]

x^5 - 9*x^3 + 58*x + 1/64*(252*x^7 + 3809*x^5 + 6666*x^3 + 8415*x)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) + 3/64*
integrate((148*x^2 - 4647)/(x^4 + 2*x^2 + 3), x)

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Fricas [B]  time = 1.7942, size = 2853, normalized size = 11.74 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^3,x, algorithm="fricas")

[Out]

1/18808834881088512*(18808834881088512*x^13 - 94044174405442560*x^11 + 601882716194832384*x^9 + 29706203590319
16864*x^7 + 10166469141273357744*x^5 + 57410392*2183743218123^(1/4)*sqrt(3)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9
)*sqrt(-66002414605209*sqrt(3) + 176883200667963)*arctan(1/863545621466021963404537403089353*sqrt(612266760452
1)*2183743218123^(3/4)*sqrt(55104008440689*x^2 + 2183743218123^(1/4)*(148*sqrt(3)*sqrt(2)*x + 4647*sqrt(2)*x)*
sqrt(-66002414605209*sqrt(3) + 176883200667963) + 55104008440689*sqrt(3))*(1549*sqrt(3) + 148)*sqrt(-660024146
05209*sqrt(3) + 176883200667963) - 1/47013582817418600331*2183743218123^(3/4)*(1549*sqrt(3)*x + 148*x)*sqrt(-6
6002414605209*sqrt(3) + 176883200667963) - 1/2*sqrt(3)*sqrt(2) + 1/2*sqrt(2)) + 57410392*2183743218123^(1/4)*s
qrt(3)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)*sqrt(-66002414605209*sqrt(3) + 176883200667963)*arctan(1/8635456214
66021963404537403089353*sqrt(6122667604521)*2183743218123^(3/4)*sqrt(55104008440689*x^2 - 2183743218123^(1/4)*
(148*sqrt(3)*sqrt(2)*x + 4647*sqrt(2)*x)*sqrt(-66002414605209*sqrt(3) + 176883200667963) + 55104008440689*sqrt
(3))*(1549*sqrt(3) + 148)*sqrt(-66002414605209*sqrt(3) + 176883200667963) - 1/47013582817418600331*21837432181
23^(3/4)*(1549*sqrt(3)*x + 148*x)*sqrt(-66002414605209*sqrt(3) + 176883200667963) + 1/2*sqrt(3)*sqrt(2) - 1/2*
sqrt(2)) + 13526491159952810208*x^3 - 2183743218123^(1/4)*(8595619*sqrt(3)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*
x^2 + 9) + 23035833*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9))*sqrt(-66002414605209*sqrt(3) + 17688320066796
3)*log(55104008440689*x^2 + 2183743218123^(1/4)*(148*sqrt(3)*sqrt(2)*x + 4647*sqrt(2)*x)*sqrt(-66002414605209*
sqrt(3) + 176883200667963) + 55104008440689*sqrt(3)) + 2183743218123^(1/4)*(8595619*sqrt(3)*sqrt(2)*(x^8 + 4*x
^6 + 10*x^4 + 12*x^2 + 9) + 23035833*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9))*sqrt(-66002414605209*sqrt(3)
+ 176883200667963)*log(55104008440689*x^2 - 2183743218123^(1/4)*(148*sqrt(3)*sqrt(2)*x + 4647*sqrt(2)*x)*sqrt
(-66002414605209*sqrt(3) + 176883200667963) + 55104008440689*sqrt(3)) + 12291279706746325584*x)/(x^8 + 4*x^6 +
10*x^4 + 12*x^2 + 9)

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Sympy [A]  time = 0.591164, size = 82, normalized size = 0.34 \begin{align*} x^{5} - 9 x^{3} + 58 x + \frac{252 x^{7} + 3809 x^{5} + 6666 x^{3} + 8415 x}{64 x^{8} + 256 x^{6} + 640 x^{4} + 768 x^{2} + 576} + 3 \operatorname{RootSum}{\left (17179869184 t^{4} - 2253289947136 t^{2} + 176883200667963, \left ( t \mapsto t \log{\left (- \frac{56941871104 t^{3}}{55104008440689} - \frac{1957224667904 t}{55104008440689} + x \right )} \right )\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**10*(5*x**6+3*x**4+x**2+4)/(x**4+2*x**2+3)**3,x)

[Out]

x**5 - 9*x**3 + 58*x + (252*x**7 + 3809*x**5 + 6666*x**3 + 8415*x)/(64*x**8 + 256*x**6 + 640*x**4 + 768*x**2 +
576) + 3*RootSum(17179869184*_t**4 - 2253289947136*_t**2 + 176883200667963, Lambda(_t, _t*log(-56941871104*_t
**3/55104008440689 - 1957224667904*_t/55104008440689 + 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^{10}}{{\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^10*(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^10/(x^4 + 2*x^2 + 3)^3, x)