3.2.17 \(\int \frac {x^{10} (4+x^2+3 x^4+5 x^6)}{(3+2 x^2+x^4)^3} \, dx\) [117]

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

[Out]

58*x-9*x^3+x^5-25/16*x*(7*x^2+15)/(x^4+2*x^2+3)^2+1/64*x*(252*x^2+3305)/(x^4+2*x^2+3)+3/256*arctan((-2*x+(-2+2
*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-8595619+7678611*3^(1/2))^(1/2)-3/256*arctan((2*x+(-2+2*3^(1/2))^(1/2))
/(2+2*3^(1/2))^(1/2))*(-8595619+7678611*3^(1/2))^(1/2)+3/512*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(8595619+7
678611*3^(1/2))^(1/2)-3/512*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(8595619+7678611*3^(1/2))^(1/2)

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Rubi [A]
time = 0.23, antiderivative size = 243, normalized size of antiderivative = 1.00, 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 = {1682, 1692, 1690, 1183, 648, 632, 210, 642} \begin {gather*} \frac {3}{256} \sqrt {7678611 \sqrt {3}-8595619} \text {ArcTan}\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} \text {ArcTan}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+x^5-9 x^3+\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 )+\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 \end {gather*}

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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

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 642

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]

Rule 648

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 1183

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 1682

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 1690

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 1692

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
]

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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.14, size = 156, normalized size = 0.64 \begin {gather*} 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 \left (4795 i+148 \sqrt {2}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{128 \sqrt {2-2 i \sqrt {2}}}+\frac {3 \left (-4795 i+148 \sqrt {2}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{128 \sqrt {2+2 i \sqrt {2}}} \end {gather*}

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 [A]
time = 0.04, size = 298, normalized size = 1.23

method	result	size
risch	\(x^{5}-9 x^{3}+58 x +\frac {\frac {63}{16} x^{7}+\frac {3809}{64} x^{5}+\frac {3333}{32} x^{3}+\frac {8415}{64} x}{\left (x^{4}+2 x^{2}+3\right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{2}+3\right )}{\sum }\frac {\left (148 \textit {\_R}^{2}-4647\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}\right )}{256}\)	\(82\)
default	\(x^{5}-9 x^{3}+58 x +\frac {\frac {63}{16} x^{7}+\frac {3809}{64} x^{5}+\frac {3333}{32} x^{3}+\frac {8415}{64} x}{\left (x^{4}+2 x^{2}+3\right )^{2}}+\frac {3 \left (1697 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+4795 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}-x \sqrt {-2+2 \sqrt {3}}\right )}{1024}+\frac {3 \left (-6196 \sqrt {3}+\frac {\left (1697 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+4795 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{256 \sqrt {2+2 \sqrt {3}}}+\frac {3 \left (-1697 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-4795 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}+x \sqrt {-2+2 \sqrt {3}}\right )}{1024}+\frac {3 \left (-6196 \sqrt {3}-\frac {\left (-1697 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-4795 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{256 \sqrt {2+2 \sqrt {3}}}\)	\(298\)

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,method=_RETURNVERBOSE)

[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+3/1024*(1697*(-2+2*3^(1/2))^(1/2)
*3^(1/2)+4795*(-2+2*3^(1/2))^(1/2))*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))+3/256*(-6196*3^(1/2)+1/2*(1697*(-2+
2*3^(1/2))^(1/2)*3^(1/2)+4795*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2)*arctan((2*x-(-2+
2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))+3/1024*(-1697*(-2+2*3^(1/2))^(1/2)*3^(1/2)-4795*(-2+2*3^(1/2))^(1/2))*l
n(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))+3/256*(-6196*3^(1/2)-1/2*(-1697*(-2+2*3^(1/2))^(1/2)*3^(1/2)-4795*(-2+2*
3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2)*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2
))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 568 vs. \(2 (190) = 380\).
time = 0.39, size = 568, normalized size = 2.34 \begin {gather*} \frac {18808834881088512 \, x^{13} - 94044174405442560 \, x^{11} + 601882716194832384 \, x^{9} + 2970620359031916864 \, x^{7} + 10166469141273357744 \, x^{5} + 57410392 \cdot 2183743218123^{\frac {1}{4}} \sqrt {3} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {-66002414605209 \, \sqrt {3} + 176883200667963} \arctan \left (\frac {1}{863545621466021963404537403089353} \cdot 2183743218123^{\frac {3}{4}} \sqrt {7176299} \sqrt {853179} \sqrt {55104008440689 \, x^{2} + 2183743218123^{\frac {1}{4}} {\left (148 \, \sqrt {3} \sqrt {2} x + 4647 \, \sqrt {2} x\right )} \sqrt {-66002414605209 \, \sqrt {3} + 176883200667963} + 55104008440689 \, \sqrt {3}} {\left (1549 \, \sqrt {3} + 148\right )} \sqrt {-66002414605209 \, \sqrt {3} + 176883200667963} - \frac {1}{47013582817418600331} \cdot 2183743218123^{\frac {3}{4}} {\left (1549 \, \sqrt {3} x + 148 \, x\right )} \sqrt {-66002414605209 \, \sqrt {3} + 176883200667963} - \frac {1}{2} \, \sqrt {3} \sqrt {2} + \frac {1}{2} \, \sqrt {2}\right ) + 57410392 \cdot 2183743218123^{\frac {1}{4}} \sqrt {3} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {-66002414605209 \, \sqrt {3} + 176883200667963} \arctan \left (\frac {1}{863545621466021963404537403089353} \cdot 2183743218123^{\frac {3}{4}} \sqrt {7176299} \sqrt {853179} \sqrt {55104008440689 \, x^{2} - 2183743218123^{\frac {1}{4}} {\left (148 \, \sqrt {3} \sqrt {2} x + 4647 \, \sqrt {2} x\right )} \sqrt {-66002414605209 \, \sqrt {3} + 176883200667963} + 55104008440689 \, \sqrt {3}} {\left (1549 \, \sqrt {3} + 148\right )} \sqrt {-66002414605209 \, \sqrt {3} + 176883200667963} - \frac {1}{47013582817418600331} \cdot 2183743218123^{\frac {3}{4}} {\left (1549 \, \sqrt {3} x + 148 \, x\right )} \sqrt {-66002414605209 \, \sqrt {3} + 176883200667963} + \frac {1}{2} \, \sqrt {3} \sqrt {2} - \frac {1}{2} \, \sqrt {2}\right ) + 13526491159952810208 \, x^{3} - 2183743218123^{\frac {1}{4}} {\left (8595619 \, \sqrt {3} \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} + 23035833 \, \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}\right )} \sqrt {-66002414605209 \, \sqrt {3} + 176883200667963} \log \left (530649602003889 \, x^{2} + \frac {69107499}{7176299} \cdot 2183743218123^{\frac {1}{4}} {\left (148 \, \sqrt {3} \sqrt {2} x + 4647 \, \sqrt {2} x\right )} \sqrt {-66002414605209 \, \sqrt {3} + 176883200667963} + 530649602003889 \, \sqrt {3}\right ) + 2183743218123^{\frac {1}{4}} {\left (8595619 \, \sqrt {3} \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} + 23035833 \, \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}\right )} \sqrt {-66002414605209 \, \sqrt {3} + 176883200667963} \log \left (530649602003889 \, x^{2} - \frac {69107499}{7176299} \cdot 2183743218123^{\frac {1}{4}} {\left (148 \, \sqrt {3} \sqrt {2} x + 4647 \, \sqrt {2} x\right )} \sqrt {-66002414605209 \, \sqrt {3} + 176883200667963} + 530649602003889 \, \sqrt {3}\right ) + 12291279706746325584 \, x}{18808834881088512 \, {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} \end {gather*}

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*2183743218123^(3/
4)*sqrt(7176299)*sqrt(853179)*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(-66
002414605209*sqrt(3) + 176883200667963) - 1/47013582817418600331*2183743218123^(3/4)*(1549*sqrt(3)*x + 148*x)*
sqrt(-66002414605209*sqrt(3) + 176883200667963) - 1/2*sqrt(3)*sqrt(2) + 1/2*sqrt(2)) + 57410392*2183743218123^
(1/4)*sqrt(3)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)*sqrt(-66002414605209*sqrt(3) + 176883200667963)*arctan(1/863
545621466021963404537403089353*2183743218123^(3/4)*sqrt(7176299)*sqrt(853179)*sqrt(55104008440689*x^2 - 218374
3218123^(1/4)*(148*sqrt(3)*sqrt(2)*x + 4647*sqrt(2)*x)*sqrt(-66002414605209*sqrt(3) + 176883200667963) + 55104
008440689*sqrt(3))*(1549*sqrt(3) + 148)*sqrt(-66002414605209*sqrt(3) + 176883200667963) - 1/470135828174186003
31*2183743218123^(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) +
176883200667963)*log(530649602003889*x^2 + 69107499/7176299*2183743218123^(1/4)*(148*sqrt(3)*sqrt(2)*x + 4647*
sqrt(2)*x)*sqrt(-66002414605209*sqrt(3) + 176883200667963) + 530649602003889*sqrt(3)) + 2183743218123^(1/4)*(8
595619*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(530649602003889*x^2 - 69107499/7176299*2183743218123^
(1/4)*(148*sqrt(3)*sqrt(2)*x + 4647*sqrt(2)*x)*sqrt(-66002414605209*sqrt(3) + 176883200667963) + 5306496020038
89*sqrt(3)) + 12291279706746325584*x)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1204 vs. \(2 (221) = 442\).
time = 0.72, size = 1204, normalized size = 4.95 \begin {gather*} 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 \sqrt {\frac {8595619}{262144} + \frac {7678611 \sqrt {3}}{262144}} \log {\left (x^{2} + x \left (- \frac {6788 \sqrt {3} \sqrt {8595619 + 7678611 \sqrt {3}}}{7176299} - \frac {2313785528 \sqrt {8595619 + 7678611 \sqrt {3}}}{18368002813563} + \frac {1697 \sqrt {2} \sqrt {8595619 + 7678611 \sqrt {3}} \sqrt {66002414605209 \sqrt {3} + 125383933330562}}{18368002813563}\right ) - \frac {1218095240252468879279 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562}}{1012150582077174852410264907} - \frac {134353410196228 \sqrt {6} \sqrt {66002414605209 \sqrt {3} + 125383933330562}}{395442840668908030011} + \frac {18391902996311867463806959889}{1012150582077174852410264907} + \frac {5204579286823805792980 \sqrt {3}}{395442840668908030011} \right )} + 3 \sqrt {\frac {8595619}{262144} + \frac {7678611 \sqrt {3}}{262144}} \log {\left (x^{2} + x \left (- \frac {1697 \sqrt {2} \sqrt {8595619 + 7678611 \sqrt {3}} \sqrt {66002414605209 \sqrt {3} + 125383933330562}}{18368002813563} + \frac {2313785528 \sqrt {8595619 + 7678611 \sqrt {3}}}{18368002813563} + \frac {6788 \sqrt {3} \sqrt {8595619 + 7678611 \sqrt {3}}}{7176299}\right ) - \frac {1218095240252468879279 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562}}{1012150582077174852410264907} - \frac {134353410196228 \sqrt {6} \sqrt {66002414605209 \sqrt {3} + 125383933330562}}{395442840668908030011} + \frac {18391902996311867463806959889}{1012150582077174852410264907} + \frac {5204579286823805792980 \sqrt {3}}{395442840668908030011} \right )} - 2 \sqrt {- \frac {9 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562}}{131072} + \frac {77360571}{262144} + \frac {207322497 \sqrt {3}}{262144}} \operatorname {atan}{\left (\frac {110208016881378 x}{22232174302 \sqrt {- 2 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} + 8595619 + 23035833 \sqrt {3}} + 1697 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} \sqrt {- 2 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} + 8595619 + 23035833 \sqrt {3}}} - \frac {52122411468 \sqrt {3} \sqrt {8595619 + 7678611 \sqrt {3}}}{22232174302 \sqrt {- 2 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} + 8595619 + 23035833 \sqrt {3}} + 1697 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} \sqrt {- 2 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} + 8595619 + 23035833 \sqrt {3}}} - \frac {6941356584 \sqrt {8595619 + 7678611 \sqrt {3}}}{22232174302 \sqrt {- 2 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} + 8595619 + 23035833 \sqrt {3}} + 1697 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} \sqrt {- 2 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} + 8595619 + 23035833 \sqrt {3}}} + \frac {5091 \sqrt {2} \sqrt {8595619 + 7678611 \sqrt {3}} \sqrt {66002414605209 \sqrt {3} + 125383933330562}}{22232174302 \sqrt {- 2 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} + 8595619 + 23035833 \sqrt {3}} + 1697 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} \sqrt {- 2 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} + 8595619 + 23035833 \sqrt {3}}} \right )} - 2 \sqrt {- \frac {9 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562}}{131072} + \frac {77360571}{262144} + \frac {207322497 \sqrt {3}}{262144}} \operatorname {atan}{\left (\frac {110208016881378 x}{22232174302 \sqrt {- 2 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} + 8595619 + 23035833 \sqrt {3}} + 1697 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} \sqrt {- 2 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} + 8595619 + 23035833 \sqrt {3}}} - \frac {5091 \sqrt {2} \sqrt {8595619 + 7678611 \sqrt {3}} \sqrt {66002414605209 \sqrt {3} + 125383933330562}}{22232174302 \sqrt {- 2 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} + 8595619 + 23035833 \sqrt {3}} + 1697 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} \sqrt {- 2 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} + 8595619 + 23035833 \sqrt {3}}} + \frac {6941356584 \sqrt {8595619 + 7678611 \sqrt {3}}}{22232174302 \sqrt {- 2 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} + 8595619 + 23035833 \sqrt {3}} + 1697 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} \sqrt {- 2 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} + 8595619 + 23035833 \sqrt {3}}} + \frac {52122411468 \sqrt {3} \sqrt {8595619 + 7678611 \sqrt {3}}}{22232174302 \sqrt {- 2 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} + 8595619 + 23035833 \sqrt {3}} + 1697 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} \sqrt {- 2 \sqrt {2} \sqrt {66002414605209 \sqrt {3} + 125383933330562} + 8595619 + 23035833 \sqrt {3}}} \right )} \end {gather*}

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*sqrt(8595619/262144 + 7678611*sqrt(3)/262144)*log(x**2 + x*(-6788*sqrt(3)*sqrt(8595619 + 7678611*sqr
t(3))/7176299 - 2313785528*sqrt(8595619 + 7678611*sqrt(3))/18368002813563 + 1697*sqrt(2)*sqrt(8595619 + 767861
1*sqrt(3))*sqrt(66002414605209*sqrt(3) + 125383933330562)/18368002813563) - 1218095240252468879279*sqrt(2)*sqr
t(66002414605209*sqrt(3) + 125383933330562)/1012150582077174852410264907 - 134353410196228*sqrt(6)*sqrt(660024
14605209*sqrt(3) + 125383933330562)/395442840668908030011 + 18391902996311867463806959889/10121505820771748524
10264907 + 5204579286823805792980*sqrt(3)/395442840668908030011) + 3*sqrt(8595619/262144 + 7678611*sqrt(3)/262
144)*log(x**2 + x*(-1697*sqrt(2)*sqrt(8595619 + 7678611*sqrt(3))*sqrt(66002414605209*sqrt(3) + 125383933330562
)/18368002813563 + 2313785528*sqrt(8595619 + 7678611*sqrt(3))/18368002813563 + 6788*sqrt(3)*sqrt(8595619 + 767
8611*sqrt(3))/7176299) - 1218095240252468879279*sqrt(2)*sqrt(66002414605209*sqrt(3) + 125383933330562)/1012150
582077174852410264907 - 134353410196228*sqrt(6)*sqrt(66002414605209*sqrt(3) + 125383933330562)/395442840668908
030011 + 18391902996311867463806959889/1012150582077174852410264907 + 5204579286823805792980*sqrt(3)/395442840
668908030011) - 2*sqrt(-9*sqrt(2)*sqrt(66002414605209*sqrt(3) + 125383933330562)/131072 + 77360571/262144 + 20
7322497*sqrt(3)/262144)*atan(110208016881378*x/(22232174302*sqrt(-2*sqrt(2)*sqrt(66002414605209*sqrt(3) + 1253
83933330562) + 8595619 + 23035833*sqrt(3)) + 1697*sqrt(2)*sqrt(66002414605209*sqrt(3) + 125383933330562)*sqrt(
-2*sqrt(2)*sqrt(66002414605209*sqrt(3) + 125383933330562) + 8595619 + 23035833*sqrt(3))) - 52122411468*sqrt(3)
*sqrt(8595619 + 7678611*sqrt(3))/(22232174302*sqrt(-2*sqrt(2)*sqrt(66002414605209*sqrt(3) + 125383933330562) +
8595619 + 23035833*sqrt(3)) + 1697*sqrt(2)*sqrt(66002414605209*sqrt(3) + 125383933330562)*sqrt(-2*sqrt(2)*sqr
t(66002414605209*sqrt(3) + 125383933330562) + 8595619 + 23035833*sqrt(3))) - 6941356584*sqrt(8595619 + 7678611
*sqrt(3))/(22232174302*sqrt(-2*sqrt(2)*sqrt(66002414605209*sqrt(3) + 125383933330562) + 8595619 + 23035833*sqr
t(3)) + 1697*sqrt(2)*sqrt(66002414605209*sqrt(3) + 125383933330562)*sqrt(-2*sqrt(2)*sqrt(66002414605209*sqrt(3
) + 125383933330562) + 8595619 + 23035833*sqrt(3))) + 5091*sqrt(2)*sqrt(8595619 + 7678611*sqrt(3))*sqrt(660024
14605209*sqrt(3) + 125383933330562)/(22232174302*sqrt(-2*sqrt(2)*sqrt(66002414605209*sqrt(3) + 125383933330562
) + 8595619 + 23035833*sqrt(3)) + 1697*sqrt(2)*sqrt(66002414605209*sqrt(3) + 125383933330562)*sqrt(-2*sqrt(2)*
sqrt(66002414605209*sqrt(3) + 125383933330562) + 8595619 + 23035833*sqrt(3)))) - 2*sqrt(-9*sqrt(2)*sqrt(660024
14605209*sqrt(3) + 125383933330562)/131072 + 77360571/262144 + 207322497*sqrt(3)/262144)*atan(110208016881378*
x/(22232174302*sqrt(-2*sqrt(2)*sqrt(66002414605209*sqrt(3) + 125383933330562) + 8595619 + 23035833*sqrt(3)) +
1697*sqrt(2)*sqrt(66002414605209*sqrt(3) + 125383933330562)*sqrt(-2*sqrt(2)*sqrt(66002414605209*sqrt(3) + 1253
83933330562) + 8595619 + 23035833*sqrt(3))) - 5091*sqrt(2)*sqrt(8595619 + 7678611*sqrt(3))*sqrt(66002414605209
*sqrt(3) + 125383933330562)/(22232174302*sqrt(-2*sqrt(2)*sqrt(66002414605209*sqrt(3) + 125383933330562) + 8595
619 + 23035833*sqrt(3)) + 1697*sqrt(2)*sqrt(66002414605209*sqrt(3) + 125383933330562)*sqrt(-2*sqrt(2)*sqrt(660
02414605209*sqrt(3) + 125383933330562) + 8595619 + 23035833*sqrt(3))) + 6941356584*sqrt(8595619 + 7678611*sqrt
(3))/(22232174302*sqrt(-2*sqrt(2)*sqrt(66002414605209*sqrt(3) + 125383933330562) + 8595619 + 23035833*sqrt(3))
+ 1697*sqrt(2)*sqrt(66002414605209*sqrt(3) + 125383933330562)*sqrt(-2*sqrt(2)*sqrt(66002414605209*sqrt(3) + 1
25383933330562) + 8595619 + 23035833*sqrt(3))) + 52122411468*sqrt(3)*sqrt(8595619 + 7678611*sqrt(3))/(22232174
302*sqrt(-2*sqrt(2)*sqrt(66002414605209*sqrt(3) + 125383933330562) + 8595619 + 23035833*sqrt(3)) + 1697*sqrt(2
)*sqrt(66002414605209*sqrt(3) + 125383933330562)*sqrt(-2*sqrt(2)*sqrt(66002414605209*sqrt(3) + 125383933330562
) + 8595619 + 23035833*sqrt(3))))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 588 vs. \(2 (190) = 380\).
time = 5.73, size = 588, normalized size = 2.42 \begin {gather*} x^{5} - 9 \, x^{3} - \frac {1}{13824} \, \sqrt {2} {\left (37 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 666 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 666 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 37 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 41823 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} - 41823 \cdot 3^{\frac {1}{4}} \sqrt {-6 \, \sqrt {3} + 18}\right )} \arctan \left (\frac {3^{\frac {3}{4}} {\left (x + 3^{\frac {1}{4}} \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}\right )}}{3 \, \sqrt {\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}}\right ) - \frac {1}{13824} \, \sqrt {2} {\left (37 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 666 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 666 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 37 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 41823 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} - 41823 \cdot 3^{\frac {1}{4}} \sqrt {-6 \, \sqrt {3} + 18}\right )} \arctan \left (\frac {3^{\frac {3}{4}} {\left (x - 3^{\frac {1}{4}} \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}\right )}}{3 \, \sqrt {\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}}\right ) - \frac {1}{27648} \, \sqrt {2} {\left (666 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 37 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 37 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 666 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} + 41823 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} + 41823 \cdot 3^{\frac {1}{4}} \sqrt {6 \, \sqrt {3} + 18}\right )} \log \left (x^{2} + 2 \cdot 3^{\frac {1}{4}} x \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}} + \sqrt {3}\right ) + \frac {1}{27648} \, \sqrt {2} {\left (666 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 37 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 37 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 666 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} + 41823 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} + 41823 \cdot 3^{\frac {1}{4}} \sqrt {6 \, \sqrt {3} + 18}\right )} \log \left (x^{2} - 2 \cdot 3^{\frac {1}{4}} x \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}} + \sqrt {3}\right ) + 58 \, x + \frac {252 \, x^{7} + 3809 \, x^{5} + 6666 \, x^{3} + 8415 \, x}{64 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}} \end {gather*}

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]

x^5 - 9*x^3 - 1/13824*sqrt(2)*(37*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 666*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3)
+ 18)*(sqrt(3) - 3) - 666*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 37*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) + 4
1823*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) - 41823*3^(1/4)*sqrt(-6*sqrt(3) + 18))*arctan(1/3*3^(3/4)*(x + 3^(1/
4)*sqrt(-1/6*sqrt(3) + 1/2))/sqrt(1/6*sqrt(3) + 1/2)) - 1/13824*sqrt(2)*(37*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(
3/2) + 666*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 666*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18
) + 37*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) + 41823*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) - 41823*3^(1/4)*sqrt(-6*sq
rt(3) + 18))*arctan(1/3*3^(3/4)*(x - 3^(1/4)*sqrt(-1/6*sqrt(3) + 1/2))/sqrt(1/6*sqrt(3) + 1/2)) - 1/27648*sqrt
(2)*(666*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 37*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 37
*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 666*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) + 41823*3^(1/4)*sqrt(2)*sqrt(
-6*sqrt(3) + 18) + 41823*3^(1/4)*sqrt(6*sqrt(3) + 18))*log(x^2 + 2*3^(1/4)*x*sqrt(-1/6*sqrt(3) + 1/2) + sqrt(3
)) + 1/27648*sqrt(2)*(666*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 37*3^(3/4)*sqrt(2)*(-6*sqrt(3)
+ 18)^(3/2) + 37*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 666*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) + 41823*3^(1
/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) + 41823*3^(1/4)*sqrt(6*sqrt(3) + 18))*log(x^2 - 2*3^(1/4)*x*sqrt(-1/6*sqrt(3
) + 1/2) + sqrt(3)) + 58*x + 1/64*(252*x^7 + 3809*x^5 + 6666*x^3 + 8415*x)/(x^4 + 2*x^2 + 3)^2

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Mupad [B]
time = 0.11, size = 184, normalized size = 0.76 \begin {gather*} 58\,x+\frac {\frac {63\,x^7}{16}+\frac {3809\,x^5}{64}+\frac {3333\,x^3}{32}+\frac {8415\,x}{64}}{x^8+4\,x^6+10\,x^4+12\,x^2+9}-9\,x^3+x^5-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {17191238-\sqrt {2}\,14352598{}\mathrm {i}}\,193760073{}\mathrm {i}}{131072\,\left (-\frac {986432531643}{131072}+\frac {\sqrt {2}\,900403059231{}\mathrm {i}}{131072}\right )}-\frac {193760073\,\sqrt {2}\,x\,\sqrt {17191238-\sqrt {2}\,14352598{}\mathrm {i}}}{262144\,\left (-\frac {986432531643}{131072}+\frac {\sqrt {2}\,900403059231{}\mathrm {i}}{131072}\right )}\right )\,\sqrt {17191238-\sqrt {2}\,14352598{}\mathrm {i}}\,3{}\mathrm {i}}{256}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {17191238+\sqrt {2}\,14352598{}\mathrm {i}}\,193760073{}\mathrm {i}}{131072\,\left (\frac {986432531643}{131072}+\frac {\sqrt {2}\,900403059231{}\mathrm {i}}{131072}\right )}+\frac {193760073\,\sqrt {2}\,x\,\sqrt {17191238+\sqrt {2}\,14352598{}\mathrm {i}}}{262144\,\left (\frac {986432531643}{131072}+\frac {\sqrt {2}\,900403059231{}\mathrm {i}}{131072}\right )}\right )\,\sqrt {17191238+\sqrt {2}\,14352598{}\mathrm {i}}\,3{}\mathrm {i}}{256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

58*x - (atan((x*(17191238 - 2^(1/2)*14352598i)^(1/2)*193760073i)/(131072*((2^(1/2)*900403059231i)/131072 - 986
432531643/131072)) - (193760073*2^(1/2)*x*(17191238 - 2^(1/2)*14352598i)^(1/2))/(262144*((2^(1/2)*900403059231
i)/131072 - 986432531643/131072)))*(17191238 - 2^(1/2)*14352598i)^(1/2)*3i)/256 + (atan((x*(2^(1/2)*14352598i
+ 17191238)^(1/2)*193760073i)/(131072*((2^(1/2)*900403059231i)/131072 + 986432531643/131072)) + (193760073*2^(
1/2)*x*(2^(1/2)*14352598i + 17191238)^(1/2))/(262144*((2^(1/2)*900403059231i)/131072 + 986432531643/131072)))*
(2^(1/2)*14352598i + 17191238)^(1/2)*3i)/256 + ((8415*x)/64 + (3333*x^3)/32 + (3809*x^5)/64 + (63*x^7)/16)/(12
*x^2 + 10*x^4 + 4*x^6 + x^8 + 9) - 9*x^3 + x^5

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