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

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

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Rubi [A]  time = 0.30, antiderivative size = 235, 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 = {1668, 1678, 1676, 1169, 634, 618, 204, 628} \begin {gather*} \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 ) \end {gather*}

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 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 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 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]

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 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 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 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 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
]

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*}

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Mathematica [C]  time = 0.32, size = 138, normalized size = 0.59 \begin {gather*} \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 ) \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.39, size = 551, normalized size = 2.34 \begin {gather*} \frac {23795867690357760 \, x^{9} + 125374477893572448 \, x^{7} + 304066571830852752 \, x^{5} - 10534088 \cdot 4152675581883^{\frac {1}{4}} \sqrt {3} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {973721762751 \, \sqrt {3} + 4152675581883} \arctan \left (\frac {1}{8471206900375217227324302495633} \cdot 4152675581883^{\frac {3}{4}} \sqrt {516403378697} \sqrt {4647630408273 \, x^{2} + 4152675581883^{\frac {1}{4}} {\left (1322 \, \sqrt {3} \sqrt {2} x - 1137 \, \sqrt {2} x\right )} \sqrt {973721762751 \, \sqrt {3} + 4152675581883} + 4647630408273 \, \sqrt {3}} \sqrt {973721762751 \, \sqrt {3} + 4152675581883} {\left (379 \, \sqrt {3} - 1322\right )} - \frac {1}{5468081251875840963} \cdot 4152675581883^{\frac {3}{4}} {\left (379 \, \sqrt {3} x - 1322 \, x\right )} \sqrt {973721762751 \, \sqrt {3} + 4152675581883} + \frac {1}{2} \, \sqrt {3} \sqrt {2} - \frac {1}{2} \, \sqrt {2}\right ) - 10534088 \cdot 4152675581883^{\frac {1}{4}} \sqrt {3} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {973721762751 \, \sqrt {3} + 4152675581883} \arctan \left (\frac {1}{8471206900375217227324302495633} \cdot 4152675581883^{\frac {3}{4}} \sqrt {516403378697} \sqrt {4647630408273 \, x^{2} - 4152675581883^{\frac {1}{4}} {\left (1322 \, \sqrt {3} \sqrt {2} x - 1137 \, \sqrt {2} x\right )} \sqrt {973721762751 \, \sqrt {3} + 4152675581883} + 4647630408273 \, \sqrt {3}} \sqrt {973721762751 \, \sqrt {3} + 4152675581883} {\left (379 \, \sqrt {3} - 1322\right )} - \frac {1}{5468081251875840963} \cdot 4152675581883^{\frac {3}{4}} {\left (379 \, \sqrt {3} x - 1322 \, x\right )} \sqrt {973721762751 \, \sqrt {3} + 4152675581883} - \frac {1}{2} \, \sqrt {3} \sqrt {2} + \frac {1}{2} \, \sqrt {2}\right ) + 380138986353465216 \, x^{3} - 4152675581883^{\frac {1}{4}} {\left (827621 \, \sqrt {3} \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} - 3529593 \, \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}\right )} \sqrt {973721762751 \, \sqrt {3} + 4152675581883} \log \left (4647630408273 \, x^{2} + 4152675581883^{\frac {1}{4}} {\left (1322 \, \sqrt {3} \sqrt {2} x - 1137 \, \sqrt {2} x\right )} \sqrt {973721762751 \, \sqrt {3} + 4152675581883} + 4647630408273 \, \sqrt {3}\right ) + 4152675581883^{\frac {1}{4}} {\left (827621 \, \sqrt {3} \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} - 3529593 \, \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}\right )} \sqrt {973721762751 \, \sqrt {3} + 4152675581883} \log \left (4647630408273 \, x^{2} - 4152675581883^{\frac {1}{4}} {\left (1322 \, \sqrt {3} \sqrt {2} x - 1137 \, \sqrt {2} x\right )} \sqrt {973721762751 \, \sqrt {3} + 4152675581883} + 4647630408273 \, \sqrt {3}\right ) + 253649077161907248 \, x}{4759173538071552 \, {\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^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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giac [B]  time = 2.61, size = 580, normalized size = 2.47 \begin {gather*} \frac {1}{82944} \, \sqrt {2} {\left (661 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 11898 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 11898 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 661 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 20466 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 20466 \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}{82944} \, \sqrt {2} {\left (661 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 11898 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 11898 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 661 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 20466 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 20466 \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}{165888} \, \sqrt {2} {\left (11898 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 661 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 661 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 11898 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 20466 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 20466 \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}{165888} \, \sqrt {2} {\left (11898 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 661 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 661 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 11898 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 20466 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 20466 \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 ) + 5 \, x + \frac {406 \, x^{7} + 889 \, x^{5} + 1272 \, x^{3} + 531 \, 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^6*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^3,x, algorithm="giac")

[Out]

1/82944*sqrt(2)*(661*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 11898*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt
(3) - 3) - 11898*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 661*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 20466*3^(
1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) + 20466*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)) + 1/82944*sqrt(2)*(661*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) +
11898*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 11898*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) +
661*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 20466*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) + 20466*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)) + 1/165888*sqrt(
2)*(11898*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 661*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) +
661*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 11898*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 20466*3^(1/4)*sqrt(2)*
sqrt(-6*sqrt(3) + 18) - 20466*3^(1/4)*sqrt(6*sqrt(3) + 18))*log(x^2 + 2*3^(1/4)*x*sqrt(-1/6*sqrt(3) + 1/2) + s
qrt(3)) - 1/165888*sqrt(2)*(11898*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 661*3^(3/4)*sqrt(2)*(-
6*sqrt(3) + 18)^(3/2) + 661*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 11898*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3)
- 20466*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) - 20466*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)) + 5*x + 1/64*(406*x^7 + 889*x^5 + 1272*x^3 + 531*x)/(x^4 + 2*x^2 + 3)^2

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maple [B]  time = 0.04, size = 422, normalized size = 1.80 \begin {gather*} 5 x -\frac {943 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{512 \sqrt {2+2 \sqrt {3}}}-\frac {185 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{512 \sqrt {2+2 \sqrt {3}}}-\frac {379 \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{64 \sqrt {2+2 \sqrt {3}}}-\frac {943 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{512 \sqrt {2+2 \sqrt {3}}}-\frac {185 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{512 \sqrt {2+2 \sqrt {3}}}-\frac {379 \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{64 \sqrt {2+2 \sqrt {3}}}-\frac {943 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{1024}-\frac {185 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{1024}+\frac {943 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{1024}+\frac {185 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{1024}-\frac {-\frac {203}{32} x^{7}-\frac {889}{64} x^{5}-\frac {159}{8} x^{3}-\frac {531}{64} x}{\left (x^{4}+2 x^{2}+3\right )^{2}} \end {gather*}

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*(-2+2*3^(1/2))^(1/2)*3^(1/2)*ln(x^2-(
-2+2*3^(1/2))^(1/2)*x+3^(1/2))-185/1024*(-2+2*3^(1/2))^(1/2)*ln(x^2-(-2+2*3^(1/2))^(1/2)*x+3^(1/2))-943/512/(2
+2*3^(1/2))^(1/2)*(-2+2*3^(1/2))*3^(1/2)*arctan((2*x-(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))-185/512/(2+2*3
^(1/2))^(1/2)*(-2+2*3^(1/2))*arctan((2*x-(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))-379/64/(2+2*3^(1/2))^(1/2)
*3^(1/2)*arctan((2*x-(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))+943/1024*(-2+2*3^(1/2))^(1/2)*3^(1/2)*ln(x^2+(
-2+2*3^(1/2))^(1/2)*x+3^(1/2))+185/1024*(-2+2*3^(1/2))^(1/2)*ln(x^2+(-2+2*3^(1/2))^(1/2)*x+3^(1/2))-943/512/(2
+2*3^(1/2))^(1/2)*(-2+2*3^(1/2))*3^(1/2)*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))-185/512/(2+2*3
^(1/2))^(1/2)*(-2+2*3^(1/2))*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))-379/64/(2+2*3^(1/2))^(1/2)
*3^(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*} 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 {gather*}

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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mupad [B]  time = 0.99, size = 176, normalized size = 0.75 \begin {gather*} 5\,x+\frac {\frac {203\,x^7}{32}+\frac {889\,x^5}{64}+\frac {159\,x^3}{8}+\frac {531\,x}{64}}{x^8+4\,x^6+10\,x^4+12\,x^2+9}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-1655242-\sqrt {2}\,2633522{}\mathrm {i}}\,1316761{}\mathrm {i}}{131072\,\left (-\frac {3725116869}{131072}+\frac {\sqrt {2}\,1497157257{}\mathrm {i}}{131072}\right )}+\frac {1316761\,\sqrt {2}\,x\,\sqrt {-1655242-\sqrt {2}\,2633522{}\mathrm {i}}}{262144\,\left (-\frac {3725116869}{131072}+\frac {\sqrt {2}\,1497157257{}\mathrm {i}}{131072}\right )}\right )\,\sqrt {-1655242-\sqrt {2}\,2633522{}\mathrm {i}}\,1{}\mathrm {i}}{256}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-1655242+\sqrt {2}\,2633522{}\mathrm {i}}\,1316761{}\mathrm {i}}{131072\,\left (\frac {3725116869}{131072}+\frac {\sqrt {2}\,1497157257{}\mathrm {i}}{131072}\right )}-\frac {1316761\,\sqrt {2}\,x\,\sqrt {-1655242+\sqrt {2}\,2633522{}\mathrm {i}}}{262144\,\left (\frac {3725116869}{131072}+\frac {\sqrt {2}\,1497157257{}\mathrm {i}}{131072}\right )}\right )\,\sqrt {-1655242+\sqrt {2}\,2633522{}\mathrm {i}}\,1{}\mathrm {i}}{256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*x + (atan((x*(- 2^(1/2)*2633522i - 1655242)^(1/2)*1316761i)/(131072*((2^(1/2)*1497157257i)/131072 - 37251168
69/131072)) + (1316761*2^(1/2)*x*(- 2^(1/2)*2633522i - 1655242)^(1/2))/(262144*((2^(1/2)*1497157257i)/131072 -
3725116869/131072)))*(- 2^(1/2)*2633522i - 1655242)^(1/2)*1i)/256 - (atan((x*(2^(1/2)*2633522i - 1655242)^(1/
2)*1316761i)/(131072*((2^(1/2)*1497157257i)/131072 + 3725116869/131072)) - (1316761*2^(1/2)*x*(2^(1/2)*2633522
i - 1655242)^(1/2))/(262144*((2^(1/2)*1497157257i)/131072 + 3725116869/131072)))*(2^(1/2)*2633522i - 1655242)^
(1/2)*1i)/256 + ((531*x)/64 + (159*x^3)/8 + (889*x^5)/64 + (203*x^7)/32)/(12*x^2 + 10*x^4 + 4*x^6 + x^8 + 9)

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sympy [A]  time = 0.66, size = 71, normalized size = 0.30 \begin {gather*} 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 {gather*}

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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