\(\int \frac {x^8 (4+x^2+3 x^4+5 x^6)}{(3+2 x^2+x^4)^3} \, dx\) [118]

Optimal result

Integrand size = 31, antiderivative size = 242 \[ \int \frac {x^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=-27 x+\frac {5 x^3}{3}+\frac {25 x \left (3+5 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac {x \left (1468+835 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac {21}{256} \sqrt {34271+22721 \sqrt {3}} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {21}{256} \sqrt {34271+22721 \sqrt {3}} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {21}{512} \sqrt {-34271+22721 \sqrt {3}} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {21}{512} \sqrt {-34271+22721 \sqrt {3}} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right ) \]

[Out]

-27*x+5/3*x^3+25/16*x*(5*x^2+3)/(x^4+2*x^2+3)^2-1/64*x*(835*x^2+1468)/(x^4+2*x^2+3)-21/512*ln(x^2+3^(1/2)-x*(-
2+2*3^(1/2))^(1/2))*(-34271+22721*3^(1/2))^(1/2)+21/512*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-34271+22721*3
^(1/2))^(1/2)-21/256*arctan((-2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(34271+22721*3^(1/2))^(1/2)+21/25
6*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(34271+22721*3^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {1682, 1692, 1690, 1183, 648, 632, 210, 642} \[ \int \frac {x^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=-\frac {21}{256} \sqrt {34271+22721 \sqrt {3}} \arctan \left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {21}{256} \sqrt {34271+22721 \sqrt {3}} \arctan \left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {5 x^3}{3}-\frac {21}{512} \sqrt {22721 \sqrt {3}-34271} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {21}{512} \sqrt {22721 \sqrt {3}-34271} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {\left (835 x^2+1468\right ) x}{64 \left (x^4+2 x^2+3\right )}+\frac {25 \left (5 x^2+3\right ) x}{16 \left (x^4+2 x^2+3\right )^2}-27 x \]

[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 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*} \text {integral}& = \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 ) \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 (21 \sqrt {51217+28496 \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 ) \\ & = -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] (verified)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.64 \[ \int \frac {x^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=-27 x+\frac {5 x^3}{3}+\frac {25 x \left (3+5 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac {x \left (1468+835 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac {21 \left (-175 i+137 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{128 \sqrt {2-2 i \sqrt {2}}}+\frac {21 \left (175 i+137 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{128 \sqrt {2+2 i \sqrt {2}}} \]

[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]])

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.33

method	result	size
risch	\(\frac {5 x^{3}}{3}-27 x +\frac {-\frac {835}{64} x^{7}-\frac {1569}{32} x^{5}-\frac {4941}{64} x^{3}-\frac {513}{8} x}{\left (x^{4}+2 x^{2}+3\right )^{2}}+\frac {21 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{2}+3\right )}{\sum }\frac {\left (137 \textit {\_R}^{2}+312\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}\right )}{256}\)	\(79\)
default	\(\frac {5 x^{3}}{3}-27 x +\frac {-\frac {835}{64} x^{7}-\frac {1569}{32} x^{5}-\frac {4941}{64} x^{3}-\frac {513}{8} x}{\left (x^{4}+2 x^{2}+3\right )^{2}}+\frac {21 \left (33 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-175 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}-x \sqrt {-2+2 \sqrt {3}}\right )}{1024}+\frac {21 \left (416 \sqrt {3}+\frac {\left (33 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-175 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{256 \sqrt {2+2 \sqrt {3}}}+\frac {21 \left (-33 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+175 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}+x \sqrt {-2+2 \sqrt {3}}\right )}{1024}+\frac {21 \left (416 \sqrt {3}-\frac {\left (-33 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+175 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{256 \sqrt {2+2 \sqrt {3}}}\)	\(295\)

[In]

int(x^8*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^3,x,method=_RETURNVERBOSE)

[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+21/256*sum((137*_R^2+312)/(_R^3+_R)
*ln(x-_R),_R=RootOf(_Z^4+2*_Z^2+3))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.15 \[ \int \frac {x^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=\frac {2560 \, x^{11} - 31232 \, x^{9} - 160328 \, x^{7} - 459312 \, x^{5} - 593208 \, x^{3} + 3 \, \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {6032439 i \, \sqrt {2} - 15113511} \log \left ({\left (104 \, \sqrt {2} - 33 i\right )} \sqrt {6032439 i \, \sqrt {2} - 15113511} + 477141 \, x\right ) - 3 \, \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {6032439 i \, \sqrt {2} - 15113511} \log \left (-{\left (104 \, \sqrt {2} - 33 i\right )} \sqrt {6032439 i \, \sqrt {2} - 15113511} + 477141 \, x\right ) + 3 \, \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {-6032439 i \, \sqrt {2} - 15113511} \log \left ({\left (104 \, \sqrt {2} + 33 i\right )} \sqrt {-6032439 i \, \sqrt {2} - 15113511} + 477141 \, x\right ) - 3 \, \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {-6032439 i \, \sqrt {2} - 15113511} \log \left (-{\left (104 \, \sqrt {2} + 33 i\right )} \sqrt {-6032439 i \, \sqrt {2} - 15113511} + 477141 \, x\right ) - 471744 \, x}{1536 \, {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} \]

[In]

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

[Out]

1/1536*(2560*x^11 - 31232*x^9 - 160328*x^7 - 459312*x^5 - 593208*x^3 + 3*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^
2 + 9)*sqrt(6032439*I*sqrt(2) - 15113511)*log((104*sqrt(2) - 33*I)*sqrt(6032439*I*sqrt(2) - 15113511) + 477141
*x) - 3*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)*sqrt(6032439*I*sqrt(2) - 15113511)*log(-(104*sqrt(2) - 33*
I)*sqrt(6032439*I*sqrt(2) - 15113511) + 477141*x) + 3*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)*sqrt(-603243
9*I*sqrt(2) - 15113511)*log((104*sqrt(2) + 33*I)*sqrt(-6032439*I*sqrt(2) - 15113511) + 477141*x) - 3*sqrt(2)*(
x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)*sqrt(-6032439*I*sqrt(2) - 15113511)*log(-(104*sqrt(2) + 33*I)*sqrt(-6032439
*I*sqrt(2) - 15113511) + 477141*x) - 471744*x)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)

Sympy [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.34 \[ \int \frac {x^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=\frac {5 x^{3}}{3} - 27 x + \frac {- 835 x^{7} - 3138 x^{5} - 4941 x^{3} - 4104 x}{64 x^{8} + 256 x^{6} + 640 x^{4} + 768 x^{2} + 576} + 21 \operatorname {RootSum} {\left (17179869184 t^{4} + 8983937024 t^{2} + 1548731523, \left ( t \mapsto t \log {\left (- \frac {1107296256 t^{3}}{310800559} + \frac {438857984 t}{310800559} + x \right )} \right )\right )} \]

[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**3/31080055
9 + 438857984*_t/310800559 + x)))

Maxima [F]

\[ \int \frac {x^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=\int { \frac {{\left (5 \, x^{6} + 3 \, x^{4} + x^{2} + 4\right )} x^{8}}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{3}} \,d x } \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 585 vs. \(2 (187) = 374\).

Time = 0.75 (sec) , antiderivative size = 585, normalized size of antiderivative = 2.42 \[ \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 {5}{3} \, x^{3} - \frac {7}{55296} \, \sqrt {2} {\left (137 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2466 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 2466 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 137 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 11232 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 11232 \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 {7}{55296} \, \sqrt {2} {\left (137 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2466 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 2466 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 137 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 11232 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 11232 \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 {7}{110592} \, \sqrt {2} {\left (2466 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 137 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 137 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2466 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 11232 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 11232 \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 {7}{110592} \, \sqrt {2} {\left (2466 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 137 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 137 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2466 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 11232 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 11232 \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 ) - 27 \, x - \frac {835 \, x^{7} + 3138 \, x^{5} + 4941 \, x^{3} + 4104 \, x}{64 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}} \]

[In]

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

[Out]

5/3*x^3 - 7/55296*sqrt(2)*(137*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 2466*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) +
18)*(sqrt(3) - 3) - 2466*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 137*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 1
1232*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) + 11232*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)) - 7/55296*sqrt(2)*(137*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^
(3/2) + 2466*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 2466*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) +
18) + 137*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 11232*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) + 11232*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)) - 7/110592
*sqrt(2)*(2466*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 137*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/
2) + 137*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 2466*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 11232*3^(1/4)*sqrt
(2)*sqrt(-6*sqrt(3) + 18) - 11232*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)) + 7/110592*sqrt(2)*(2466*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 137*3^(3/4)*sqrt(2)
*(-6*sqrt(3) + 18)^(3/2) + 137*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 2466*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3
) - 11232*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) - 11232*3^(1/4)*sqrt(6*sqrt(3) + 18))*log(x^2 - 2*3^(1/4)*x*sq
rt(-1/6*sqrt(3) + 1/2) + sqrt(3)) - 27*x - 1/64*(835*x^7 + 3138*x^5 + 4941*x^3 + 4104*x)/(x^4 + 2*x^2 + 3)^2

Mupad [B] (verification not implemented)

Time = 8.90 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.75 \[ \int \frac {x^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=\frac {5\,x^3}{3}-\frac {\frac {835\,x^7}{64}+\frac {1569\,x^5}{32}+\frac {4941\,x^3}{64}+\frac {513\,x}{8}}{x^8+4\,x^6+10\,x^4+12\,x^2+9}-27\,x+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-68542-\sqrt {2}\,27358{}\mathrm {i}}\,126681219{}\mathrm {i}}{131072\,\left (\frac {12541440681}{131072}+\frac {\sqrt {2}\,4940567541{}\mathrm {i}}{16384}\right )}-\frac {126681219\,\sqrt {2}\,x\,\sqrt {-68542-\sqrt {2}\,27358{}\mathrm {i}}}{262144\,\left (\frac {12541440681}{131072}+\frac {\sqrt {2}\,4940567541{}\mathrm {i}}{16384}\right )}\right )\,\sqrt {-68542-\sqrt {2}\,27358{}\mathrm {i}}\,21{}\mathrm {i}}{256}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-68542+\sqrt {2}\,27358{}\mathrm {i}}\,126681219{}\mathrm {i}}{131072\,\left (-\frac {12541440681}{131072}+\frac {\sqrt {2}\,4940567541{}\mathrm {i}}{16384}\right )}+\frac {126681219\,\sqrt {2}\,x\,\sqrt {-68542+\sqrt {2}\,27358{}\mathrm {i}}}{262144\,\left (-\frac {12541440681}{131072}+\frac {\sqrt {2}\,4940567541{}\mathrm {i}}{16384}\right )}\right )\,\sqrt {-68542+\sqrt {2}\,27358{}\mathrm {i}}\,21{}\mathrm {i}}{256} \]

[In]

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

[Out]

(atan((x*(- 2^(1/2)*27358i - 68542)^(1/2)*126681219i)/(131072*((2^(1/2)*4940567541i)/16384 + 12541440681/13107
2)) - (126681219*2^(1/2)*x*(- 2^(1/2)*27358i - 68542)^(1/2))/(262144*((2^(1/2)*4940567541i)/16384 + 1254144068
1/131072)))*(- 2^(1/2)*27358i - 68542)^(1/2)*21i)/256 - ((513*x)/8 + (4941*x^3)/64 + (1569*x^5)/32 + (835*x^7)
/64)/(12*x^2 + 10*x^4 + 4*x^6 + x^8 + 9) - 27*x - (atan((x*(2^(1/2)*27358i - 68542)^(1/2)*126681219i)/(131072*
((2^(1/2)*4940567541i)/16384 - 12541440681/131072)) + (126681219*2^(1/2)*x*(2^(1/2)*27358i - 68542)^(1/2))/(26
2144*((2^(1/2)*4940567541i)/16384 - 12541440681/131072)))*(2^(1/2)*27358i - 68542)^(1/2)*21i)/256 + (5*x^3)/3