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

Optimal. Leaf size=237 \[ x^5-\frac {17 x^3}{3}+\frac {3}{32} \sqrt {\frac {3}{2} \left (8669+5011 \sqrt {3}\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {3}{32} \sqrt {\frac {3}{2} \left (8669+5011 \sqrt {3}\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {25 \left (3-x^2\right ) x}{8 \left (x^4+2 x^2+3\right )}+19 x+\frac {3}{16} \sqrt {\frac {3}{2} \left (5011 \sqrt {3}-8669\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {3}{16} \sqrt {\frac {3}{2} \left (5011 \sqrt {3}-8669\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right ) \]

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Rubi [A]  time = 0.29, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {1668, 1676, 1169, 634, 618, 204, 628} \begin {gather*} x^5-\frac {17 x^3}{3}+\frac {25 \left (3-x^2\right ) x}{8 \left (x^4+2 x^2+3\right )}+\frac {3}{32} \sqrt {\frac {3}{2} \left (8669+5011 \sqrt {3}\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {3}{32} \sqrt {\frac {3}{2} \left (8669+5011 \sqrt {3}\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+19 x+\frac {3}{16} \sqrt {\frac {3}{2} \left (5011 \sqrt {3}-8669\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {3}{16} \sqrt {\frac {3}{2} \left (5011 \sqrt {3}-8669\right )} \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)^2,x]

[Out]

19*x - (17*x^3)/3 + x^5 + (25*x*(3 - x^2))/(8*(3 + 2*x^2 + x^4)) + (3*Sqrt[(3*(-8669 + 5011*Sqrt[3]))/2]*ArcTa
n[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/16 - (3*Sqrt[(3*(-8669 + 5011*Sqrt[3]))/2]*ArcTan[(Sq
rt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/16 + (3*Sqrt[(3*(8669 + 5011*Sqrt[3]))/2]*Log[Sqrt[3] - Sq
rt[2*(-1 + Sqrt[3])]*x + x^2])/32 - (3*Sqrt[(3*(8669 + 5011*Sqrt[3]))/2]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*
x + x^2])/32

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

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 )^2} \, dx &=\frac {25 x \left (3-x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{48} \int \frac {-450+1050 x^2-336 x^6+240 x^8}{3+2 x^2+x^4} \, dx\\ &=\frac {25 x \left (3-x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{48} \int \left (912-816 x^2+240 x^4-\frac {54 \left (59-31 x^2\right )}{3+2 x^2+x^4}\right ) \, dx\\ &=19 x-\frac {17 x^3}{3}+x^5+\frac {25 x \left (3-x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {9}{8} \int \frac {59-31 x^2}{3+2 x^2+x^4} \, dx\\ &=19 x-\frac {17 x^3}{3}+x^5+\frac {25 x \left (3-x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {1}{32} \left (3 \sqrt {3 \left (1+\sqrt {3}\right )}\right ) \int \frac {59 \sqrt {2 \left (-1+\sqrt {3}\right )}-\left (59+31 \sqrt {3}\right ) x}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx-\frac {1}{32} \left (3 \sqrt {3 \left (1+\sqrt {3}\right )}\right ) \int \frac {59 \sqrt {2 \left (-1+\sqrt {3}\right )}+\left (59+31 \sqrt {3}\right ) x}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx\\ &=19 x-\frac {17 x^3}{3}+x^5+\frac {25 x \left (3-x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {1}{16} \left (3 \sqrt {\frac {3}{2} \left (3182-1829 \sqrt {3}\right )}\right ) \int \frac {1}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx-\frac {1}{16} \left (3 \sqrt {\frac {3}{2} \left (3182-1829 \sqrt {3}\right )}\right ) \int \frac {1}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx+\frac {1}{32} \left (3 \sqrt {\frac {3}{2} \left (8669+5011 \sqrt {3}\right )}\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}{32} \left (3 \sqrt {\frac {3}{2} \left (8669+5011 \sqrt {3}\right )}\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\\ &=19 x-\frac {17 x^3}{3}+x^5+\frac {25 x \left (3-x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {3}{32} \sqrt {\frac {3}{2} \left (8669+5011 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )-\frac {3}{32} \sqrt {\frac {3}{2} \left (8669+5011 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{8} \left (3 \sqrt {\frac {3}{2} \left (3182-1829 \sqrt {3}\right )}\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}{8} \left (3 \sqrt {\frac {3}{2} \left (3182-1829 \sqrt {3}\right )}\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 )\\ &=19 x-\frac {17 x^3}{3}+x^5+\frac {25 x \left (3-x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {3}{16} \sqrt {\frac {3}{2} \left (-8669+5011 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {3}{16} \sqrt {\frac {3}{2} \left (-8669+5011 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {3}{32} \sqrt {\frac {3}{2} \left (8669+5011 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )-\frac {3}{32} \sqrt {\frac {3}{2} \left (8669+5011 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )\\ \end {align*}

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Mathematica [C]  time = 0.16, size = 132, normalized size = 0.56 \begin {gather*} x^5-\frac {17 x^3}{3}-\frac {25 \left (x^2-3\right ) x}{8 \left (x^4+2 x^2+3\right )}+19 x+\frac {9 \left (31 \sqrt {2}+90 i\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{16 \sqrt {2-2 i \sqrt {2}}}+\frac {9 \left (31 \sqrt {2}-90 i\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{16 \sqrt {2+2 i \sqrt {2}}} \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)^2,x]

[Out]

19*x - (17*x^3)/3 + x^5 - (25*x*(-3 + x^2))/(8*(3 + 2*x^2 + x^4)) + (9*(90*I + 31*Sqrt[2])*ArcTan[x/Sqrt[1 - I
*Sqrt[2]]])/(16*Sqrt[2 - (2*I)*Sqrt[2]]) + (9*(-90*I + 31*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(16*Sqrt[2 +
(2*I)*Sqrt[2]])

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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 )^2} \, 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)^2,x]

[Out]

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

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fricas [B]  time = 1.30, size = 476, normalized size = 2.01 \begin {gather*} \frac {287671488 \, x^{9} - 1054795456 \, x^{7} + 3068495872 \, x^{5} + 3588 \cdot 677973267^{\frac {1}{4}} \sqrt {3} \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {-43440359 \, \sqrt {3} + 75330363} \arctan \left (\frac {1}{1822344999502852422} \cdot 677973267^{\frac {3}{4}} \sqrt {4494867} \sqrt {4494867 \, x^{2} + 677973267^{\frac {1}{4}} {\left (31 \, \sqrt {3} x + 59 \, x\right )} \sqrt {-43440359 \, \sqrt {3} + 75330363} + 4494867 \, \sqrt {3}} {\left (59 \, \sqrt {3} \sqrt {2} + 93 \, \sqrt {2}\right )} \sqrt {-43440359 \, \sqrt {3} + 75330363} - \frac {1}{405428013666} \cdot 677973267^{\frac {3}{4}} {\left (59 \, \sqrt {3} \sqrt {2} x + 93 \, \sqrt {2} x\right )} \sqrt {-43440359 \, \sqrt {3} + 75330363} - \frac {1}{2} \, \sqrt {3} \sqrt {2} + \frac {1}{2} \, \sqrt {2}\right ) + 3588 \cdot 677973267^{\frac {1}{4}} \sqrt {3} \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {-43440359 \, \sqrt {3} + 75330363} \arctan \left (\frac {1}{1822344999502852422} \cdot 677973267^{\frac {3}{4}} \sqrt {4494867} \sqrt {4494867 \, x^{2} - 677973267^{\frac {1}{4}} {\left (31 \, \sqrt {3} x + 59 \, x\right )} \sqrt {-43440359 \, \sqrt {3} + 75330363} + 4494867 \, \sqrt {3}} {\left (59 \, \sqrt {3} \sqrt {2} + 93 \, \sqrt {2}\right )} \sqrt {-43440359 \, \sqrt {3} + 75330363} - \frac {1}{405428013666} \cdot 677973267^{\frac {3}{4}} {\left (59 \, \sqrt {3} \sqrt {2} x + 93 \, \sqrt {2} x\right )} \sqrt {-43440359 \, \sqrt {3} + 75330363} + \frac {1}{2} \, \sqrt {3} \sqrt {2} - \frac {1}{2} \, \sqrt {2}\right ) + 5142127848 \, x^{3} - 3 \cdot 677973267^{\frac {1}{4}} {\left (15033 \, x^{4} + 30066 \, x^{2} + 8669 \, \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 3\right )} + 45099\right )} \sqrt {-43440359 \, \sqrt {3} + 75330363} \log \left (4494867 \, x^{2} + 677973267^{\frac {1}{4}} {\left (31 \, \sqrt {3} x + 59 \, x\right )} \sqrt {-43440359 \, \sqrt {3} + 75330363} + 4494867 \, \sqrt {3}\right ) + 3 \cdot 677973267^{\frac {1}{4}} {\left (15033 \, x^{4} + 30066 \, x^{2} + 8669 \, \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 3\right )} + 45099\right )} \sqrt {-43440359 \, \sqrt {3} + 75330363} \log \left (4494867 \, x^{2} - 677973267^{\frac {1}{4}} {\left (31 \, \sqrt {3} x + 59 \, x\right )} \sqrt {-43440359 \, \sqrt {3} + 75330363} + 4494867 \, \sqrt {3}\right ) + 19094195016 \, x}{287671488 \, {\left (x^{4} + 2 \, x^{2} + 3\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)^2,x, algorithm="fricas")

[Out]

1/287671488*(287671488*x^9 - 1054795456*x^7 + 3068495872*x^5 + 3588*677973267^(1/4)*sqrt(3)*sqrt(2)*(x^4 + 2*x
^2 + 3)*sqrt(-43440359*sqrt(3) + 75330363)*arctan(1/1822344999502852422*677973267^(3/4)*sqrt(4494867)*sqrt(449
4867*x^2 + 677973267^(1/4)*(31*sqrt(3)*x + 59*x)*sqrt(-43440359*sqrt(3) + 75330363) + 4494867*sqrt(3))*(59*sqr
t(3)*sqrt(2) + 93*sqrt(2))*sqrt(-43440359*sqrt(3) + 75330363) - 1/405428013666*677973267^(3/4)*(59*sqrt(3)*sqr
t(2)*x + 93*sqrt(2)*x)*sqrt(-43440359*sqrt(3) + 75330363) - 1/2*sqrt(3)*sqrt(2) + 1/2*sqrt(2)) + 3588*67797326
7^(1/4)*sqrt(3)*sqrt(2)*(x^4 + 2*x^2 + 3)*sqrt(-43440359*sqrt(3) + 75330363)*arctan(1/1822344999502852422*6779
73267^(3/4)*sqrt(4494867)*sqrt(4494867*x^2 - 677973267^(1/4)*(31*sqrt(3)*x + 59*x)*sqrt(-43440359*sqrt(3) + 75
330363) + 4494867*sqrt(3))*(59*sqrt(3)*sqrt(2) + 93*sqrt(2))*sqrt(-43440359*sqrt(3) + 75330363) - 1/4054280136
66*677973267^(3/4)*(59*sqrt(3)*sqrt(2)*x + 93*sqrt(2)*x)*sqrt(-43440359*sqrt(3) + 75330363) + 1/2*sqrt(3)*sqrt
(2) - 1/2*sqrt(2)) + 5142127848*x^3 - 3*677973267^(1/4)*(15033*x^4 + 30066*x^2 + 8669*sqrt(3)*(x^4 + 2*x^2 + 3
) + 45099)*sqrt(-43440359*sqrt(3) + 75330363)*log(4494867*x^2 + 677973267^(1/4)*(31*sqrt(3)*x + 59*x)*sqrt(-43
440359*sqrt(3) + 75330363) + 4494867*sqrt(3)) + 3*677973267^(1/4)*(15033*x^4 + 30066*x^2 + 8669*sqrt(3)*(x^4 +
2*x^2 + 3) + 45099)*sqrt(-43440359*sqrt(3) + 75330363)*log(4494867*x^2 - 677973267^(1/4)*(31*sqrt(3)*x + 59*x
)*sqrt(-43440359*sqrt(3) + 75330363) + 4494867*sqrt(3)) + 19094195016*x)/(x^4 + 2*x^2 + 3)

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giac [B]  time = 1.85, size = 576, normalized size = 2.43 \begin {gather*} x^{5} - \frac {17}{3} \, x^{3} - \frac {1}{2304} \, \sqrt {2} {\left (31 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 558 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 558 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 31 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2124 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} - 2124 \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}{2304} \, \sqrt {2} {\left (31 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 558 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 558 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 31 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2124 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} - 2124 \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}{4608} \, \sqrt {2} {\left (558 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 31 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 31 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 558 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} + 2124 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} + 2124 \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}{4608} \, \sqrt {2} {\left (558 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 31 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 31 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 558 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} + 2124 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} + 2124 \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 ) + 19 \, x - \frac {25 \, {\left (x^{3} - 3 \, x\right )}}{8 \, {\left (x^{4} + 2 \, x^{2} + 3\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)^2,x, algorithm="giac")

[Out]

x^5 - 17/3*x^3 - 1/2304*sqrt(2)*(31*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 558*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3
) + 18)*(sqrt(3) - 3) - 558*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 31*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) +
2124*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) - 2124*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)) - 1/2304*sqrt(2)*(31*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3
/2) + 558*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 558*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18)
+ 31*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) + 2124*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) - 2124*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)) - 1/4608*sqrt(2)*
(558*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 31*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 31*3^(
3/4)*(6*sqrt(3) + 18)^(3/2) + 558*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) + 2124*3^(1/4)*sqrt(2)*sqrt(-6*sq
rt(3) + 18) + 2124*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
/4608*sqrt(2)*(558*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 31*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^
(3/2) + 31*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 558*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) + 2124*3^(1/4)*sqrt
(2)*sqrt(-6*sqrt(3) + 18) + 2124*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)) + 19*x - 25/8*(x^3 - 3*x)/(x^4 + 2*x^2 + 3)

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maple [B]  time = 0.03, size = 419, normalized size = 1.77 \begin {gather*} x^{5}-\frac {17 x^{3}}{3}+19 x +\frac {57 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{8 \sqrt {2+2 \sqrt {3}}}+\frac {405 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{32 \sqrt {2+2 \sqrt {3}}}-\frac {177 \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{8 \sqrt {2+2 \sqrt {3}}}+\frac {57 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{8 \sqrt {2+2 \sqrt {3}}}+\frac {405 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{32 \sqrt {2+2 \sqrt {3}}}-\frac {177 \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{8 \sqrt {2+2 \sqrt {3}}}+\frac {57 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{16}+\frac {405 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{64}-\frac {57 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{16}-\frac {405 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{64}+\frac {-\frac {25}{8} x^{3}+\frac {75}{8} x}{x^{4}+2 x^{2}+3} \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)^2,x)

[Out]

x^5-17/3*x^3+19*x+(-25/8*x^3+75/8*x)/(x^4+2*x^2+3)+57/16*(-2+2*3^(1/2))^(1/2)*3^(1/2)*ln(x^2-(-2+2*3^(1/2))^(1
/2)*x+3^(1/2))+405/64*(-2+2*3^(1/2))^(1/2)*ln(x^2-(-2+2*3^(1/2))^(1/2)*x+3^(1/2))+57/8/(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))+405/32/(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))-177/8/(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))-57/16*(-2+2*3^(1/2))^(1/2)*3^(1/2)*ln(x^2+(-2+2*3^(1/2))^(1/2)*x+3^(1
/2))-405/64*(-2+2*3^(1/2))^(1/2)*ln(x^2+(-2+2*3^(1/2))^(1/2)*x+3^(1/2))+57/8/(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))+405/32/(2+2*3^(1/2))^(1/2)*(-2+2*3^(1/2))*arc
tan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))-177/8/(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*} x^{5} - \frac {17}{3} \, x^{3} + 19 \, x - \frac {25 \, {\left (x^{3} - 3 \, x\right )}}{8 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac {9}{8} \, \int \frac {31 \, x^{2} - 59}{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)^2,x, algorithm="maxima")

[Out]

x^5 - 17/3*x^3 + 19*x - 25/8*(x^3 - 3*x)/(x^4 + 2*x^2 + 3) + 9/8*integrate((31*x^2 - 59)/(x^4 + 2*x^2 + 3), x)

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mupad [B]  time = 0.94, size = 164, normalized size = 0.69 \begin {gather*} 19\,x+\frac {\frac {75\,x}{8}-\frac {25\,x^3}{8}}{x^4+2\,x^2+3}-\frac {17\,x^3}{3}+x^5-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {26007-\sqrt {2}\,897{}\mathrm {i}}\,24219{}\mathrm {i}}{64\,\left (-\frac {1380483}{16}+\frac {\sqrt {2}\,4286763{}\mathrm {i}}{128}\right )}-\frac {24219\,\sqrt {2}\,x\,\sqrt {26007-\sqrt {2}\,897{}\mathrm {i}}}{128\,\left (-\frac {1380483}{16}+\frac {\sqrt {2}\,4286763{}\mathrm {i}}{128}\right )}\right )\,\sqrt {26007-\sqrt {2}\,897{}\mathrm {i}}\,3{}\mathrm {i}}{16}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {26007+\sqrt {2}\,897{}\mathrm {i}}\,24219{}\mathrm {i}}{64\,\left (\frac {1380483}{16}+\frac {\sqrt {2}\,4286763{}\mathrm {i}}{128}\right )}+\frac {24219\,\sqrt {2}\,x\,\sqrt {26007+\sqrt {2}\,897{}\mathrm {i}}}{128\,\left (\frac {1380483}{16}+\frac {\sqrt {2}\,4286763{}\mathrm {i}}{128}\right )}\right )\,\sqrt {26007+\sqrt {2}\,897{}\mathrm {i}}\,3{}\mathrm {i}}{16} \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)^2,x)

[Out]

19*x + ((75*x)/8 - (25*x^3)/8)/(2*x^2 + x^4 + 3) - (atan((x*(26007 - 2^(1/2)*897i)^(1/2)*24219i)/(64*((2^(1/2)
*4286763i)/128 - 1380483/16)) - (24219*2^(1/2)*x*(26007 - 2^(1/2)*897i)^(1/2))/(128*((2^(1/2)*4286763i)/128 -
1380483/16)))*(26007 - 2^(1/2)*897i)^(1/2)*3i)/16 + (atan((x*(2^(1/2)*897i + 26007)^(1/2)*24219i)/(64*((2^(1/2
)*4286763i)/128 + 1380483/16)) + (24219*2^(1/2)*x*(2^(1/2)*897i + 26007)^(1/2))/(128*((2^(1/2)*4286763i)/128 +
1380483/16)))*(2^(1/2)*897i + 26007)^(1/2)*3i)/16 - (17*x^3)/3 + x^5

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sympy [B]  time = 1.36, size = 1205, normalized size = 5.08

result too large to display

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)**2,x)

[Out]

x**5 - 17*x**3/3 + 19*x + (-25*x**3 + 75*x)/(8*x**4 + 16*x**2 + 24) - 3*sqrt(26007/2048 + 15033*sqrt(3)/2048)*
log(x**2 + x*(-304*sqrt(2)*sqrt(8669 + 5011*sqrt(3))/299 - 433349*sqrt(6)*sqrt(8669 + 5011*sqrt(3))/1498289 +
152*sqrt(3)*sqrt(8669 + 5011*sqrt(3))*sqrt(43440359*sqrt(3) + 75240962)/1498289) - 2882918249387*sqrt(2)*sqrt(
43440359*sqrt(3) + 75240962)/2244869927521 - 993398584*sqrt(6)*sqrt(43440359*sqrt(3) + 75240962)/1343965233 +
49936376949404567/2244869927521 + 17261871038090*sqrt(3)/1343965233) + 3*sqrt(26007/2048 + 15033*sqrt(3)/2048)
*log(x**2 + x*(-152*sqrt(3)*sqrt(8669 + 5011*sqrt(3))*sqrt(43440359*sqrt(3) + 75240962)/1498289 + 433349*sqrt(
6)*sqrt(8669 + 5011*sqrt(3))/1498289 + 304*sqrt(2)*sqrt(8669 + 5011*sqrt(3))/299) - 2882918249387*sqrt(2)*sqrt
(43440359*sqrt(3) + 75240962)/2244869927521 - 993398584*sqrt(6)*sqrt(43440359*sqrt(3) + 75240962)/1343965233 +
49936376949404567/2244869927521 + 17261871038090*sqrt(3)/1343965233) - 2*sqrt(-27*sqrt(2)*sqrt(43440359*sqrt(
3) + 75240962)/1024 + 234063/2048 + 405891*sqrt(3)/2048)*atan(2996578*sqrt(3)*x/(17641*sqrt(2)*sqrt(-2*sqrt(2)
*sqrt(43440359*sqrt(3) + 75240962) + 8669 + 15033*sqrt(3)) + 152*sqrt(43440359*sqrt(3) + 75240962)*sqrt(-2*sqr
t(2)*sqrt(43440359*sqrt(3) + 75240962) + 8669 + 15033*sqrt(3))) - 1523344*sqrt(6)*sqrt(8669 + 5011*sqrt(3))/(1
7641*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(43440359*sqrt(3) + 75240962) + 8669 + 15033*sqrt(3)) + 152*sqrt(43440359*sqr
t(3) + 75240962)*sqrt(-2*sqrt(2)*sqrt(43440359*sqrt(3) + 75240962) + 8669 + 15033*sqrt(3))) - 1300047*sqrt(2)*
sqrt(8669 + 5011*sqrt(3))/(17641*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(43440359*sqrt(3) + 75240962) + 8669 + 15033*sqrt
(3)) + 152*sqrt(43440359*sqrt(3) + 75240962)*sqrt(-2*sqrt(2)*sqrt(43440359*sqrt(3) + 75240962) + 8669 + 15033*
sqrt(3))) + 456*sqrt(8669 + 5011*sqrt(3))*sqrt(43440359*sqrt(3) + 75240962)/(17641*sqrt(2)*sqrt(-2*sqrt(2)*sqr
t(43440359*sqrt(3) + 75240962) + 8669 + 15033*sqrt(3)) + 152*sqrt(43440359*sqrt(3) + 75240962)*sqrt(-2*sqrt(2)
*sqrt(43440359*sqrt(3) + 75240962) + 8669 + 15033*sqrt(3)))) - 2*sqrt(-27*sqrt(2)*sqrt(43440359*sqrt(3) + 7524
0962)/1024 + 234063/2048 + 405891*sqrt(3)/2048)*atan(2996578*sqrt(3)*x/(17641*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(434
40359*sqrt(3) + 75240962) + 8669 + 15033*sqrt(3)) + 152*sqrt(43440359*sqrt(3) + 75240962)*sqrt(-2*sqrt(2)*sqrt
(43440359*sqrt(3) + 75240962) + 8669 + 15033*sqrt(3))) - 456*sqrt(8669 + 5011*sqrt(3))*sqrt(43440359*sqrt(3) +
75240962)/(17641*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(43440359*sqrt(3) + 75240962) + 8669 + 15033*sqrt(3)) + 152*sqrt
(43440359*sqrt(3) + 75240962)*sqrt(-2*sqrt(2)*sqrt(43440359*sqrt(3) + 75240962) + 8669 + 15033*sqrt(3))) + 130
0047*sqrt(2)*sqrt(8669 + 5011*sqrt(3))/(17641*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(43440359*sqrt(3) + 75240962) + 8669
+ 15033*sqrt(3)) + 152*sqrt(43440359*sqrt(3) + 75240962)*sqrt(-2*sqrt(2)*sqrt(43440359*sqrt(3) + 75240962) +
8669 + 15033*sqrt(3))) + 1523344*sqrt(6)*sqrt(8669 + 5011*sqrt(3))/(17641*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(4344035
9*sqrt(3) + 75240962) + 8669 + 15033*sqrt(3)) + 152*sqrt(43440359*sqrt(3) + 75240962)*sqrt(-2*sqrt(2)*sqrt(434
40359*sqrt(3) + 75240962) + 8669 + 15033*sqrt(3))))

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